ECOLOGICAL NETWORKS On the structural stability … · RESEARCH ARTICLE ECOLOGICAL NETWORKS On the structural stabilityof mutualistic systems Rudolf P. Rohr,1,2* Serguei Saavedra,1*

RESEARCH

INTRODUCTION Several major develop-

ments in theoretical ecology have relied

on either dynamical stability or numeri-

cal simulations but oftentimes they have

found contradictory results This is partly a

result of not rigorously checking either the

assumption that a steady state is feasiblemdash

meaning all species have constant and

positive abundancesmdashor the dependence

of results to model parameterization

Here we extend the concept of structural

stability to community ecology in order to

account for these two problems Specifi-

cally we studied the set of conditions lead-

ing to the stable coexistence of all species

within a community This shifts the ques-

tion from asking whether we can find a

feasible equilibrium point for a fixed set

of parameter values to asking how large

is the range of parameter values that are

compatible with the stable coexistence of

all species

RATIONALE We begin by disentangling

the conditions of global stability from the

conditions of feasibility of a steady state in

ecological systems To quantify the domain

of stable coexistence

we first find its center

(the structural vec-

tor of intrinsic growth

rates) Next we deter-

mine the boundaries

of such a domain by

quantifying the amount of variation from

the structural vector tolerated before one

species goes extinct Through this two-step

approach we disentangle the effects of the

size of the feasibility domain from how

close a solution is to its boundary which

is at the heart of previous contradictory

results We illustrate our method by explor-

ing how the observed architecture of mutu-

alistic networks between plants and their

pollinators or seed dispersers affects their

domain of stable coexistence

RESULTS First we determined the net-

work architecture that maximizes the

structural stability of mutualistic systems

This corresponds to networks with a maxi-

mal level of nestedness a small trade-off

between the number and intensity of inter-

actions a species has and a high level of

mutualistic strength within the constraints

of global stability Second we found that

the large majority of observed mutual-

istic networks are close to this optimum

network architecture maximizing the

range of parameters that are compatible

with species coexistence

CONCLUSION Structural stability has

played a major role in several fields such

as evolutionary developmental biology in

which it has brought the view that some

morphological structures are more com-

mon than others because they are compat-

ible with a wider range of developmental

conditions In community ecology structural

stability is the sort of framework needed to

study the consequences of global environ-

mental changemdashby definition large and di-

rectionalmdashon species coexistence Structural

stability will serve to assess both the range of

variability a given community can withstand

and why some community patterns are more

widespread than others

On the structural stability of mutualistic systemsRudolf P Rohr12 Serguei Saavedra1 Jordi Bascompte1

RESEARCH ARTICLE SUMMARY

1Integrative Ecology Group Estacioacuten Bioloacutegica de DontildeanandashConsejo Superior de Investigaciones Cientiacuteficas (EBD-CSIC) Calle Ameacuterico Vespucio sn E-41092 Sevilla Spain2Unit of Ecology and Evolution Department of Biology University of Fribourg Chemin du Museacutee 10 CH-1700 Fribourg Switzerland

Corresponding author E-mail bascompteebdcsices Cite this article as R Rohr et al Science 345 1253497 (2014) DOI 101126science1253497

Read the full article at httpdxdoiorg101126science1253497

ON OUR WEBSITE

The architecture of plant-animal mutualistic networks modulates the range of condi-

tions leading to the stable coexistence of all species The area of the dif erent domains

represents the structural stability of a model of mutualistic communities with a given network

architecture The nested networks observed in naturemdashillustrated here by the network at the

bottommdashlead to a maxi mum structural stability

rsquo to

Animalsrsquo tolerated conditions

ECOLOGICAL NETWORKS

416 25 JULY 2014 bull VOL 345 ISSUE 6195 sciencemagorg SCIENCE

Published by AAAS

RESEARCH ARTICLE

ECOLOGICAL NETWORKS

On the structural stability ofmutualistic systemsRudolf P Rohr12 Serguei Saavedra1 Jordi Bascompte1dagger

In theoretical ecology traditional studies based on dynamical stability and numericalsimulations have not found a unified answer to the effect of network architecture oncommunity persistence Here we introduce a mathematical framework based on theconcept of structural stability to explain such a disparity of results We investigated therange of conditions necessary for the stable coexistence of all species in mutualisticsystems We show that the apparently contradictory conclusions reached by previousstudies arise as a consequence of overseeing either the necessary conditions forpersistence or its dependence on model parameterizationWe show that observed networkarchitectures maximize the range of conditions for species coexistence We discuss theapplicability of structural stability to study other types of interspecific interactions

Aprevailing question in ecology [particularlysince Mayrsquos seminal work in the early 1970s(1)] is whether given an observed numberof species and their interactions there areways to organize those interactions that

lead to more persistent communities Conven-tionally studies addressing this question haveeither looked into local stability or used numer-ical simulations (2ndash4) However these studieshave not yet found a unified answer (1 5ndash12)Therefore the current challenge is to develop ageneral framework in order to provide a uni-fied assessment of the implications of the archi-tectural patterns of the networks we observe innature

Main approaches in theoretical ecology

Dynamical stability and feasibility

Studies based on the mathematical notions oflocal stability D-stability and global stabilityhave advanced our knowledge on what makesecological communities stable In particular thesestudies explore how interaction strengths needto be distributed across species so that an as-sumed feasible equilibrium point can be stable(1ndash4 13ndash17) By definition a feasible equilibriumpoint is that in which all species have a constantpositive abundance across time A negative abun-dance makes no sense biologically and an abun-dance of zero would correspond to an extinctspeciesThe dynamical stability of a feasible equilib-

rium point corresponds to the conditions underwhich the system returns to the equilibriumpoint

after a perturbation in species abundance Localstability for instance looks at whether a systemwill return to an assumed feasible equilibriumafter an infinitesimally small perturbation (1ndash3 13)D-stability in turn looks at the local