Research ArticleNavigating Deeply Uncertain Tradeoffs in HarvestedPredator-Prey Systems

Antonia Hadjimichael 1 Patrick M Reed1 and Julianne D Quinn2

1School of Civil and Environmental Engineering Cornell University Ithaca NY USA2Department of Engineering Systems and the Environment University of Virginia Charlottesville VA USA

Correspondence should be addressed to Antonia Hadjimichael ah986cornelledu

Received 14 October 2019 Accepted 21 November 2019 Published 27 February 2020

Guest Editor Viet-(anh Pham

Copyright copy 2020 Antonia Hadjimichael et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Multiple fisheries have collapsed as a result of overfishing and strong limitations in our knowledge of system conditions andconsequential ecological interactions Fishery managers need to establish harvesting strategies that balance economic benefitsagainst ecological objectives including avoiding unintended catastrophic consequences Our results show that classical as-sumptions for fisheries management can yield severe instabilities in our quantified views of socioecological tradeoffs making theirability to inform stakeholder preferences questionable(e complex ecological interactions implied by different parameterizationsof such systems yield highly complex and nonlinear dynamic properties with multiple distinct basins of attraction We show thatsmall changes in our deeply uncertain representations of predator-prey systems can fundamentally shift their dynamics and thevalidity of candidate management strategies for harvest Insights from this study highlight the importance of ensuring modelscapture deep uncertainties as well as a breadth of financial and ecological criteria when searching for robust management optionsfor resilient fisheries

1 Introduction

One in four fisheries has collapsed in the latter half of the20th century [1] In the northwest Atlantic most Atlantic cod(Gadus morhua) stocks collapsed in the early 1990s in adecline that was considered to be sudden and unexpected atthe time [2] (e collapse has since been attributed to thedecades-long overexploitation of the system at unsustainablelevels [3] and changes in ocean climate conditions [4]Overlooked changes in environmental conditions and sys-tem interactions led to the collapse of the sardine (Sardinopssagax) and anchovy (Engraulis encrasicolus) fisheries in thenorthern Benguela ecosystem off the coast of Namibia in the1970s As both species are energy-rich prey their collapseculminated in significant population declines for theirpredators as well [5] In the Volga River the construction ofdams has interrupted spawning migrations and reducedhabitat sizes resulting in the collapse of inconnu (Stenodusleucichthys) beluga (Huso huso) Russian sturgeon (A

gueldenstaedtii) and herrings (Clupea harengus) [6] (esecatastrophic events have been attributed to imprudent hu-man action on marine and freshwater ecosystems and todeep uncertainty in system conditions and poorly under-stood interactions [7 8] Deep uncertainty refers to situa-tions where parameters and relationships describing thesystem can be complex and difficult to estimate from em-pirical data and experts cannot agree on probability densityfunctions to describe them or on the relationships them-selves [9 10]

Such is the case for predator-prey theory in the trophicecology field (e standard theory of predator-prey inter-actions has largely been based on the LotkandashVolterraequations that describe a system of two differential equationswith a simple relation of proportionality between preyconsumption and predator growth (e core assumptions ofthis model (prey growth and trophic function) have beenchallenged by multiple authors on both empirical andtheoretical bases (eg [11ndash13]) Arditi and Ginzburg [14]

HindawiComplexityVolume 2020 Article ID 4170453 18 pageshttpsdoiorg10115520204170453

argued that the trophic function should account for theintricacies of the predation process at the macroscale (iethat the predators need to share the prey available to them)(is proposition (the ldquoratio-dependentrdquo trophic function)sparked a strong debate [15ndash18] that appeared to come to itsconclusion with the general agreement that a range ofpredator interference levels can be found in nature (ie apredator-dependent trophic function with case-specificlevels of interference [19])

In this study we consider a fishery management problemwhere a fleet must develop a harvesting strategy that bal-ances profits with the ecological stability of a predator-preysystem We use the predator-dependent predator-preysystem of equations proposed by Arditi and Akccedilakaya [20]that includes parameter m for the level of predator inter-ference Accounting for predator interaction (m) and timeadaptive human prey harvesting (zt) the following discrete-time forms of the predator-prey system equations aredefined

xt+1 xt + bxt 1 minusxt

K1113874 1113875 minus

αxtyt

ymt + αhxt

minus zt+1xt1113890 1113891 middot eϵx

yt+1 yt +cαxtyt

ymt + αhxt

minus dyt1113890 1113891 middot eϵy

(1)

where x is prey abundance y is predator abundance z is thefraction of harvested prey and t isin 1 2 is the time indexin years b is the prey growth rate and K is its environmentalcarrying capacity α represents the rate at which the predatorencounters the prey h is the time it needs to consume theprey and c is the rate at which it converts the consumed preyto new predators d is the predator death rate and m is thelevel of predator interference Parameterizing the level ofpredator interference in this way allows us to move betweenthe two contested equation forms (predator dependenceversus not) and examine the effects of this type of parametricuncertainty on management tradeoffs Furthermore this isthe first study to use this specific form of predator-preyrelationships and pair it with adaptive human harvesting ofprey Environmental stochasticity is included using ldquoprocessnoiserdquo factors ϵx and ϵy modeled as coming from a log-normal distribution ϵi sim Normal(00 σ2i ) [21ndash23] Addi-tional information on the model and its parameterization isprovided in Section 2 detailed parameter descriptions unitsand default values are listed in Table 1

Different system parameterizations can have profoundimplications on the ways the system behaves Even in sys-tems without human disruption (zt 0) the parametervalues and population interactions can affect system dy-namics in complex ways changing for example the pres-ence and stability of equilibria between the species(Figure 1) Focusing on the zero isoclines (ie the black linesdesignating the prey and predator population levels thatresult in either of the species having a zero growth rate) wecan identify equilibria (at the intersection of the two zeroisoclines) For the prey-dependent model if the nontrivial(coexistence) equilibrium is stable it is also a globalattractor which is the specific value that the system tends to

evolve toward irrespective of initial conditions (Figure 1(a))If the equilibrium is unstable then the global attractor of thesystem is a stable limit cycle (Figure 1(d)) (ie a closedtrajectory in the phase space with at least one other trajectoryspiraling into it [24]) For the generalized predator-de-pendent model when the nontrivial (coexistence) equilib-rium is stable it is also a global attractor (Figure 1(b)) Forthis model if the nontrivial equilibrium is unstable it caneither lead to stable limit cycles (Figure 1(c)) or deterministicextinction (Figure 1(e)) When predator interference is high(mgt 1) and there is sufficient carrying capacity (K) there aretwo nontrivial equilibria only one of which is stable(Figure 1(f )) (e instability of the nontrivial equilibriumcan therefore lead to sustained population oscillations orextinction depending on the model parameters and theinitial conditions [24] As a result the complex ecologicalinteractions implied by each model formulation are non-linear and yield highly complex dynamical impacts on theecosystem leading to distinct basins of attraction [25 26](e complex dynamics of these systems can be further seenin the bifurcation diagrams in Figure S1 of the Supple-mentary Material for all respective systems presented inFigure 1 (e diagrams are plotted with respect to parameterm and demonstrate both the fixed points and the periodicorbits possible under different parameterizations Specifi-cally for the systems presented in Figures 1(b)ndash1(d) de-creasing values of m change the stability of the equilibriumpoint resulting in periodic behavior

Precise estimates of predator interference predationand growth and death rates are difficult to estimate fromempirical data especially for nonartificial environments[19 20] As a result the form of functional responses for alarge number of species remains unknown [27] (is isconcerning as the modern ecological paradigm highlightsthe significance of species interactions to the management ofpopulations communities and ecosystems [28ndash30] (econsequences of such deeply uncertain ecological dynamicson the management objectives can therefore be significantpotentially leading to unattained harvesting profits orworse unintentional population collapse

In this study we quantify and analyze the tradeoffs ofmanaging (harvesting) a two-species fishery governed by apredator-prey relationship and assess how deep uncertaintyin population interactions can affect the managementtradeoffs (is application expands on the work by [31 32]and [33] that also sought to identify management tradeoffsresulting from a socioecological system as well as the im-plications of deep uncertainty in the systemrsquos parameters Asillustrated in Figure 1 the system investigated in this ap-plication exhibits a much richer variety of dynamics in waysthat may potentially alter the topological characteristics ofthe attained tradeoffs (eg if the parameterization shifts in amanner that changes the stability of an equilibriumFigures 1(b) and 1(c)) Furthermore the system consideredhere includes an additional dimension increasing thecomplexity of maximizing economic benefits without in-advertently driving either of the species to collapse (e preycan be overconsumed by the predator and harvester thepredator can collapse if there is not enough available prey

2 Complexity

Finally we demonstrate how the concept of multiobjectiverobustness can be valuable for selecting harvesting policiesthat avoid triggering potentially catastrophic consequencesin harvested fisheries

(is study aims to bridge and complement the twointellectual threads shaping the management of fisherieseconomics and biology As recounted by [34] seminal workby [35 36] helped ground a biological rationale for themanagement of fisheries by illuminating the connectionsbetween fishing effort mortality and dynamics From theeconomic perspective early work by [37 38] established anoperational foundation for the management of fisheriesbased on capital theory and investment concepts thataddressed resource conservation by establishing an objectivethat ought to be pursued by management Optimal controltheory methods [39 40] complemented these efforts bydescribing optimal action paths to achieve this objective Inrecent decades progressive discoveries on the importance ofcomplex multispecies relationships trophic connectivityand ecosystem interactions have questioned the traditionaland to this day predominant view of fisheries managementthat of a single-species- and single-objective-based controlthat establishes a ldquomaximum sustainable yieldrdquo or an ldquoal-lowable biological catchrdquo [29 41 42] Work by [43 44] andothers demonstrated that single-species assessments andmanagement controls may indeed produce misleadingpredictions and destructive impacts on ecosystems withmultispecies interactions and in marine environments allspecies have predator-prey relationships with other speciesin their ecosystem

On these grounds ecosystem-based fishery manage-ment [29 45] has been promoted as the new paradigm forfisheries management advocating for the considerationof multiple species and values However explicit incor-poration of nonharvesting values has been limited infisheries management studies In cases where multispe-cies interactions are considered [46 47] the inclusion ofnonharvesting values has been hindered by the fact that

these commodities are not typically traded in markets(is might lead to the underappreciation or even com-plete omission of the nonmarket values of environmentaland ecosystem goods and services from studies aiming toprovide support for socioecological systems management[48] Such formulations might consequently result in theinappropriate suggestion of strategies that promote re-source exploitation and degrade the ecosystem and itsprovisions [49 50] Broadening the set of fishery man-agement objectives to include nonharvesting values couldresult in fundamentally different management strategies[47 51 52] (e latest review of fisheriesrsquo decision-making applications using multiple criteria [53] identi-fied several studies that considered either several speciesor several objectives (including nonharvesting) How-ever all of the presented approaches (multiattributeutility linear and nonlinear goal programming andweighted goal programming) collapsed these multipleobjectives into one using an a-priori formulation of thepreferences of the stakeholders to be included in themodel in the form of weights (ese weights may notaccurately reflect the true stakeholder preferences es-pecially before exploring the wider set of possible optionsand in cases of nonlinear threshold responses in theobjective space [54ndash56] Furthermore specifying specificgoals or weights before the search may miss potentialsolutions that are of interest by unnecessarily limiting thesearch space [57] To the best of the authorsrsquo knowledgethere has been one study that applied heuristic globaloptimization for the identification of managementstrategies for a fishery without collapsing objectives intoone Mardle et al [58] (e application used the GEN-OCOP III genetic algorithm [59] albeit inappropriatelywith a search population and number of function eval-uations that were too small for the problem at hand (iswork aims to expand on the current literature by opti-mizing state-dependent adaptive harvesting strategiesthat explicitly consider a broad range of objectives

Table 1 Description values in the assumed state of the world (SOW) and ranges of sampled uncertain model parameters (e fisheryharvesting plans are optimized to a system assumed to be described by the base values and then re-evaluated in 4000 alternative SOWsgenerated by a Latin Hypercube Sample across the parameter ranges listed in the table (minimum and maximum)

Parameter Description Unit Base value Minimum Maximumα Rate at which the prey is available to the predator 1time 0005 0002 2b Prey growth rate 1time 05 0005 1

c Rate with which consumed prey is converted topredator abundance massmasslowast 05 02 1

d Predator death rate 1time 01 005 02

h Handling time (time each predator needs to consumethe caught prey) time 01 0001 1

K Prey carrying capacity given its environmentalconditions masslowast 2000 100 5000

m Predator interference parameter massmasslowast 07 01 15

σx

Standard deviation of stochastic noise in preypopulation mass2lowast 0004 0001 001

σy

Standard deviation of stochastic noise in predatorpopulation mass2lowast 0004 0001 001

lowast(e units of these parameters depend on the units used to measure prey (x) and predator (y) abundance If prey and predator abundance is measured involume then these units would equivalently change

Complexity 3

(including nonharvesting) in a multiobjective optimi-zation framework We believe this is the first applicationof stochastic multiobjective control for the identificationof robust harvesting strategies for a fishery

(e rest of this manuscript is organized as follows InSection 2 we first explain the system under study and discussthe presence and stability of equilibria We then detail howwe used multiobjective optimization to identify candidate

Pred

ator

abun

danc

ePred iso

Prey iso0

100

200

300

400

200 400 600 8000Prey abundance

(a)

Pred iso

Prey iso

Pred

ator

abun

danc

e

1000 2000 3000 4000 5000 6000 7000 800000

50

100

150

200

250

Prey abundance

(b)

Pred iso

Prey isoPred

ator

abun

danc

e

200 400 600 80000

200

400

600

800

1000

1200

Prey abundance

(c)

Pred

ator

abun

danc

e Pred iso

Prey iso

0

100

200

300

400

100 200 300 400 5000Prey abundance

(d)

Pred

ator

abun

danc

e

0

100

200

300

400

500

600

100 200 300 400 500 600 7000Prey abundance

(e)

Pred

ator

abun

danc

e

Pred iso

Prey iso

500 1500 2500 35001000 30002000 400000

500

1000

1500

2000

2500

400

300

Prey abundance

(f )

Figure 1 Direction fields and trajectories for different parameterizations of the predator-prey system Black lines indicate zero-isoclinestheir intersections indicate nontrivial equilibria (a) Prey-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 05 h 01 K 500 m 0) (b) Predator-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 01 h 01 K 2000 m 07) (c) Predator-dependent system with unstable equilibrium and limit cyclesas the global attractor (α 0047 b 0877 c 0666 d 0094 h 0306 K 189372 m 0465) (d) Prey-dependent system withunstable equilibrium and limit cycles as the global attractor (α 0005 b 05 c 05 d 01 h 01 K 2000 m 0) (e) Predator-dependent system with unstable equilibrium and deterministic extinction (α 1775 b 0389 c 0441 d 0083 h 0941K 446507 m 0107) (f ) Predator-dependent system with two equilibria and no global attractor (α 0796 b 0215 c 0565d 0137 h 0472 K 485848 m 121) All systems assume no process noise

4 Complexity

harvesting strategies for five management objectives whileconstraining strategies to avoid predator collapse Finally weexplain how the robustness of each candidate strategy wasassessed using several satisficing criteria In Section 3 wepresent the potential values achieved in each objective byeach of the candidate management strategies and demon-strate the significant instability of these tradeoffs whenconsidering uncertainty in parameter values We then ex-plore how interactions between parameters affect systemstability and consequently the attainment of managementobjectives including avoiding predator collapse Lastly weshow how alternative preferences in management strategymay affect the system and potentially avoid populationcollapse in unstable systems

2 Methods

21 System Equilibria and Stability A generalized predator-dependent predator-prey system of equations has beenmodified for the purposes of this study to account for humanaction by means of harvesting

dx

dt bx 1 minus

x

K1113874 1113875 minus

αxy

ym + αhxminus zx

dy

dt

cαxy

ym + αhxminus dy

(2)

where z describes the harvesting effort performed by thefleet(e parameter values describing the system are listed inTable 1 and represent our best current state of knowledgeSince the system in this study represents a stylistic examplethese values were not derived from a specific empiricalsystem but were based on values and ranges that appear inmultiple literature sources (eg [21ndash24]) In an unharvestedsystem for the nontrivial (coexistence) equilibrium to existthe following equation must hold [24]

cgt h d (3)

A mathematical proof of how this condition also holdsfor a system with harvested prey is provided in the Sup-plementary Material In an unharvested system for thenontrivial equilibrium to be stable the following equationmust hold

α(hK)1minus m lt(b)

m (4)

Biologically when (3) holds predators convert theconsumed prey to new predators at a higher rate than therate of their death d and handling time h (ie their losses intime and energy) When (4) holds the prey isocline (detailedin the Supplementary Material) decreases as a function of xstabilizing the system [24] For a system where prey isharvested (ie the additional parameter z only decreases theprey level) the condition must also hold as the prey isoclinecan only further decrease as a function of x (is is alsodemonstrated experimentally by our study

We use the stochastic closed loop control method directpolicy search (DPS) [60] also known as parameterization-simulation-optimization to identify harvesting policies(is

allows for the use of a state-aware control rule that maps theprey population level (xt) to the harvesting effort at the nexttime step (zt+1) instead of optimizing all individual har-vesting efforts(e following sections describe this approachin detail beginning with the management objectives thepolicies that were optimized and the algorithm used

22 Optimization ofHarvesting Strategies (e optimizationis aimed at determining dynamic harvesting policies thatdescribe how much prey to harvest over time in order tooptimize five objectives and meet the specified constraint(e objectives are designed to address financial goals andto ensure that the fish population is maintained at naturallevels this gives rise to tradeoffs between candidateharvesting strategies We identify a set of ldquonondominatedrdquosolutions which is comprised of the harvesting strategiesthat perform better than any other strategy in at least oneof the five objectives (e nondominated solutionscompose optimal tradeoffs where improvement in anysingle objective comes at the cost of degraded perfor-mance in one or more of the remaining objectives (eobjectives and constraint are described in more detailbelow

221 Maximize Harvesting Discounted Profits (Net PresentValue) (e net present value of harvesting profits for eachrealization of environmental stochasticity is given by1113936

Tminus 1t0 zt+1ixti(1 + δ)t for Tyears where δ is the discount rate

used to convert future benefits to present value xti is preyabundance at the tth year of the ith realization and zt+1i isthe harvesting effort performed for that prey (e expectedharvesting discounted profits O1 are estimated as the averageacross N realizations

O1 1N

1113944

N

i11113944

Tminus 1

t0

zt+1ixti

(1 + δ)t⎛⎝ ⎞⎠ (5)

where δ 005

222 Minimize Prey Population Deficit (e prey pop-ulation deficit for each realization of environmental sto-chasticity is given by (K minus xti)K where K is the preycarrying capacity (ie the maximum population abundancethat can be achieved if the prey is not subjected to predationor harvest) (e expected prey population deficit O2 is es-timated as the average deficit over all time steps averagedacross N realizations

O2 1N

1113944

N

i1

1T

1113944

T

i1

K minus xti

K1113888 1113889⎛⎝ ⎞⎠ (6)

When policies are re-evaluated or reoptimized in sys-tems with different parameter combinations (as elaboratedin Section 26) the respective value of K is adjusted ac-cordingly so as to reflect the deficit of population as it relatesto the carrying capacity implied by the new set ofparameters

Complexity 5

223 Minimize Longest Duration of Consecutive LowHarvest Given operational costs the fishery managerswould like to avoid long durations of consecutively lowharvest defined by a minimum harvest limit (e maxduration of low harvest is given by bothzt lt limit and zt+1 lt limit holding for all t in a realization of Tyears (e expected worst case of consecutive low harvest O3is defined as the average of the max duration across Nrealizations

O3 1N

1113944

N

i1maxT ϕti1113872 11138731113872 1113873

where

ϕti

ϕtminus 1i + 1 if zt lt limit

0 otherwise

⎧⎪⎨

⎪⎩

ϕ1i

1 if z1 lt limit

0 otherwise

⎧⎪⎨

⎪⎩

(7)

where limit 5

224 Maximize Worst Harvest Instance Given operationalcosts the fishery managers would like to avoid exposure tofinancial risks Variance-minimizing strategies have beenwidely employed in the literature to model behavior underrisk authors have noted however that the costs of risksassociated with uncertainty often depend on higher ordermoments such as skewness and kurtosis (tail events in thedistribution) [61] (is was approximated in this analysis bymaximizing the worst harvest instance as well as minimizingthe variance of harvest in every realization (O5 explained inthe next section) (e worst harvest instance is approxi-mated here by the 1st percentile of harvest for every T-yearrealization(e expected worst harvest instance is calculatedas the average 1st percentile across N realizations

O4 1N

1113944

N

i1percentileT zt+1ixti 11113872 11138731113872 1113873 (8)

225 Minimize Harvest Variance More traditionally en-countered in the literature is the minimization of the var-iance of deviations from the expected profits [61 62] (isobjective has been approximated by estimating the varianceof the obtained harvest in every T year realization and av-eraging across all N realizations

O5 1N

1113944

N

i1VarT zt+1ixti1113872 11138731113872 1113873 (9)

23 Avoid Collapse of Predator Population Considering theunharvested predator population as a valuable species the

population of which the managers would like to maintainthe optimization is also subject to a constraint

1N

1113944

N

i1ψti1113872 1113873 0

where

ψti

1 if yti lt 1

0 otherwise

⎧⎪⎨

⎪⎩

forallt isin T andforalli isin N

(10)

24 Formulation of Harvesting Policy For the purposes ofthis study candidate DPS control rules were used to map thecurrent levels of prey population (xt) to the harvesting effortat the next time step (zt+1) (e optimization was aimed atidentifying the parameters describing the control rulesinstead of the harvesting efforts themselves allowing forstate-based feedback control strategies (e control ruleswere in the form of Gaussian radial basis functions (RBFs)following the formulation by [63] (e optimization prob-lem was formulated as follows

MinimizeF zx( 1113857 minus O1 O2 O3 minus O4 O5( 1113857

z z1 z2 zT( 1113857

zt+1 1113944n

i1wi exp minus

xt( 1113857 minus ci

bi

1113888 1113889

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

(11)

where i 1 2 n and n is the number of RBFs used in thefunction mapping in this study n 2 wi is the weight of theith RBF and the weights are formulated so as to be positive(ie wi gt 0 foralli) and sum to one (ie 1113936

ni1wi 1) ci and bi are

the center and radius of the ith RBF All wi ci and bi isin [0 1]We used two RBFs and one input in this study resulting insix decision variables that need to be optimized for thecontrol rule mapping current prey population to nextharvest More inputs can be used in the policy formulationbut were omitted from equation (10) for the purposes ofsimplicity Note that with the application of these controlrules the harvesting effort zt+1 will not necessarily be thesame in each of the N realizations as the harvesting action isinformed by the respective level of prey xt which is subjectto the environmental stochasticity at each realizationFurthermore this formulation assumes a perfectly accuratemeasurement of the prey level at all times as well as aperfectly accurate execution of the harvesting strategy(esesimplifying assumptions have the benefit in this study ofdemonstrating the severe difficulty of the class of fisheriesmanagement problems even in optimistic formulations ofavailable information

25 Optimization Algorithm (e multiobjective formula-tion was solved for N 100 realizations of randomly

6 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

Finally we demonstrate how the concept of multiobjectiverobustness can be valuable for selecting harvesting policiesthat avoid triggering potentially catastrophic consequencesin harvested fisheries

(is study aims to bridge and complement the twointellectual threads shaping the management of fisherieseconomics and biology As recounted by [34] seminal workby [35 36] helped ground a biological rationale for themanagement of fisheries by illuminating the connectionsbetween fishing effort mortality and dynamics From theeconomic perspective early work by [37 38] established anoperational foundation for the management of fisheriesbased on capital theory and investment concepts thataddressed resource conservation by establishing an objectivethat ought to be pursued by management Optimal controltheory methods [39 40] complemented these efforts bydescribing optimal action paths to achieve this objective Inrecent decades progressive discoveries on the importance ofcomplex multispecies relationships trophic connectivityand ecosystem interactions have questioned the traditionaland to this day predominant view of fisheries managementthat of a single-species- and single-objective-based controlthat establishes a ldquomaximum sustainable yieldrdquo or an ldquoal-lowable biological catchrdquo [29 41 42] Work by [43 44] andothers demonstrated that single-species assessments andmanagement controls may indeed produce misleadingpredictions and destructive impacts on ecosystems withmultispecies interactions and in marine environments allspecies have predator-prey relationships with other speciesin their ecosystem

On these grounds ecosystem-based fishery manage-ment [29 45] has been promoted as the new paradigm forfisheries management advocating for the considerationof multiple species and values However explicit incor-poration of nonharvesting values has been limited infisheries management studies In cases where multispe-cies interactions are considered [46 47] the inclusion ofnonharvesting values has been hindered by the fact that

these commodities are not typically traded in markets(is might lead to the underappreciation or even com-plete omission of the nonmarket values of environmentaland ecosystem goods and services from studies aiming toprovide support for socioecological systems management[48] Such formulations might consequently result in theinappropriate suggestion of strategies that promote re-source exploitation and degrade the ecosystem and itsprovisions [49 50] Broadening the set of fishery man-agement objectives to include nonharvesting values couldresult in fundamentally different management strategies[47 51 52] (e latest review of fisheriesrsquo decision-making applications using multiple criteria [53] identi-fied several studies that considered either several speciesor several objectives (including nonharvesting) How-ever all of the presented approaches (multiattributeutility linear and nonlinear goal programming andweighted goal programming) collapsed these multipleobjectives into one using an a-priori formulation of thepreferences of the stakeholders to be included in themodel in the form of weights (ese weights may notaccurately reflect the true stakeholder preferences es-pecially before exploring the wider set of possible optionsand in cases of nonlinear threshold responses in theobjective space [54ndash56] Furthermore specifying specificgoals or weights before the search may miss potentialsolutions that are of interest by unnecessarily limiting thesearch space [57] To the best of the authorsrsquo knowledgethere has been one study that applied heuristic globaloptimization for the identification of managementstrategies for a fishery without collapsing objectives intoone Mardle et al [58] (e application used the GEN-OCOP III genetic algorithm [59] albeit inappropriatelywith a search population and number of function eval-uations that were too small for the problem at hand (iswork aims to expand on the current literature by opti-mizing state-dependent adaptive harvesting strategiesthat explicitly consider a broad range of objectives

Table 1 Description values in the assumed state of the world (SOW) and ranges of sampled uncertain model parameters (e fisheryharvesting plans are optimized to a system assumed to be described by the base values and then re-evaluated in 4000 alternative SOWsgenerated by a Latin Hypercube Sample across the parameter ranges listed in the table (minimum and maximum)

Parameter Description Unit Base value Minimum Maximumα Rate at which the prey is available to the predator 1time 0005 0002 2b Prey growth rate 1time 05 0005 1

c Rate with which consumed prey is converted topredator abundance massmasslowast 05 02 1

d Predator death rate 1time 01 005 02

h Handling time (time each predator needs to consumethe caught prey) time 01 0001 1

K Prey carrying capacity given its environmentalconditions masslowast 2000 100 5000

m Predator interference parameter massmasslowast 07 01 15

σx

Standard deviation of stochastic noise in preypopulation mass2lowast 0004 0001 001

σy

Standard deviation of stochastic noise in predatorpopulation mass2lowast 0004 0001 001

lowast(e units of these parameters depend on the units used to measure prey (x) and predator (y) abundance If prey and predator abundance is measured involume then these units would equivalently change

Complexity 3

(including nonharvesting) in a multiobjective optimi-zation framework We believe this is the first applicationof stochastic multiobjective control for the identificationof robust harvesting strategies for a fishery

(e rest of this manuscript is organized as follows InSection 2 we first explain the system under study and discussthe presence and stability of equilibria We then detail howwe used multiobjective optimization to identify candidate

Pred

ator

abun

danc

ePred iso

Prey iso0

100

200

300

400

200 400 600 8000Prey abundance

(a)

Pred iso

Prey iso

Pred

ator

abun

danc

e

1000 2000 3000 4000 5000 6000 7000 800000

50

100

150

200

250

Prey abundance

(b)

Pred iso

Prey isoPred

ator

abun

danc

e

200 400 600 80000

200

400

600

800

1000

1200

Prey abundance

(c)

Pred

ator

abun

danc

e Pred iso

Prey iso

0

100

200

300

400

100 200 300 400 5000Prey abundance

(d)

Pred

ator

abun

danc

e

0

100

200

300

400

500

600

100 200 300 400 500 600 7000Prey abundance

(e)

Pred

ator

abun

danc

e

Pred iso

Prey iso

500 1500 2500 35001000 30002000 400000

500

1000

1500

2000

2500

400

300

Prey abundance

(f )

Figure 1 Direction fields and trajectories for different parameterizations of the predator-prey system Black lines indicate zero-isoclinestheir intersections indicate nontrivial equilibria (a) Prey-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 05 h 01 K 500 m 0) (b) Predator-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 01 h 01 K 2000 m 07) (c) Predator-dependent system with unstable equilibrium and limit cyclesas the global attractor (α 0047 b 0877 c 0666 d 0094 h 0306 K 189372 m 0465) (d) Prey-dependent system withunstable equilibrium and limit cycles as the global attractor (α 0005 b 05 c 05 d 01 h 01 K 2000 m 0) (e) Predator-dependent system with unstable equilibrium and deterministic extinction (α 1775 b 0389 c 0441 d 0083 h 0941K 446507 m 0107) (f ) Predator-dependent system with two equilibria and no global attractor (α 0796 b 0215 c 0565d 0137 h 0472 K 485848 m 121) All systems assume no process noise

4 Complexity

harvesting strategies for five management objectives whileconstraining strategies to avoid predator collapse Finally weexplain how the robustness of each candidate strategy wasassessed using several satisficing criteria In Section 3 wepresent the potential values achieved in each objective byeach of the candidate management strategies and demon-strate the significant instability of these tradeoffs whenconsidering uncertainty in parameter values We then ex-plore how interactions between parameters affect systemstability and consequently the attainment of managementobjectives including avoiding predator collapse Lastly weshow how alternative preferences in management strategymay affect the system and potentially avoid populationcollapse in unstable systems

2 Methods

21 System Equilibria and Stability A generalized predator-dependent predator-prey system of equations has beenmodified for the purposes of this study to account for humanaction by means of harvesting

dx

dt bx 1 minus

x

K1113874 1113875 minus

αxy

ym + αhxminus zx

dy

dt

cαxy

ym + αhxminus dy

(2)

where z describes the harvesting effort performed by thefleet(e parameter values describing the system are listed inTable 1 and represent our best current state of knowledgeSince the system in this study represents a stylistic examplethese values were not derived from a specific empiricalsystem but were based on values and ranges that appear inmultiple literature sources (eg [21ndash24]) In an unharvestedsystem for the nontrivial (coexistence) equilibrium to existthe following equation must hold [24]

cgt h d (3)

A mathematical proof of how this condition also holdsfor a system with harvested prey is provided in the Sup-plementary Material In an unharvested system for thenontrivial equilibrium to be stable the following equationmust hold

α(hK)1minus m lt(b)

m (4)

Biologically when (3) holds predators convert theconsumed prey to new predators at a higher rate than therate of their death d and handling time h (ie their losses intime and energy) When (4) holds the prey isocline (detailedin the Supplementary Material) decreases as a function of xstabilizing the system [24] For a system where prey isharvested (ie the additional parameter z only decreases theprey level) the condition must also hold as the prey isoclinecan only further decrease as a function of x (is is alsodemonstrated experimentally by our study

We use the stochastic closed loop control method directpolicy search (DPS) [60] also known as parameterization-simulation-optimization to identify harvesting policies(is

allows for the use of a state-aware control rule that maps theprey population level (xt) to the harvesting effort at the nexttime step (zt+1) instead of optimizing all individual har-vesting efforts(e following sections describe this approachin detail beginning with the management objectives thepolicies that were optimized and the algorithm used

22 Optimization ofHarvesting Strategies (e optimizationis aimed at determining dynamic harvesting policies thatdescribe how much prey to harvest over time in order tooptimize five objectives and meet the specified constraint(e objectives are designed to address financial goals andto ensure that the fish population is maintained at naturallevels this gives rise to tradeoffs between candidateharvesting strategies We identify a set of ldquonondominatedrdquosolutions which is comprised of the harvesting strategiesthat perform better than any other strategy in at least oneof the five objectives (e nondominated solutionscompose optimal tradeoffs where improvement in anysingle objective comes at the cost of degraded perfor-mance in one or more of the remaining objectives (eobjectives and constraint are described in more detailbelow

221 Maximize Harvesting Discounted Profits (Net PresentValue) (e net present value of harvesting profits for eachrealization of environmental stochasticity is given by1113936

Tminus 1t0 zt+1ixti(1 + δ)t for Tyears where δ is the discount rate

used to convert future benefits to present value xti is preyabundance at the tth year of the ith realization and zt+1i isthe harvesting effort performed for that prey (e expectedharvesting discounted profits O1 are estimated as the averageacross N realizations

O1 1N

1113944

N

i11113944

Tminus 1

t0

zt+1ixti

(1 + δ)t⎛⎝ ⎞⎠ (5)

where δ 005

222 Minimize Prey Population Deficit (e prey pop-ulation deficit for each realization of environmental sto-chasticity is given by (K minus xti)K where K is the preycarrying capacity (ie the maximum population abundancethat can be achieved if the prey is not subjected to predationor harvest) (e expected prey population deficit O2 is es-timated as the average deficit over all time steps averagedacross N realizations

O2 1N

1113944

N

i1

1T

1113944

T

i1

K minus xti

K1113888 1113889⎛⎝ ⎞⎠ (6)

When policies are re-evaluated or reoptimized in sys-tems with different parameter combinations (as elaboratedin Section 26) the respective value of K is adjusted ac-cordingly so as to reflect the deficit of population as it relatesto the carrying capacity implied by the new set ofparameters

Complexity 5

223 Minimize Longest Duration of Consecutive LowHarvest Given operational costs the fishery managerswould like to avoid long durations of consecutively lowharvest defined by a minimum harvest limit (e maxduration of low harvest is given by bothzt lt limit and zt+1 lt limit holding for all t in a realization of Tyears (e expected worst case of consecutive low harvest O3is defined as the average of the max duration across Nrealizations

O3 1N

1113944

N

i1maxT ϕti1113872 11138731113872 1113873

where

ϕti

ϕtminus 1i + 1 if zt lt limit

0 otherwise

⎧⎪⎨

⎪⎩

ϕ1i

1 if z1 lt limit

0 otherwise

⎧⎪⎨

⎪⎩

(7)

where limit 5

224 Maximize Worst Harvest Instance Given operationalcosts the fishery managers would like to avoid exposure tofinancial risks Variance-minimizing strategies have beenwidely employed in the literature to model behavior underrisk authors have noted however that the costs of risksassociated with uncertainty often depend on higher ordermoments such as skewness and kurtosis (tail events in thedistribution) [61] (is was approximated in this analysis bymaximizing the worst harvest instance as well as minimizingthe variance of harvest in every realization (O5 explained inthe next section) (e worst harvest instance is approxi-mated here by the 1st percentile of harvest for every T-yearrealization(e expected worst harvest instance is calculatedas the average 1st percentile across N realizations

O4 1N

1113944

N

i1percentileT zt+1ixti 11113872 11138731113872 1113873 (8)

225 Minimize Harvest Variance More traditionally en-countered in the literature is the minimization of the var-iance of deviations from the expected profits [61 62] (isobjective has been approximated by estimating the varianceof the obtained harvest in every T year realization and av-eraging across all N realizations

O5 1N

1113944

N

i1VarT zt+1ixti1113872 11138731113872 1113873 (9)

23 Avoid Collapse of Predator Population Considering theunharvested predator population as a valuable species the

population of which the managers would like to maintainthe optimization is also subject to a constraint

1N

1113944

N

i1ψti1113872 1113873 0

where

ψti

1 if yti lt 1

0 otherwise

⎧⎪⎨

⎪⎩

forallt isin T andforalli isin N

(10)

24 Formulation of Harvesting Policy For the purposes ofthis study candidate DPS control rules were used to map thecurrent levels of prey population (xt) to the harvesting effortat the next time step (zt+1) (e optimization was aimed atidentifying the parameters describing the control rulesinstead of the harvesting efforts themselves allowing forstate-based feedback control strategies (e control ruleswere in the form of Gaussian radial basis functions (RBFs)following the formulation by [63] (e optimization prob-lem was formulated as follows

MinimizeF zx( 1113857 minus O1 O2 O3 minus O4 O5( 1113857

z z1 z2 zT( 1113857

zt+1 1113944n

i1wi exp minus

xt( 1113857 minus ci

bi

1113888 1113889

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

(11)

where i 1 2 n and n is the number of RBFs used in thefunction mapping in this study n 2 wi is the weight of theith RBF and the weights are formulated so as to be positive(ie wi gt 0 foralli) and sum to one (ie 1113936

ni1wi 1) ci and bi are

the center and radius of the ith RBF All wi ci and bi isin [0 1]We used two RBFs and one input in this study resulting insix decision variables that need to be optimized for thecontrol rule mapping current prey population to nextharvest More inputs can be used in the policy formulationbut were omitted from equation (10) for the purposes ofsimplicity Note that with the application of these controlrules the harvesting effort zt+1 will not necessarily be thesame in each of the N realizations as the harvesting action isinformed by the respective level of prey xt which is subjectto the environmental stochasticity at each realizationFurthermore this formulation assumes a perfectly accuratemeasurement of the prey level at all times as well as aperfectly accurate execution of the harvesting strategy(esesimplifying assumptions have the benefit in this study ofdemonstrating the severe difficulty of the class of fisheriesmanagement problems even in optimistic formulations ofavailable information

25 Optimization Algorithm (e multiobjective formula-tion was solved for N 100 realizations of randomly

6 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

(including nonharvesting) in a multiobjective optimi-zation framework We believe this is the first applicationof stochastic multiobjective control for the identificationof robust harvesting strategies for a fishery

(e rest of this manuscript is organized as follows InSection 2 we first explain the system under study and discussthe presence and stability of equilibria We then detail howwe used multiobjective optimization to identify candidate

Pred

ator

abun

danc

ePred iso

Prey iso0

100

200

300

400

200 400 600 8000Prey abundance

(a)

Pred iso

Prey iso

Pred

ator

abun

danc

e

1000 2000 3000 4000 5000 6000 7000 800000

50

100

150

200

250

Prey abundance

(b)

Pred iso

Prey isoPred

ator

abun

danc

e

200 400 600 80000

200

400

600

800

1000

1200

Prey abundance

(c)

Pred

ator

abun

danc

e Pred iso

Prey iso

0

100

200

300

400

100 200 300 400 5000Prey abundance

(d)

Pred

ator

abun

danc

e

0

100

200

300

400

500

600

100 200 300 400 500 600 7000Prey abundance

(e)

Pred

ator

abun

danc

e

Pred iso

Prey iso

500 1500 2500 35001000 30002000 400000

500

1000

1500

2000

2500

400

300

Prey abundance

(f )

Figure 1 Direction fields and trajectories for different parameterizations of the predator-prey system Black lines indicate zero-isoclinestheir intersections indicate nontrivial equilibria (a) Prey-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 05 h 01 K 500 m 0) (b) Predator-dependent system with a global attractor stable equilibrium(α 0005 b 05 c 05 d 01 h 01 K 2000 m 07) (c) Predator-dependent system with unstable equilibrium and limit cyclesas the global attractor (α 0047 b 0877 c 0666 d 0094 h 0306 K 189372 m 0465) (d) Prey-dependent system withunstable equilibrium and limit cycles as the global attractor (α 0005 b 05 c 05 d 01 h 01 K 2000 m 0) (e) Predator-dependent system with unstable equilibrium and deterministic extinction (α 1775 b 0389 c 0441 d 0083 h 0941K 446507 m 0107) (f ) Predator-dependent system with two equilibria and no global attractor (α 0796 b 0215 c 0565d 0137 h 0472 K 485848 m 121) All systems assume no process noise

4 Complexity

harvesting strategies for five management objectives whileconstraining strategies to avoid predator collapse Finally weexplain how the robustness of each candidate strategy wasassessed using several satisficing criteria In Section 3 wepresent the potential values achieved in each objective byeach of the candidate management strategies and demon-strate the significant instability of these tradeoffs whenconsidering uncertainty in parameter values We then ex-plore how interactions between parameters affect systemstability and consequently the attainment of managementobjectives including avoiding predator collapse Lastly weshow how alternative preferences in management strategymay affect the system and potentially avoid populationcollapse in unstable systems

2 Methods

21 System Equilibria and Stability A generalized predator-dependent predator-prey system of equations has beenmodified for the purposes of this study to account for humanaction by means of harvesting

dx

dt bx 1 minus

x

K1113874 1113875 minus

αxy

ym + αhxminus zx

dy

dt

cαxy

ym + αhxminus dy

(2)

where z describes the harvesting effort performed by thefleet(e parameter values describing the system are listed inTable 1 and represent our best current state of knowledgeSince the system in this study represents a stylistic examplethese values were not derived from a specific empiricalsystem but were based on values and ranges that appear inmultiple literature sources (eg [21ndash24]) In an unharvestedsystem for the nontrivial (coexistence) equilibrium to existthe following equation must hold [24]

cgt h d (3)

A mathematical proof of how this condition also holdsfor a system with harvested prey is provided in the Sup-plementary Material In an unharvested system for thenontrivial equilibrium to be stable the following equationmust hold

α(hK)1minus m lt(b)

m (4)

Biologically when (3) holds predators convert theconsumed prey to new predators at a higher rate than therate of their death d and handling time h (ie their losses intime and energy) When (4) holds the prey isocline (detailedin the Supplementary Material) decreases as a function of xstabilizing the system [24] For a system where prey isharvested (ie the additional parameter z only decreases theprey level) the condition must also hold as the prey isoclinecan only further decrease as a function of x (is is alsodemonstrated experimentally by our study

We use the stochastic closed loop control method directpolicy search (DPS) [60] also known as parameterization-simulation-optimization to identify harvesting policies(is

allows for the use of a state-aware control rule that maps theprey population level (xt) to the harvesting effort at the nexttime step (zt+1) instead of optimizing all individual har-vesting efforts(e following sections describe this approachin detail beginning with the management objectives thepolicies that were optimized and the algorithm used

22 Optimization ofHarvesting Strategies (e optimizationis aimed at determining dynamic harvesting policies thatdescribe how much prey to harvest over time in order tooptimize five objectives and meet the specified constraint(e objectives are designed to address financial goals andto ensure that the fish population is maintained at naturallevels this gives rise to tradeoffs between candidateharvesting strategies We identify a set of ldquonondominatedrdquosolutions which is comprised of the harvesting strategiesthat perform better than any other strategy in at least oneof the five objectives (e nondominated solutionscompose optimal tradeoffs where improvement in anysingle objective comes at the cost of degraded perfor-mance in one or more of the remaining objectives (eobjectives and constraint are described in more detailbelow

221 Maximize Harvesting Discounted Profits (Net PresentValue) (e net present value of harvesting profits for eachrealization of environmental stochasticity is given by1113936

Tminus 1t0 zt+1ixti(1 + δ)t for Tyears where δ is the discount rate

used to convert future benefits to present value xti is preyabundance at the tth year of the ith realization and zt+1i isthe harvesting effort performed for that prey (e expectedharvesting discounted profits O1 are estimated as the averageacross N realizations

O1 1N

1113944

N

i11113944

Tminus 1

t0

zt+1ixti

(1 + δ)t⎛⎝ ⎞⎠ (5)

where δ 005

222 Minimize Prey Population Deficit (e prey pop-ulation deficit for each realization of environmental sto-chasticity is given by (K minus xti)K where K is the preycarrying capacity (ie the maximum population abundancethat can be achieved if the prey is not subjected to predationor harvest) (e expected prey population deficit O2 is es-timated as the average deficit over all time steps averagedacross N realizations

O2 1N

1113944

N

i1

1T

1113944

T

i1

K minus xti

K1113888 1113889⎛⎝ ⎞⎠ (6)

When policies are re-evaluated or reoptimized in sys-tems with different parameter combinations (as elaboratedin Section 26) the respective value of K is adjusted ac-cordingly so as to reflect the deficit of population as it relatesto the carrying capacity implied by the new set ofparameters

Complexity 5

223 Minimize Longest Duration of Consecutive LowHarvest Given operational costs the fishery managerswould like to avoid long durations of consecutively lowharvest defined by a minimum harvest limit (e maxduration of low harvest is given by bothzt lt limit and zt+1 lt limit holding for all t in a realization of Tyears (e expected worst case of consecutive low harvest O3is defined as the average of the max duration across Nrealizations

O3 1N

1113944

N

i1maxT ϕti1113872 11138731113872 1113873

where

ϕti

ϕtminus 1i + 1 if zt lt limit

0 otherwise

⎧⎪⎨

⎪⎩

ϕ1i

1 if z1 lt limit

0 otherwise

⎧⎪⎨

⎪⎩

(7)

where limit 5

224 Maximize Worst Harvest Instance Given operationalcosts the fishery managers would like to avoid exposure tofinancial risks Variance-minimizing strategies have beenwidely employed in the literature to model behavior underrisk authors have noted however that the costs of risksassociated with uncertainty often depend on higher ordermoments such as skewness and kurtosis (tail events in thedistribution) [61] (is was approximated in this analysis bymaximizing the worst harvest instance as well as minimizingthe variance of harvest in every realization (O5 explained inthe next section) (e worst harvest instance is approxi-mated here by the 1st percentile of harvest for every T-yearrealization(e expected worst harvest instance is calculatedas the average 1st percentile across N realizations

O4 1N

1113944

N

i1percentileT zt+1ixti 11113872 11138731113872 1113873 (8)

225 Minimize Harvest Variance More traditionally en-countered in the literature is the minimization of the var-iance of deviations from the expected profits [61 62] (isobjective has been approximated by estimating the varianceof the obtained harvest in every T year realization and av-eraging across all N realizations

O5 1N

1113944

N

i1VarT zt+1ixti1113872 11138731113872 1113873 (9)

23 Avoid Collapse of Predator Population Considering theunharvested predator population as a valuable species the

population of which the managers would like to maintainthe optimization is also subject to a constraint

1N

1113944

N

i1ψti1113872 1113873 0

where

ψti

1 if yti lt 1

0 otherwise

⎧⎪⎨

⎪⎩

forallt isin T andforalli isin N

(10)

24 Formulation of Harvesting Policy For the purposes ofthis study candidate DPS control rules were used to map thecurrent levels of prey population (xt) to the harvesting effortat the next time step (zt+1) (e optimization was aimed atidentifying the parameters describing the control rulesinstead of the harvesting efforts themselves allowing forstate-based feedback control strategies (e control ruleswere in the form of Gaussian radial basis functions (RBFs)following the formulation by [63] (e optimization prob-lem was formulated as follows

MinimizeF zx( 1113857 minus O1 O2 O3 minus O4 O5( 1113857

z z1 z2 zT( 1113857

zt+1 1113944n

i1wi exp minus

xt( 1113857 minus ci

bi

1113888 1113889

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

(11)

where i 1 2 n and n is the number of RBFs used in thefunction mapping in this study n 2 wi is the weight of theith RBF and the weights are formulated so as to be positive(ie wi gt 0 foralli) and sum to one (ie 1113936

ni1wi 1) ci and bi are

the center and radius of the ith RBF All wi ci and bi isin [0 1]We used two RBFs and one input in this study resulting insix decision variables that need to be optimized for thecontrol rule mapping current prey population to nextharvest More inputs can be used in the policy formulationbut were omitted from equation (10) for the purposes ofsimplicity Note that with the application of these controlrules the harvesting effort zt+1 will not necessarily be thesame in each of the N realizations as the harvesting action isinformed by the respective level of prey xt which is subjectto the environmental stochasticity at each realizationFurthermore this formulation assumes a perfectly accuratemeasurement of the prey level at all times as well as aperfectly accurate execution of the harvesting strategy(esesimplifying assumptions have the benefit in this study ofdemonstrating the severe difficulty of the class of fisheriesmanagement problems even in optimistic formulations ofavailable information

25 Optimization Algorithm (e multiobjective formula-tion was solved for N 100 realizations of randomly

6 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

harvesting strategies for five management objectives whileconstraining strategies to avoid predator collapse Finally weexplain how the robustness of each candidate strategy wasassessed using several satisficing criteria In Section 3 wepresent the potential values achieved in each objective byeach of the candidate management strategies and demon-strate the significant instability of these tradeoffs whenconsidering uncertainty in parameter values We then ex-plore how interactions between parameters affect systemstability and consequently the attainment of managementobjectives including avoiding predator collapse Lastly weshow how alternative preferences in management strategymay affect the system and potentially avoid populationcollapse in unstable systems

2 Methods

21 System Equilibria and Stability A generalized predator-dependent predator-prey system of equations has beenmodified for the purposes of this study to account for humanaction by means of harvesting

dx

dt bx 1 minus

x

K1113874 1113875 minus

αxy

ym + αhxminus zx

dy

dt

cαxy

ym + αhxminus dy

(2)

where z describes the harvesting effort performed by thefleet(e parameter values describing the system are listed inTable 1 and represent our best current state of knowledgeSince the system in this study represents a stylistic examplethese values were not derived from a specific empiricalsystem but were based on values and ranges that appear inmultiple literature sources (eg [21ndash24]) In an unharvestedsystem for the nontrivial (coexistence) equilibrium to existthe following equation must hold [24]

cgt h d (3)

A mathematical proof of how this condition also holdsfor a system with harvested prey is provided in the Sup-plementary Material In an unharvested system for thenontrivial equilibrium to be stable the following equationmust hold

α(hK)1minus m lt(b)

m (4)

Biologically when (3) holds predators convert theconsumed prey to new predators at a higher rate than therate of their death d and handling time h (ie their losses intime and energy) When (4) holds the prey isocline (detailedin the Supplementary Material) decreases as a function of xstabilizing the system [24] For a system where prey isharvested (ie the additional parameter z only decreases theprey level) the condition must also hold as the prey isoclinecan only further decrease as a function of x (is is alsodemonstrated experimentally by our study

We use the stochastic closed loop control method directpolicy search (DPS) [60] also known as parameterization-simulation-optimization to identify harvesting policies(is

allows for the use of a state-aware control rule that maps theprey population level (xt) to the harvesting effort at the nexttime step (zt+1) instead of optimizing all individual har-vesting efforts(e following sections describe this approachin detail beginning with the management objectives thepolicies that were optimized and the algorithm used

22 Optimization ofHarvesting Strategies (e optimizationis aimed at determining dynamic harvesting policies thatdescribe how much prey to harvest over time in order tooptimize five objectives and meet the specified constraint(e objectives are designed to address financial goals andto ensure that the fish population is maintained at naturallevels this gives rise to tradeoffs between candidateharvesting strategies We identify a set of ldquonondominatedrdquosolutions which is comprised of the harvesting strategiesthat perform better than any other strategy in at least oneof the five objectives (e nondominated solutionscompose optimal tradeoffs where improvement in anysingle objective comes at the cost of degraded perfor-mance in one or more of the remaining objectives (eobjectives and constraint are described in more detailbelow

221 Maximize Harvesting Discounted Profits (Net PresentValue) (e net present value of harvesting profits for eachrealization of environmental stochasticity is given by1113936

Tminus 1t0 zt+1ixti(1 + δ)t for Tyears where δ is the discount rate

used to convert future benefits to present value xti is preyabundance at the tth year of the ith realization and zt+1i isthe harvesting effort performed for that prey (e expectedharvesting discounted profits O1 are estimated as the averageacross N realizations

O1 1N

1113944

N

i11113944

Tminus 1

t0

zt+1ixti

(1 + δ)t⎛⎝ ⎞⎠ (5)

where δ 005

222 Minimize Prey Population Deficit (e prey pop-ulation deficit for each realization of environmental sto-chasticity is given by (K minus xti)K where K is the preycarrying capacity (ie the maximum population abundancethat can be achieved if the prey is not subjected to predationor harvest) (e expected prey population deficit O2 is es-timated as the average deficit over all time steps averagedacross N realizations

O2 1N

1113944

N

i1

1T

1113944

T

i1

K minus xti

K1113888 1113889⎛⎝ ⎞⎠ (6)

When policies are re-evaluated or reoptimized in sys-tems with different parameter combinations (as elaboratedin Section 26) the respective value of K is adjusted ac-cordingly so as to reflect the deficit of population as it relatesto the carrying capacity implied by the new set ofparameters

Complexity 5

223 Minimize Longest Duration of Consecutive LowHarvest Given operational costs the fishery managerswould like to avoid long durations of consecutively lowharvest defined by a minimum harvest limit (e maxduration of low harvest is given by bothzt lt limit and zt+1 lt limit holding for all t in a realization of Tyears (e expected worst case of consecutive low harvest O3is defined as the average of the max duration across Nrealizations

O3 1N

1113944

N

i1maxT ϕti1113872 11138731113872 1113873

where

ϕti

ϕtminus 1i + 1 if zt lt limit

0 otherwise

⎧⎪⎨

⎪⎩

ϕ1i

1 if z1 lt limit

0 otherwise

⎧⎪⎨

⎪⎩

(7)

where limit 5

224 Maximize Worst Harvest Instance Given operationalcosts the fishery managers would like to avoid exposure tofinancial risks Variance-minimizing strategies have beenwidely employed in the literature to model behavior underrisk authors have noted however that the costs of risksassociated with uncertainty often depend on higher ordermoments such as skewness and kurtosis (tail events in thedistribution) [61] (is was approximated in this analysis bymaximizing the worst harvest instance as well as minimizingthe variance of harvest in every realization (O5 explained inthe next section) (e worst harvest instance is approxi-mated here by the 1st percentile of harvest for every T-yearrealization(e expected worst harvest instance is calculatedas the average 1st percentile across N realizations

O4 1N

1113944

N

i1percentileT zt+1ixti 11113872 11138731113872 1113873 (8)

225 Minimize Harvest Variance More traditionally en-countered in the literature is the minimization of the var-iance of deviations from the expected profits [61 62] (isobjective has been approximated by estimating the varianceof the obtained harvest in every T year realization and av-eraging across all N realizations

O5 1N

1113944

N

i1VarT zt+1ixti1113872 11138731113872 1113873 (9)

23 Avoid Collapse of Predator Population Considering theunharvested predator population as a valuable species the

population of which the managers would like to maintainthe optimization is also subject to a constraint

1N

1113944

N

i1ψti1113872 1113873 0

where

ψti

1 if yti lt 1

0 otherwise

⎧⎪⎨

⎪⎩

forallt isin T andforalli isin N

(10)

24 Formulation of Harvesting Policy For the purposes ofthis study candidate DPS control rules were used to map thecurrent levels of prey population (xt) to the harvesting effortat the next time step (zt+1) (e optimization was aimed atidentifying the parameters describing the control rulesinstead of the harvesting efforts themselves allowing forstate-based feedback control strategies (e control ruleswere in the form of Gaussian radial basis functions (RBFs)following the formulation by [63] (e optimization prob-lem was formulated as follows

MinimizeF zx( 1113857 minus O1 O2 O3 minus O4 O5( 1113857

z z1 z2 zT( 1113857

zt+1 1113944n

i1wi exp minus

xt( 1113857 minus ci

bi

1113888 1113889

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

(11)

where i 1 2 n and n is the number of RBFs used in thefunction mapping in this study n 2 wi is the weight of theith RBF and the weights are formulated so as to be positive(ie wi gt 0 foralli) and sum to one (ie 1113936

ni1wi 1) ci and bi are

the center and radius of the ith RBF All wi ci and bi isin [0 1]We used two RBFs and one input in this study resulting insix decision variables that need to be optimized for thecontrol rule mapping current prey population to nextharvest More inputs can be used in the policy formulationbut were omitted from equation (10) for the purposes ofsimplicity Note that with the application of these controlrules the harvesting effort zt+1 will not necessarily be thesame in each of the N realizations as the harvesting action isinformed by the respective level of prey xt which is subjectto the environmental stochasticity at each realizationFurthermore this formulation assumes a perfectly accuratemeasurement of the prey level at all times as well as aperfectly accurate execution of the harvesting strategy(esesimplifying assumptions have the benefit in this study ofdemonstrating the severe difficulty of the class of fisheriesmanagement problems even in optimistic formulations ofavailable information

25 Optimization Algorithm (e multiobjective formula-tion was solved for N 100 realizations of randomly

6 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

223 Minimize Longest Duration of Consecutive LowHarvest Given operational costs the fishery managerswould like to avoid long durations of consecutively lowharvest defined by a minimum harvest limit (e maxduration of low harvest is given by bothzt lt limit and zt+1 lt limit holding for all t in a realization of Tyears (e expected worst case of consecutive low harvest O3is defined as the average of the max duration across Nrealizations

O3 1N

1113944

N

i1maxT ϕti1113872 11138731113872 1113873

where

ϕti

ϕtminus 1i + 1 if zt lt limit

0 otherwise

⎧⎪⎨

⎪⎩

ϕ1i

1 if z1 lt limit

0 otherwise

⎧⎪⎨

⎪⎩

(7)

where limit 5

224 Maximize Worst Harvest Instance Given operationalcosts the fishery managers would like to avoid exposure tofinancial risks Variance-minimizing strategies have beenwidely employed in the literature to model behavior underrisk authors have noted however that the costs of risksassociated with uncertainty often depend on higher ordermoments such as skewness and kurtosis (tail events in thedistribution) [61] (is was approximated in this analysis bymaximizing the worst harvest instance as well as minimizingthe variance of harvest in every realization (O5 explained inthe next section) (e worst harvest instance is approxi-mated here by the 1st percentile of harvest for every T-yearrealization(e expected worst harvest instance is calculatedas the average 1st percentile across N realizations

O4 1N

1113944

N

i1percentileT zt+1ixti 11113872 11138731113872 1113873 (8)

225 Minimize Harvest Variance More traditionally en-countered in the literature is the minimization of the var-iance of deviations from the expected profits [61 62] (isobjective has been approximated by estimating the varianceof the obtained harvest in every T year realization and av-eraging across all N realizations

O5 1N

1113944

N

i1VarT zt+1ixti1113872 11138731113872 1113873 (9)

23 Avoid Collapse of Predator Population Considering theunharvested predator population as a valuable species the

population of which the managers would like to maintainthe optimization is also subject to a constraint

1N

1113944

N

i1ψti1113872 1113873 0

where

ψti

1 if yti lt 1

0 otherwise

⎧⎪⎨

⎪⎩

forallt isin T andforalli isin N

(10)

24 Formulation of Harvesting Policy For the purposes ofthis study candidate DPS control rules were used to map thecurrent levels of prey population (xt) to the harvesting effortat the next time step (zt+1) (e optimization was aimed atidentifying the parameters describing the control rulesinstead of the harvesting efforts themselves allowing forstate-based feedback control strategies (e control ruleswere in the form of Gaussian radial basis functions (RBFs)following the formulation by [63] (e optimization prob-lem was formulated as follows

MinimizeF zx( 1113857 minus O1 O2 O3 minus O4 O5( 1113857

z z1 z2 zT( 1113857

zt+1 1113944n

i1wi exp minus

xt( 1113857 minus ci

bi

1113888 1113889

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

(11)

where i 1 2 n and n is the number of RBFs used in thefunction mapping in this study n 2 wi is the weight of theith RBF and the weights are formulated so as to be positive(ie wi gt 0 foralli) and sum to one (ie 1113936

ni1wi 1) ci and bi are

the center and radius of the ith RBF All wi ci and bi isin [0 1]We used two RBFs and one input in this study resulting insix decision variables that need to be optimized for thecontrol rule mapping current prey population to nextharvest More inputs can be used in the policy formulationbut were omitted from equation (10) for the purposes ofsimplicity Note that with the application of these controlrules the harvesting effort zt+1 will not necessarily be thesame in each of the N realizations as the harvesting action isinformed by the respective level of prey xt which is subjectto the environmental stochasticity at each realizationFurthermore this formulation assumes a perfectly accuratemeasurement of the prey level at all times as well as aperfectly accurate execution of the harvesting strategy(esesimplifying assumptions have the benefit in this study ofdemonstrating the severe difficulty of the class of fisheriesmanagement problems even in optimistic formulations ofavailable information

25 Optimization Algorithm (e multiobjective formula-tion was solved for N 100 realizations of randomly

6 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

generated environmental stochasticity with each realizationspanningT 100 years Initial prey and predator populationvalues were assumed to be x0 2000 and y0 250 (eparameter values for the assumed state of the world (SOW)are listed in Table 1 (e default SOW is assumed to have asingle stable equilibrium that is a global attractor (e pa-rameters of the formulated policy were optimized using theBorg multiobjective evolutionary algorithm (MOEA)MOEAs are heuristic optimization algorithms that follow aniterative search procedure to adapt and evolve a populationof possible solutions toward an optimized set To do soMOEAs apply various probabilistic operators for mutationmating archiving and selection [64]

(e Borg MOEA has been designed for the optimizationof a wide array of many-objective multimodal problems [65](e Borg MOEA is a stochastic population-based searchalgorithm Multiple diagnostic studies have demonstrated itscapacity to perform consistently well across multiple chal-lenging applications [64] Its success in identifying highquality Pareto-approximate solution sets has been attributedto its use of ϵ-dominance archiving ϵ-progress and itsautoadaptive exploitation of multiple search operators basedon their success in generating high quality solutions [65] (enumerical precision required for evaluating each objective ϵis specified by the decision maker creating multidimensionalϵ-boxes for ranking and archiving solutions as the algorithmcompletes its search [65] Additionally ϵ-values provide acontrol on the resolution of the Pareto-approximate set (eg[66])(e standardmeans of specifying ϵ-values is to considerthe significance of precision in each objective(e ϵ-values forthe five objectives were set as follows O1 ϵ 5 O2 ϵ 0005O3 ϵ1 O4 ϵ1 and O5 ϵ 5 (e Borg MOEA wasimplemented using its default parameter values as recom-mended by [65] Since Borg is a stochastic optimization al-gorithm its search is affected by the random seed thatinitializes the population and impacts its stochastic operators(e optimization problem was hence solved with 20 randomseed runs each of which exploited 3000 function evaluations

26 Robustness Analysis As illustrated in Figure 1 the pa-rameters describing the predator-prey system can fundamen-tally shift its dynamics and by extension the harvesting strategyone should choose to employ Given the limitations of ourknowledge of the parameters describing the modeled system[19 20 27] decision makers may consider identifying potentialpolicies that continue to perform satisfactorily when operatedunder a broad range of alternative system characteristics alsoreferred to as SOWs Such policies are termed ldquorobustrdquo and canbe identified using various metrics available in the decisionanalysis literature [67ndash69] (e domain criterion satisficingmeasure [70] quantifies the fraction of potential SOWs inwhich a solution meets a desired performance (eg a set ofcriteria) and has been widely used in the robustness literatureWhen compared against other satisficing- and regret-basedmeasures thismetric has been found to identify solutions in linewith the stakeholdersrsquo performance criteria [67] We generated4000 alternative SOWs using a Latin Hypercube Sample acrossthe ranges of the deeply uncertain parameters listed in Table 1

assuming uniformparameter distributions(eprey populationat each SOW was initialized at the respective sampled K-simulated unharvested population trajectories of both prey andpredator for all SOWs can be found in Figure S3 in the Sup-plementary Material (e purpose of the exploratory parameterranges is to encompass the best available nominal estimateswhile sampling broadly enough in their feasible ranges todiscover consequential impacts(e intent of this approach is toshift focus from predicting system conditions to instead dis-covering scenarios that are consequential to the decisionmakers[71ndash73] Our multivariate satisficing robustness measurequantifies the percentage () of the SOWs in which harvestmanagement is possible (ie there is no deterministic extinc-tion) that meet the following requirements

(1) Net Present Value (NPV) ge 1 500(2) Prey population deficit (Prey Deficit) le 05(3) Longest duration of low harvest (Low Harvest Du-

ration) le 5(4) Worst harvest instance (Worst Harvest) ge 50(5) Harvest variance le 2300(6) Duration of predator population collapse (Predator

Collapse) le 1

(ese performance criteria should be elicited by relevantstakeholders in a real world application In this exploratorywork they are set so as to reflect possible performance levelsexpected by decision makers managing this system Criteria1 3 4 and 5 are each met by at least 75 of the solutions inthe assumed SOW Criterion 2 was set based on critical fishbiomass levels for sustainable yields as reported in theliterature [74] Criterion 6 was met by all solutions in theassumed SOW as it was defined as a constraint

Even though more robust solutions may be discovered ifthe optimization is conducted over a large ensemble ofdeeply uncertain SOWs said robustness might also result inincreased losses (regrets) in the assumed SOW as well as inSOWs that are never encountered [31] Instead in this studywe aim to establish a baseline of performance in the assumedSOW representing our best current state of knowledge andthen re-evaluate this performance across a subjective andwide enough range of SOWs (is allows us to both min-imize regret in the assumed SOW and also highlight theexistence of areas in the parametric space where manage-ment plans might fail despite their highly adaptive andoptimistic design due to their ignorance of a fundamentalshift in system dynamics Lastly this intentionally broadensemble of parameter combinations produces SOWs whereextinction of one or both species is unavoidable (detailed inthe following section) Such SOWs were omitted from therobustness analysis of the candidate solutions so as to onlyevaluate them in contexts where the choice of strategy ac-tually matters and affects the outcomes

3 Results and Discussion

(is section is organized as follows with Figure 2 sum-marizing the motivation and main findings of each section

Complexity 7

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

In Section 31 we first present and discuss the objectivevalues attained in the assumed SOW presented in Figure 3highlighting the strong tradeoffs between NPV and PreyDeficit objectives In Section 32 we explore the impacts ofdeep uncertainty on the inferred tradeoffs for four otherpotential SOWs Figures 4 and 5 demonstrate severe in-stabilities in these objective values when strategies are re-evaluated in different SOWs Further Figure 5 shows thatwhen compared with solutions identified assuming perfectknowledge of the SOW there are significant losses in thepopulations of the two species despite the highly adaptiveharvesting policies In Section 33 we explore how systemstability is generally affected by parametric uncertainty bylooking at the SOWs leading to predator population col-lapse In Figure 6 we show that there is a multidimensionalthreshold surface dividing sustained harvesting and recoveryfrom fishery collapse where there are no managementtradeoffs to be attained Finally in Section 34 we posit thatthe concept of multiobjective robustness can be a valuabledriver for selecting management strategies that meet nec-essary system performance and avoid system collapse Wepresent the robustness values of each candidate managementstrategy in Figure 7 and the implications of choosing two ofthem for the two fish populations in Figure 8 Figure 2summarizes the methodological approach of this study withthe motivation and main findings of each section

31 Fishery Management Tradeoffs in Assumed State of theWorld (e general challenge considered by fishery man-agement is how to best exploit the system through harvestingwhile balancing multiple societal objectives with respect toenvironmental sustainability and profit Figure 3 presents aparallel axis plot of the objective values achieved by eachoptimized solution in the assumed SOW (ie the set ofmodel parameters describing the system to which thepolicies were optimized)(e performance on each objectiveis represented by a vertical axis (e points where each linecrosses a vertical axis represent the average performancevalue for that objective across 100 realizations of environ-mental stochasticity An upward shift in one of the verticalaxes indicates increased preference in the equivalent ob-jective performance All lines have been shaded according totheir performance on the NPV objective(e plot is orientedsuch that the ideal solution would be a dark horizontal linecrossing the top of each vertical axis Diagonal intersectinglines indicate pairwise tradeoffs between two objectiveswhere improving the performance in one objective is onlypossible with reduced performance in the other One shouldnote that the identified solutions carry no stakeholderpreference (or weight) towards the objectives but have beeninstead identified by searching the space of possibilities aswidely as possible Presenting objective performance andtradeoffs in such a format allows and facilitates an a-pos-teriori elicitation and negotiation of preferences by thedecision makers [76] For example in systems or decision-making contexts where species conservation is valued morethan harvest profits one can express such weighting byimposing upward moving limits on the respective axes (is

preference elicitation process should be iterative allowingfor stakeholders with diverging preferences to evaluate theperformance and appropriateness of the candidate solutionsfor the system at hand

As indicated by the intersecting lines and the switch inthe color gradient there appears to be strong tradeoffsbetween the NPV and Prey Deficit objectives as well asbetween the Prey Deficit and the Low Harvest Durationobjectives NPV-maximizing policies do so by harvestinglarge parts of the available prey population depleting it fromits natural levels by 582 on average across all realizationsSimilarly policies that minimize the Low Harvest Durationto zero (ie harvesting at least 5 of the available prey at alltimes) also severely deplete the prey population In contrastpolicies that minimize the Prey Deficit to 5 achieve verylow values in the NPV objective (indicated by their very lightcolor) as well as in the Low Harvest Duration objectiveLooking at the two objectives incorporated to minimizefinancial risk Worst Harvest and Harvest Variance one maynote that some solutions ranking poorly with regards to their

Figure 2 Main findings on the implications of deep uncertainty onthis predator-prey system Latin numerals indicate the Results andDiscussion sections in which each step is presented See the maintext for further description

8 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

Worst Harvest perform very well in minimizing HarvestVariance (ese solutions also achieve low NPVs (as indi-cated by their light color) suggesting that the low variance isachieved by simply consistently harvesting very little at everytime step (is observation is consistent with past robustoptimization studies (eg [77 78]) that have found vari-ance-minimizing objectives penalizing outcomes both aboveand below the mean (ie both higher and lower profits whenonly lower profits are of concern)

(e identified solutions have been optimized under theassumption that the abstracted harvesting agent has access toan accurate reading of the prey population at each timestep(ie ldquoerror-free informationrdquo) Furthermore the solutionsand tradeoffs that arise as a result of this formulation onlytake into account uncertainty in the form of environmentalstochasticity (ϵi) which is traditionally assumed to be wellcharacterized (ese assumptions are commonly employedin the literature [52 79 80] and are highlighted here toemphasize the highly optimistic nature of such optimizedstrategies even when they account for standard sources ofenvironmental uncertainty In addition the harvestingpolicies assume no incidental by-catch of the predator Ourintent is to illustrate that even in a highly optimistic har-vested predator-prey management context with a well-in-formed and highly adaptive harvesting agent deepuncertainties that are traditionally neglected pose severechallenges

32 Tradeoff Implications of Deep Uncertainty Figure 3presents the objective values achieved by each policy under

a SOW representing our best knowledge of the system apredator-dependent system with a stable global attractor thedynamics of which are presented in Figure 1(b) As previouslyelaborated precise estimates of growth and death rates pre-dation and interference are often difficult to estimate fromempirical data [19 20] Given this uncertainty in the parameterestimates Figure 1(c) presents the dynamics of a predator-dependent system with a now unstable equilibrium resultingfrom slight shifts in our best estimates of the systemrsquos pa-rameter values Figures 1(a)ndash1(f) illustrate the wide variety ofdynamic behavior that can occur as a result of uncertainty inthe system parameterization in absence of any human dis-turbance (harvesting) Having identified the Pareto-approxi-mate set of solutions for the assumed SOW (Figure 3) Figure 4presents the same set of optimized solutions with their per-formance in a three-dimensional objective space(e solutionsin the assumed SOW are presented in blue and in light redlight orange light green and brown the plot shows how theirobjective values change when re-evaluated in four other SOWs(e parameter values for the four SOWs presented in Figure 4are provided in Table 2

When policies are re-evaluated in a SOW with de-terministic extinction (dynamics presented inFigure 1(e)) the Pareto front collapses (brown points)Population collapse is deterministic in this SOW andoccurs even without any harvest Deterministic extinc-tion has been the focus of several studies [13 24 81]particularly in the context of the prey- versus ratio-de-pendent predator-prey theory Deterministic extinctionbehavior was routinely observed [24] but could not bedescribed by the classic prey-dependent model (e

Pref

eren

ceA

vera

ge o

bjec

tive p

erfo

rman

ce

0

600

1800

3000

4200

5400

Net

pre

sent

val

ue (N

PV)

Net present value (NPV)

Prey population deficit

Longest duration of low harvest

Duration of predator population collapse

Harvest variance

058 1000 00 31842 0000Worst harvest

instance

5586 005 00 230 00 00

Figure 3 Parallel axis plot of the average objective values of solutions optimized in the assumed SOW(e points where each line (solution)intersects with a vertical axis represent the average performance value across 100 realizations of environmental stochasticity (e plot isoriented such that the ideal solution would be a horizontal line crossing the top of each vertical axis Diagonal lines indicate pairwisetradeoffs between two objectives (e color gradient is based on the performance of each solution in the NPV objective with darker colorsrepresenting a higher NPV

Complexity 9

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

behavior is explained by predators becoming more effi-cient (higher α and c and lower h values) and dying outafter exhausting the prey

(e policies were also re-evaluated in a parametricallynearby SOW with a global attractor (light orange) and in aparametrically distant SOW with a global attractor (lightgreen) both of which exhibit dynamic behavior akin to thatin the assumed SOW (presented in Figure 1(b)) Both re-evaluations examine situations where the decision makersfind themselves managing a system exhibiting similar dy-namic behavior to that assumed but having different pa-rameter values from their best estimates In the nearby SOW(light orange) a subset of candidate harvesting policies causesignificant losses in the prey population numbers (increasedvalues in prey population deficit) and predator populationcollapse (Figure 4) In the distant SOW (light green) theconditions for prey growth are more favorable higher Kallows for larger prey population and higherm stabilizes thesystem In this SOW no significant losses are exhibited in

either of the two populations and the adaptive controlpolicies that had been identified appear to outperform theexpected objective values achieved in the assumed SOWHowever this is a stable SOW favorable to higher growthand a decision maker might be inclined to inquire into whatobjective values could have been achieved having had perfectinformation about the SOW being managed In other wordsif the optimization was performed having the accuratereading of the parameters describing the system each timehow would the performance of those solutions differ

Points in dark orange and dark green in Figure 4 presentthe objective values achieved by the solutions identifiedwhen re-optimizing to the equivalent nearby and distantSOWs In the nearby stable SOW (light and dark orange)the significant losses in prey population numbers are re-duced by the solutions identified through optimization withperfect information (is is more clearly seen in theequivalent parallel axis plot (Figure 5(a)) Here the solutionsidentified in the assumed SOW and re-evaluated in the one

Net pr

esen

t valu

e (NPV

)

500

400

Wor

st ha

rves

t ins

tanc

e

300

200

100

0100 075 050

Prey population deficit025 0

4000

800012000

700

600

Solutions identified inassumed SOWSolutions re-evaluated in nearbySOW with global attractorSolutions identified in nearbySOW with global attractorSolutions re-evaluated in a deterministic extinction SOW

Solutions re-evaluated in distantSOW with global attractorSolutions identified in distantSOW with global attractorSolutions re-evaluated in SOW without global attractorSolutions identified in SOWwithout global attractor

Figure 4 Solutions as points in a 3D objective space Blue as identified in the assumed SOW brown re-evaluated in a SOW withdeterministic extinction light red re-evaluated in nearby SOW without a global attractor light orange re-evaluated in nearby SOW withglobal attractor light green re-evaluated in distant SOWwith global attractor Dark red dark green and dark orange represent the solutionsidentified when optimizing to those SOWs (e parameter values for each SOW are provided in Table 2

10 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

nearby (presented in grey) are contrasted with thoseachieved in the nearby SOW having had perfect informationduring the optimization (presented in shades of orange)(eregrets are significant in the Prey Deficit objective where thevalue of the worst-performing policy could have been re-duced from 97 to 75 with perfect information about theSOW parameterization More importantly many of thepolicies re-evaluated in this SOW fail to meet the PredatorCollapse constraint with the worst-performing among themfailing 2934 of the time Conversely all the policiesidentified with perfect information about the SOWmeet thatconstraint Looking at the distant stable SOW (light and darkgreen in Figure 4) even though the state-aware controlpolicies manage to avoid significant prey losses the regretsin this case come in the form of lower values in the NPV andWorst Harvest objectives (is is despite the fact that theharvesting agent is highly adaptive and has a perfect readingof the prey population at each timestep

When solutions are re-evaluated in a nearby SOWwithout a global attractor (light red in Figure 4) multiplebasins of attraction exist and changes in initial conditions

can lead to different equilibria (as depicted in Figure 1(f )) Inthis SOW the predators are more efficient (higher α and cvalues) but also exhibit high interference (mgt 1) which hasa stabilizing effect on their population [24] In such a systemthe decision maker harvesting the prey competes with moreefficient predators and the solutions re-evaluated in thisSOW end up significantly depleting the prey (light red inFigure 4) as well as collapsing the predator population inmost instances (grey lines in Figure 5(c)) Even if the systemdynamics are less favorable in this SOW predator pop-ulation collapse could have been entirely avoided by allsolutions if the optimization had perfect information aboutthe SOW parameters (red lines in Figure 5(c))

33 System Stability Implications of Deep UncertaintyFigures 4 and 5 present how the initial perceptions oftradeoffs in the original SOW would be misleading if thefishery was actually described by the parameterizations ofthe four other distinct SOWs selected here for the purposeof demonstration As knowledge limits may yield broad

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

00 097 1000 00 40908 29

Pref

eren

ce

3606 042 00 1167 00 00

0

1000

2000

3000

Net

pre

sent

valu

e (N

PV)

(a)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

060 1000 00 917890 0000

17862 008 00 781 00 00

0

5000

10000

15000

(b)

Net present value (NPV)

Prey populationdeficit

Longest duration of low harvest

Worst harvest instance

Duration of predator population collapse

Harvest variance

Net

pre

sent

valu

e (N

PV)

Pref

eren

ce

4311 00 00 64 00 00

098 1000 00 141741 810001000200030004000

(c)

Figure 5 Solutions re-evaluated in three SOWs and solutions optimized to those SOWs in a parallel coordinate form (a) Nearby SOWwitha global attractor Orange solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW(b) Distant SOW with a global attractor Green solutions optimized in this SOW and grey solutions identified in assumed SOW and re-evaluated in this SOW (c) Nearby SOW without a global attractor Red solutions optimized in this SOW and grey solutions identified inassumed SOW and re-evaluated in this SOW (e parameter values for each SOW are provided in Table 2

Complexity 11

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

Distant SOW with a global attractorNearby SOW with a global attractorNearby SOW with deterministic extinctionSOW without a global attractor Assumed SOW

SOW with deterministic extinctionSOW without deterministic extinction

α

b

m

Perc

enta

ge o

f sol

utio

ns w

ithou

t pre

dato

r col

laps

e

00 02 04 06 08 10 12 14

0002

0406

0810

Inequality for stability

0

20

40

60

80

100

200

175

150

125

100

075

050

025

000

Figure 6 All the sampled SOWs as they fall in the α b and m parametric space (all other parameters kept constant) (e percentage ofsolutions that do not lead to any predator collapse when applied in each SOW is represented by the color at each point Small points andlarge points indicate SOWs with and without deterministic extinction respectively (e shaded surface indicates the inequality for stability(4) with all other parameters held constant at the default values and harvesting z assumed to be zero (e SOWs presented in Figure 4 arealso contextualized in the parametric space

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re ea

ch cr

iterio

n is

met

()

Most robust in NPV - ANPVMost robust across criteria - BMO

Prey population deficit lt05

Longest duration of low harvest lt5

Worst harvest instance gt50

Duration of predator population collapse lt1

Net present value (NPV) gt1500

Harvest variance lt2300

Perc

enta

ge o

f SO

Ws w

ithou

t DE

whe

re al

l crit

eria

are m

et (

)

0

20

40

60

80

100

0

10

20

30

40

Figure 7 Parallel coordinate plot of satisficing tradeoffs for all identified solutions when applied in the alternative SOWs without de-terministic extinction Each vertical axis represents the percentage of SOWs in which each solution fulfills each criterion Each solution iscolored based on the percentage of SOWs in which it fulfills all criteria (e two highlighted solutions ANPV and BMO were identified usingtwo robustness definitions meeting criterion 1 and meeting all criteria respectively

12 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

parameter ranges it is a decision relevant concern to assesshow consequential performance changes may be for thetradeoff solutions across a broader sample of the deeplyuncertain parametric space We therefore have explored theimpacts of 4000 SOWs Figure 6 presents the sampledSOWs in the α b and m parametric space with all otherparameter values kept constant (e color of each pointindicates the percentage of the original set of tradeoff so-lutions () that do not lead to inadvertent predator collapsewithin each sampled SOW Certain parameter combinationscause deterministic extinction as populations collapse evenin the absence of harvest these SOWs are indicated by the

smaller dark red points in Figure 6 Even though the modelused in this study is in a discrete-time form the continuous-time model (equation (2) in Section 2) can be used to studythe underlying dynamics of the system (e necessarycondition for a stable global attractor equilibrium as derivedfor this generalized system with harvest (equation (4) inSection 2) and applied to the parametric space is repre-sented by the shaded surface shown in Figure 6

(e underlying dynamics of the system can shift suchthat the coexistence of the two species is deterministicallyimpossible In other words a multidimensional thresholdsurface exists that separates sustained harvesting and

Assumed SOW SOW with a global attractor SOW without a global attractor

Most robust in NPV - ANPVMost robust across criteria - BMO

Policy Trajectory

Prey abundance Prey abundance Prey abundance

Endpoint2050

of theoretical sustainable yield

Pred

ator

abun

danc

ePred iso

Prey iso

Naturalequilibrium

0

200

400

600

800

1000

1200

1400

Pred iso

Prey iso

Naturalequilibrium

Pred

ator

abun

danc

e

0

200

400

600

800

1000

1200

1400

Pred

ator

abun

danc

e

Pred iso Prey iso

Natural equilibrium

0

50

100

150

200

250

300

0 500 1500 20001000 0 500 1500 20001000 0 500 1500 2000 2500 3000 35001000

(a) (b) (c)

02 04 06 08 1000Ratio of prey harvested

Figure 8 Fish population trajectories as a result of two harvesting policies applied in the assumed SOW and two alternative SOWs(e twoapplied solutions ANPV and BMO were identified using two robustness definitions meeting criterion 1 and meeting all criteria respectivelyEach line is a system trajectory starting near the unharvested equilibrium(e color at each point of the trajectory indicates harvesting effortCritical prey population levels 50 and 20 of the biomass for theoretical sustainable yield indicate overfishing and endangermentrespectively [74 75]

Table 2 Parameter values for the assumed SOW and four distinct SOWs(e fishery harvesting policies are optimized to the assumed SOW(in blue) (e performance of the solutions in four distinct SOWs is highlighted in Figures 4ndash6

Parameters Assumed(blue)

Nearby and stable(orange)

Distant andstable (green)

Nearby andunstable (red)

Deterministicextinction (brown)

α 0005 0208 1867 0796 1775b 05 0663 0830 0215 0389c 05 0361 0542 0565 0441d 01 0095 0197 0137 0083h 01 0372 0949 0472 0941K 2000 208058 461048 485848 446507m 07 093 1442 121 0107σx 0004 0004 0002 0008 0003σy 0004 0003 0005 0009 0007

Complexity 13

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

recovery from fishery collapse where there are no man-agement tradeoffs to be attained Even though this ex-ploratory analysis sampled the parametric space uniformlyits intent has not been to assign equal probabilities to alloutcomes including the deterministic extinction cases butrather to explore broadly to discover said threshold surfacein the parametric space With regards to the parametricplausibility of such a region it could be the side effect ofpermanent change in one or more critical components in theenvironment of a species Such changes may be progressive(eg climate change [4] or habitat loss [6]) and lead topermanent changes in the growth or death rates of a speciessuch that it inevitably moves towards extinction [82] Whencollapse is not prescribed by the underlying dynamics it isevident that it can be avoided by some of the optimizedstrategies and as a result value preference in the objectivespace becomes the most decisive consideration Whenfeasible we hypothesize that the concept of multiobjectiverobustness [76 83] achieved through compromise can be avaluable driver in selecting a strategy that can avoid thecollapse of the two species

34 System Behavior as a Result of Decision PreferencesExploiting knowledge of the deterministic extinctionthreshold we shift focus to reevaluating the candidateharvesting strategies in SOWs where management tradeoffsexist and assess their robustness To do so we re-evaluateeach of the Pareto-approximate solutions presented inFigure 3 in the alternative sampled SOWs and compute theirrobustness using the domain criterion satisficing measure(is measure includes minimum performance requirementsdefined by six satisficing criteria for the managementproblemsrsquo objectives and predator population collapseconstraint (detailed in Section 2) We then calculate thepercentage of SOWs where each solution meets each of thecriteria as well as the percentage of SOWs where eachsolution meets all criteria Extinction of the two species isdeterministic in some of the SOWs and thus the domaincriterion satisficing measure was only applied in SOWs inwhich human action (and choice thereof) matters Figure 7presents the percentage of SOWs where each of the opti-mized solutions meets each of the six criteria (placed on avertical axis) Each line is shaded according to the percentageof SOWs where it meets all specified criteria As with Fig-ure 3 diagonal intersecting lines indicate pairwise tradeoffsin the robustness of the equivalent criteria For examplethere appears to be a strong tradeoff between meeting theNPV criterion in many SOWs and meeting the Prey Deficitcriterion in many SOWs Furthermore solutions most ro-bust in either of those two criteria (top-most lines crossingthe vertical axes) fail to meet at least one of the otherspecified criteria (indicated by their white color)

Two solutions are highlighted in this figure ANPV andBMO illustrating two different robustness definitions (eANPV solution meets criterion 1 (NPVge 1500) in the mostSOWs (leftmost vertical axis in Figure 7) NPV-maximizingcriteria (or similar metrics of discounted profits) are verycommon in the bioeconomics literature (eg [34 84ndash86])

typically considered as the only system objective (e BMOsolution meets all of the specified satisficing criteria in themost SOWs meaning that it exhibits the highest multi-objective robustness (indicated by the line color and colorbar on the right of Figure 7) (is solution is more in linewith newer paradigms in fisheries management with expertscalling for multiobjective perspectives and values [29 45](e concept of multiobjective robustness builds on thatperspective by identifying policies that are successful inmeeting a specified level of performance in all objectives(esolution most robust in NPV performs poorly in the preypopulation deficit and the harvest variance criteria (esolution most robust across all criteria achieves robustnessby compromising individual robustness for the NPV andlow harvest duration criteria as well as performance in theobjective space (see Supplementary Material Figure S2)

With regards to the entire set of optimized solutions andtheir application in other SOWs Figure 7 highlights thatassumptions on their stability are tenuous even in SOWswhere species coexistence and harvesting are possible (iewithout deterministic extinction) Looking at criterion 2 inparticular the most robust of the solutions in meeting thatcriterion (topmost line crossing the axis) only does so infewer than 50 of the considered SOWs without deter-ministic extinction and in doing so it harvests at very lowrates and fails to ever meet the other criteria (as indicated byits color) Solution ANPV (most robust in criterion 1) har-vests at very high rates and fails to meet the prey deficitcriterion Finally solution BMO (meeting all criteria in themost SOWs) only performs acceptably in fewer than 50 ofthe sampled SOWs In other words out of the original set ofoptimized solutions all of which adapting their harvestgiven a perfect reading of the prey population at eachtimestep most fail to meet all the criteria in other SOWs andthe most robust among them only does so in less than half ofthe sampled SOWs without deterministic extinction At thesame time when looking at criteria 1 and 2 the equivalentobjectives of which are conflicting (maximizing NPV andminimizing prey population deficit) increasing robustnesspreference for any of the two inevitably reduces robustnessin the other

Linking our observations for this socioecological systemto the broader concept of resilience we draw from itsdefinition as proposed by [87] ldquothe capacity of a system toabsorb disturbance and reorganize while undergoing changeso as to retain essentially the same function structureidentity and feedbacksrdquo While robustness is a related termit is defined as the ability of the strategies to maintain anacceptable performance (as measured by some criteria) andit more closely pertains to the decision space of the system asshaped by the decision-makersrsquo value preferences As suchthe concept can be used to link complex system dynamicspersistence and transformation to performance measures(albeit in a more narrow sense-that of feedback and control)[88] With the application of the concept of multiobjectiverobustness in this paper the identified strategy is also moreresilient across the sampled SOWs

(is is more explicitly demonstrated in Figure 8 wherewe present the trajectories of the predator-prey system as a

14 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

result of the two harvesting strategies in the assumed SOWand two alternative SOWs We include five trajectories ineach of the subplots aiming to capture imperfect readings ofthe initial population levels each starting near the natural(unharvested) equilibrium of the system Figure 8(a) pres-ents the trajectories of the two populations as they resultfrom the implementation of the two policies in the assumedSOW Solution ANPV appears to harvest at significantlyhigher levels (indicated by the color of each line segment)and reduce the prey population resulting in a reduction inthe predator population as well Solution BMO harvests atsignificantly lower levels and reduces both populations aswell albeit to levels much closer to the natural equilibriumWhen applied in a nearby SOW with a global attractor(Figure 8(b)) the harvesting actions behave similarlyshifting the prey isocline and as a result the equilibriummoves to lower population levels for the two species (thedynamics of the unharvested system are presented inFigure 1(b)) Figure 8(c) shows the two policies applied in aSOW where the coexistence equilibrium (trajectory startingpoint) is not a global attractor Here the harvesting can shiftthe system to a different basin of attraction so as to lead tothe collapse of the two populations (is is avoided by thestrategy selected for being robust across the multiple ob-jectives (BMO)

4 Conclusions

Progressive discoveries in the ecological literature havehighlighted the importance of multispecies relationships andtrophic connectivity in fisheries management resulting in anew proposed paradigm that of ecosystem-based fisherymanagement [29 45] Within this new direction multiplespecies and objective values need to be taken into account(is simple exploratory experiment was designed with thespecific aim of complementing the current fisheries man-agement literature with the inclusion of a multiobjectiveperspective for the management of a two-species fishery(eformulation of the system in such a manner allows us todemonstrate the potentially catastrophic consequences ofdeep uncertainty as manifested in our parameterizedmathematical representations of predator-prey systems thatare coupled with human actions (harvesting) and humanpreferences (multiobjective tradeoffs) For this purpose theisoclines equilibria and conditions for stability were ana-lytically derived for the harvested system and used to dis-cover regions of the deeply uncertain parametric space thatlead to system instability and fundamentally impact theestimated tradeoffs(e impacts of deep uncertainty on sucha system are significant as distinct basins of attraction can bepresent and also shift with only marginal changes in as-sumed mathematical parameterizations Human actionmanifested in the form of harvest can move the populationsinto basins of attraction where the attractor is a collapsedsystem

As a result harvesting strategies able to navigate thedynamic complexities of such a system need to be identifiedWe demonstrate that multiobjective robustness can bevaluable as a driver for identifying fishery harvest policies

that can navigate deep uncertainties in system parametersand relationships It is important to note here that theconcept of multiobjective robustness is not treated as en-tirely equivalent to resilience but rather as an alternativeprinciple by which one can operationalize the identificationof such policies in a systematic manner (e principle isapplied here with the specific aim of achieving acceptableperformance as measured by explicit criteria that are de-fined subjectively and specifically for the system at hand(ebroader concept of resilience includes a wider array of as-pects and spatial and temporal scales as well as systemboundaries [88] that multiobjective robustness (as appliedhere) does not claim to consider Nevertheless managementpolicies that are not robust fail to meet their decision-rel-evant criteria and by extension lead to a system that is alsonot resilient For these reasons we believe that the findingshave broader implications for socioecological managementin general where balancing conflicting economic and eco-logical objectives is an essential yet complex task that isfurther impeded by severe uncertainties in determiningtipping points and the consequences of crossing them [89]

(e system used in the study is specifically formulated ata reduced complexity relative to those found in the math-ematical biology literature where multispecies systems aretypical or in resource economics where operational costsandmore complex financial accounting take place Howeverthe more complex models in the respective fields are nottypically paired with each other and the impacts of deepuncertainty in the ecological system are not manifestedthrough to the decision space to assess its social implica-tions (e socioecological model employed in this studypairs the two perspectives to demonstrate significant im-plications for the decision maker even when management isoperating in the optimistic context of adaptive perfectlyinformed and perfectly implemented policies Furthermorethe harvesting policies assume no incidental by-catch norintentional harvest of the predator as the aim has been tohighlight the implications of deep uncertainty for such asystem

Be that as it may this simplicity in design bears certainlimitations that future work should aim to address First theinformation used by the state-aware control rules is limitedto the current levels of the prey fish population In theinterest of avoiding crossing tipping points critical to therecovery and sustainability of a fishery additional infor-mation can be incorporated for example predator pop-ulation levels (is would be particularly valuable for systemparameterizations that exhibit multistability (for example inFigure 1(f)) where remaining with the basin of attraction ofthe preferred equilibrium is a vital concern In a real ap-plication this would likely be accompanied by additionalcosts borne by sensing and measurements Secondly thecontrol rules mapping system state to action could takedifferent forms to more realistically represent the harvestingactions performed by fisheries for example to avoid sig-nificantly shifting the prescribed harvest between sequentialtime steps or accidentally harvesting other species (by-catch) Lastly the study assumes a perfect reading of the preypopulation at each time step as well as a perfect execution of

Complexity 15

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

the prescribed harvesting effort (ese are common as-sumptions in the literature [52 79 80] but result in highlyoptimistic quantifications of the tradeoffs and should beaddressed in future applications Inclusion of a learningprocedure about system parameters and dynamics in theform of a probabilistic approach with sequentially updatedestimates [90 91] would also be a valuable next step

Data Availability

(e predator-prey harvesting optimization code re-evalu-ation code robustness analysis and identified solutions areavailable on Github at httpsgithubcomantonia-hadGeneralized_fish_game (e optimization and Pareto-sort-ing can be replicated using the software code available for theBorg MOEA (httpborgmoeaorg) paretopy (httpsgithubcommatthewjwoodruffparetopy) and the MOEAframework (httpmoeaframeworkorg)

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this material are those of the authors anddo not necessarily reflect the views of the funding entities

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(is study was partially supported by the National ScienceFoundation (NSF) through the Network for SustainableClimate Risk Management (SCRiM) under NSF cooperativeagreement GEO-1240507 and the Penn State Center forClimate Risk Management

Supplementary Materials

S1 Appendix derivation of condition (3) S2 Appendixderivation of the prey isocline S1 Figure bifurcation dia-grams for systems presented in Figure 1 with regard topredator interference parameter m S2 Figure populationtrajectories for prey and predator populations under allsampled SOWs S3 Figure parallel axis plot of the objectivevalues achieved by each optimized solution in the assumedSOW (Supplementary Materials)

References

[1] CMullon P Freon and P Cury ldquo(e dynamics of collapse inworld fisheriesrdquo Fish and Fisheries vol 6 no 2 pp 111ndash1202005

[2] W H Lear and L S Parsons ldquoHistory andmanagement of thefishery for northern cod in NAFODivisions 2J 3K and 3Lrdquo inPerspectives on Canadian Marine Fisheries ManagementW H Lear and L S Parsons Eds vol 226 pp 55ndash90Canadian Bulletin of Fisheries and Aquatic Sciences OttawaCanada 1993

[3] J A Hutchings and R A Myers ldquoWhat can be learned fromthe collapse of a renewable resource Atlantic Cod Gadus

morhua of newfoundland and labradorrdquo Canadian Journal ofFisheries and Aquatic Sciences vol 51 no 9 pp 2126ndash21461994

[4] C Mollmann and R Diekmann ldquoChapter 4mdashmarine eco-system regime shifts induced by climate and overfishing areview for the northern hemisphererdquo in Advances in Eco-logical Research (Global Change in Multispecies Systems Part2) G Woodward U Jacob and E J OrsquoGorman Eds vol 47pp 303ndash347 Academic Press Cambridge MA USA 2012

[5] K Ludynia J-P Roux R Jones J Kemper andL G Underhill ldquoSurviving off junk low-energy prey domi-nates the diet of African penguins spheniscus demersusatMercury island Namibia between 1996 and 2009rdquo AfricanJournal of Marine Science vol 32 no 3 pp 563ndash572 2010

[6] J F Craig Freshwater Fisheries Ecology John Wiley amp SonsChichester UK 2015

[7] J Travis F C Coleman P J Auster et al ldquoIntegrating theinvisible fabric of nature into fisheries managementrdquo Pro-ceedings of the National Academy of Sciences vol 111 no 2pp 581ndash584 2014

[8] M L Pinsky and D Byler ldquoFishing fast growth and climatevariability increase the risk of collapserdquo Proceedings of theRoyal Society B Biological Sciences vol 282 no 1813 ArticleID 20151053 2015

[9] R J Lempert and M T Collins ldquoManaging the risk of un-certain threshold responses comparison of robust optimumand precautionary approachesrdquo Risk Analysis vol 27 no 4pp 1009ndash1026 2007

[10] F H Knight Risk Uncertainty and Profit Courier DoverPublications Mineola NY USA 1921

[11] M P Hassell and G C Varley ldquoNew inductive populationmodel for insect parasites and its bearing on biologicalcontrolrdquo Nature vol 223 no 5211 pp 1133ndash1137 1969

[12] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquoFe Journal ofAnimal Ecology vol 44 no 1 pp 331ndash340 1975

[13] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal of MathematicalBiology vol 36 no 4 pp 389ndash406 1998

[14] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-Dependencerdquo Journal of Feoretical Biologyvol 139 no 3 pp 311ndash326 1989

[15] G D Ruxton andW S C Gurney ldquo(e interpretation of testsfor ratio-dependencerdquoOikos vol 65 no 2 pp 334-335 1992

[16] P A Abrams ldquo(e fallacies of ldquoratio-dependentrdquo predationrdquoEcology vol 75 no 6 pp 1842ndash1850 1994

[17] O Sarnelle ldquoInferring process from pattern trophic levelabundances and imbedded interactionsrdquo Ecology vol 75no 6 pp 1835ndash1841 1994

[18] H R Akcakaya R Arditi and L R Ginzburg ldquoRatio-de-pendent predation an abstraction that worksrdquo Ecologyvol 76 no 3 pp 995ndash1004 1995

[19] P A Abrams and L R Ginzburg ldquo(e nature of predationprey dependent ratio dependent or neitherrdquo Trends inEcology amp Evolution vol 15 no 8 pp 337ndash341 2000

[20] R Arditi and H R Akccedilakaya ldquoUnderestimation of mutualinterference of predatorsrdquo Oecologia vol 83 no 3pp 358ndash361 1990

[21] S R Carpenter K L Cottingham and C A Stow ldquoFittingpredator-prey models to time series with observation errorsrdquoEcology vol 75 no 5 pp 1254ndash1264 1994

[22] C Jost and R Arditi ldquoIdentifying predator-prey processesfrom time-seriesrdquo Feoretical Population Biology vol 57no 4 pp 325ndash337 2000

16 Complexity

[23] J L Sabo ldquoStochasticity predator-prey dynamics and triggerharvest of nonnative predatorsrdquo Ecology vol 86 no 9pp 2329ndash2343 2005

[24] R Arditi J-M Callois Y Tyutyunov and C Jost ldquoDoesmutual interference always stabilize predator-prey dynamicsA comparison of modelsrdquo Comptes Rendus Biologies vol 327no 11 pp 1037ndash1057 2004

[25] Y-M Bozec S OrsquoFarrell J H Bruggemann B E Luckhurstand P J Mumby ldquoTradeoffs between fisheries harvest and theresilience of coral reefsrdquo Proceedings of the National Academyof Sciences vol 113 no 16 pp 4536ndash4541 2016

[26] J A Estes J Terborgh J S Brashares et al ldquoTrophicdowngrading of planet earthrdquo Science vol 333 no 6040pp 301ndash306 2011

[27] D B Stouffer ldquoAll ecological models are wrong but some areusefulrdquo Journal of Animal Ecology vol 88 no 2 pp 192ndash1952019

[28] S Murawski ldquoDefinitions of overfishing from an ecosystemperspectiverdquo ICES Journal of Marine Science vol 57 no 3pp 649ndash658 2000

[29] E K Pikitch C Santora E A Babcock et al ldquoECOLOGYecosystem-based fishery managementrdquo Science vol 305no 5682 pp 346-347 2004

[30] W J Sutherland R P Freckleton H C J Godfray et alldquoIdentification of 100 fundamental ecological questionsrdquoJournal of Ecology vol 101 no 1 pp 58ndash67 2013

[31] J D Quinn P M Reed and K Keller ldquoDirect policy searchfor robust multi-objective management of deeply uncertainsocio-ecological tipping pointsrdquo Environmental Modelling ampSoftware vol 92 pp 125ndash141 2017

[32] R Singh P M Reed and K Keller ldquoMany-objective robustdecision making for managing an ecosystem with a deeplyuncertain threshold responserdquo Ecology and Society vol 20no 3 pp 1ndash32 2015

[33] D Hadka J Herman P Reed and K Keller ldquoAn open sourceframework for many-objective robust decision makingrdquoEnvironmental Modelling amp Software vol 74 pp 114ndash1292015

[34] J E Wilen ldquoRenewable resource economists and policy whatdifferences have we maderdquo Journal of Environmental Eco-nomics and Management vol 39 no 3 pp 306ndash327 2000

[35] M B Schaefer ldquoSome considerations of population dynamicsand economics in relation to the management of the com-mercial marine fisheriesrdquo Journal of the Fisheries ResearchBoard of Canada vol 14 no 5 pp 669ndash681 1957

[36] R J H Beverton and S J Holt ldquoOn the dynamics of exploitedfish populationsrdquo Fisheries Investigations Series Vol 19Ministry of Agriculture FAO Rome Italy 1957

[37] A Scott Natural Resources Fe Economics of ConservationUniversity of Toronto Press Toronto Canada 1955

[38] H S Gordon ldquo(e economic theory of a common-propertyresource the fisheryrdquo Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[39] L S Pontryagin V S Boltyanskii R V Gamkrelidze andE F Mishchenko Mathematical Feory of Optimal ProcessesWiley-Interscience Hoboken NJ USA 1962

[40] R Dorfman ldquoAn economic interpretation of optimal controltheoryrdquo Fe American Economic Review vol 59 no 5pp 817ndash831 1969

[41] H Matsuda and P A Abrams ldquoMaximal yields from mul-tispecies fisheries systems rules for systems with multipletrophic levelsrdquo Ecological Applications vol 16 no 1pp 225ndash237 2006

[42] J Morishita ldquoWhat is the ecosystem approach for fisheriesmanagementrdquo Marine Policy vol 32 no 1 pp 19ndash26 2008

[43] A Hollowed N Bax R Beamish et al ldquoAre multispeciesmodels an improvement on single-species models for mea-suring fishing impacts on marine ecosystemsrdquo ICES Journalof Marine Science vol 57 no 3 pp 707ndash719 2000

[44] C J Walters V Christensen S J Martell and J F KitchellldquoPossible ecosystem impacts of applying MSY policies fromsingle-species assessmentrdquo ICES Journal of Marine Sciencevol 62 no 3 pp 558ndash568 2005

[45] S M Garcia A Zerbi C Aliaume T Do Chi and G Lasserreldquo(e ecosystem approach to fisheries issues terminologyprinciples institutional foundations implementation andoutlookrdquo Food amp Agriculture Organization (FAO) RomeItaly No 443 in FAO Fisheries Technical Paper 2003

[46] J Hoekstra and J C J M van den Bergh ldquoHarvesting andconservation in a predator-prey systemrdquo Journal of EconomicDynamics and Control vol 29 no 6 pp 1097ndash1120 2005

[47] J B Kellner J N Sanchirico A Hastings and P J MumbyldquoOptimizing for multiple species and multiple valuestradeoffs inherent in ecosystem-based fisheries managementrdquoConservation Letters vol 4 no 1 pp 21ndash30 2011

[48] R J Johnston T A Grigalunas J J Opaluch M Mazzottaand J Diamantedes ldquoValuing estuarine resource servicesusing economic and ecological models the peconic estuarysystem studyrdquo Coastal Management vol 30 no 1 pp 47ndash652002

[49] J P Hoehn and A Randall ldquoToo many proposals pass thebenefit cost testrdquo Fe American Economic Review vol 79no 3 pp 544ndash551 1989

[50] D Holland J Sanchirico R Johnston et al EconomicAnalysis for Ecosystem-Based Management Applications toMarine and Coastal Environments Routledge Abingdon UK2012 httpswwwtaylorfranciscombooks9781936331246

[51] J N Sanchirico D K Lew A C Haynie D M Kling andD F Layton ldquoConservation values in marine ecosystem-based managementrdquo Marine Policy vol 38 pp 523ndash5302013

[52] E Tromeur and N Loeuille ldquoBalancing yield with resilienceand conservation objectives in harvested predator-preycommunitiesrdquo Oikos vol 126 no 12 pp 1780ndash1789 2017

[53] P S Leung ldquoMultiple-criteria decision-making (MCDM)applications in fishery managementrdquo International Journal ofEnvironmental Technology and Management vol 6 no 1-2pp 96ndash110 2006

[54] M Zeleny ldquoCognitive equilibirum a new paradigm of de-cision makingrdquo Human Systems Management vol 8 no 3pp 185ndash188 1989

[55] R Lahdelma J P Salminen and Hokkanen ldquoUsing multi-criteria methods in environmental planning and manage-mentrdquo Environmental Management vol 26 no 6pp 595ndash605 2000

[56] M Franssen ldquoArrowrsquos theorem multi-criteria decisionproblems and multi-attribute preferences in engineeringdesignrdquo Research in Engineering Design vol 16 no 1-2pp 42ndash56 2005

[57] C Coello Coello G B Lamont and D A Van VeldhuizenEvolutionary Algorithms for Solving Multi-Objective ProblemsSpringer US New York USA 2007

[58] S J Mardle S Pascoe and M Tamiz ldquoAn investigation ofgenetic algorithms for the optimization of multi-objectivefisheries bioeconomic modelsrdquo International Transactions inOperational Research vol 7 no 1 pp 33ndash49 2000

Complexity 17

[59] Z Michalewicz and G Nazhiyath ldquoGenocop III a co-evo-lutionary algorithm for numerical optimization problemswith nonlinear constraintsrdquo in Proceedings of 1995 IEEEInternational Conference on Evolutionary Computation vol 2pp 647ndash651 Pittsburgh PA USA July 1995

[60] M T Rosenstein and A G Barto ldquoRobot weightlifting bydirect policy searchrdquo in Proceedings of the 17th InternationalJoint Conference on Artificial Intelligence (IJCAIrsquo01) vol 2pp 839ndash846 Seattle WA USA August 2001

[61] L Eeckhoudt C Gollier and H Schlesinger Economic andFinancial Decisions under Risk Princeton University PressPrinceton NJ USA 2005

[62] S A Sethi T A Branch and R Watson ldquoGlobal fisherydevelopment patterns are driven by profit but not trophiclevelrdquo Proceedings of the National Academy of Sciencesvol 107 no 27 pp 12163ndash12167 2010

[63] M Giuliani A Castelletti F Pianosi E Mason and P ReedldquoCurses tradeoffs and scalable management advancingevolutionary multiobjective direct policy search to improvewater reservoir operationsrdquo Journal of Water ResourcesPlanning and Management vol 142 no 2 Article ID04015050 2016

[64] P M Reed D Hadka J D Herman J R Kasprzyk andJ B Kollat ldquoEvolutionary multiobjective optimization inwater resources the past present and futurerdquo Advances inWater Resources vol 51 pp 438ndash456 2013

[65] D Hadka and P Reed ldquoBorg an auto-adaptive many-ob-jective evolutionary computing frameworkrdquo EvolutionaryComputation vol 21 no 2 pp 231ndash259 2013

[66] J B Kollat and P M Reed ldquoA computational scaling analysisof multiobjective evolutionary algorithms in long-termgroundwater monitoring applicationsrdquo Advances in WaterResources vol 30 no 3 pp 408ndash419 2007

[67] J D Herman P M Reed H B Zeff and G W CharacklisldquoHow should robustness be defined for water systems plan-ning under changerdquo Journal of Water Resources Planning andManagement vol 141 no 10 Article ID 04015012 2015

[68] M Giuliani and A Castelletti ldquoIs robustness really robustHow different definitions of robustness impact decision-making under climate changerdquo Climatic Change vol 135no 3-4 pp 409ndash424 2016

[69] C McPhail H R Maier J H Kwakkel M GiulianiA Castelletti and S Westra ldquoRobustness metrics how arethey calculated when should they be used and why do theygive different resultsrdquo Earthrsquos Future vol 6 no 2pp 169ndash191 2018

[70] M K Starr Product Design and DecisionFeory Prentice-HallSeries in Engineering Design Prentice-Hall Englewood CliffsNJ USA 1962

[71] S Bankes ldquoExploratory modeling for policy analysisrdquo Op-erations Research vol 41 no 3 pp 435ndash449 1993

[72] B P Bryant and R J Lempert ldquo(inking inside the box aparticipatory computer-assisted approach to scenario dis-coveryrdquo Technological Forecasting and Social Change vol 77no 1 pp 34ndash49 2010

[73] D G Groves and R J Lempert ldquoA new analytic method forfinding policy-relevant scenariosrdquo Global EnvironmentalChange vol 17 no 1 pp 73ndash85 2007

[74] B Worm R Hilborn J K Baum et al ldquoRebuilding globalfisheriesrdquo Science vol 325 no 5940 pp 578ndash585 2009

[75] M L Pinsky O P Jensen D Ricard and S R PalumbildquoUnexpected patterns of fisheries collapse in the worldrsquosoceansrdquo Proceedings of the National Academy of Sciencesvol 108 no 20 pp 8317ndash8322 2011

[76] J R Kasprzyk S Nataraj P M Reed and R J LempertldquoMany objective robust decision making for complex envi-ronmental systems undergoing changerdquo EnvironmentalModelling amp Software vol 42 pp 55ndash71 2013

[77] D W Watkins and D C McKinney ldquoFinding robust solu-tions to water resources problemsrdquo Journal of Water Re-sources Planning and Management vol 123 no 1 pp 49ndash581997

[78] J D Quinn P M Reed M Giuliani and A Castelletti ldquoRivalframings a framework for discovering how problem for-mulation uncertainties shape risk management trade-offs inwater resources systemsrdquo Water Resources Research vol 53no 8 pp 7208ndash7233 2017

[79] M R Kelly Y Xing and S Lenhart ldquoOptimal fish harvestingfor a population modeled by a nonlinear parabolic partialdifferential equationrdquo Natural Resource Modeling vol 29no 1 pp 36ndash70 2015

[80] A O Shelton J F Samhouri A C Stier and P S LevinldquoAssessing trade-offs to inform ecosystem-based fisheriesmanagement of forage fishrdquo Scientific Reports vol 4 no 1p 7110 2014

[81] C Jost O Arino and R Arditi ldquoAbout deterministic ex-tinction in ratio-dependent predator-prey modelsrdquo Bulletin ofMathematical Biology vol 61 no 1 pp 19ndash32 1999

[82] B Rieman D Lee J McIntyre K Overton and R (urowConsideration of Extinction Risks for Salmonids Fish HabitatRelationships Technical Bulletin 14 Boise ID US Departmentof Agriculture Forest Service Intermountain Research Sta-tion Washington DC USA 1993

[83] K Deb and H Gupta ldquoIntroducing robustness in multi-objective optimizationrdquo Evolutionary Computation vol 14no 4 pp 463ndash494 2006

[84] J J Deroba and J R Bence ldquoA review of harvest policiesunderstanding relative performance of control rulesrdquo Fish-eries Research vol 94 no 3 pp 210ndash223 2008

[85] F K Diekert D Oslash Hjermann E Naeligvdal and N C StensethldquoSpare the young fish optimal harvesting policies for north-east arctic codrdquo Environmental and Resource Economicsvol 47 no 4 pp 455ndash475 2010

[86] T Kumar Kar ldquoModelling and analysis of a harvested prey-predator system incorporating a prey refugerdquo Journal ofComputational and Applied Mathematics vol 185 no 1pp 19ndash33 2006

[87] C Folke S Carpenter B Walker et al ldquoRegime shiftsresilience and biodiversity in ecosystem managementrdquo An-nual Review of Ecology Evolution and Systematics vol 35no 1 pp 557ndash581 2004

[88] J Anderies C Folke B Walker and E Ostrom ldquoAligning keyconcepts for global change policy robustness resilience andsustainabilityrdquo Ecology and Society vol 18 no 2 p 8 2013

[89] TM Lenton ldquoEnvironmental tipping pointsrdquoAnnual Reviewof Environment and Resources vol 38 no 1 pp 1ndash29 2013

[90] R M Dorazio and F A Johnson ldquoBayesian inference anddecision theorymdasha framework for decision making in naturalresource managementrdquo Ecological Applications vol 13 no 2pp 556ndash563 2003

[91] R Singh J D Quinn P M Reed and K Keller ldquoSkill (or lackthereof) of data-model fusion techniques to provide an earlywarning signal for an approaching tipping pointrdquo PLoS Onevol 13 no 2 Article ID e0191768 2018

18 Complexity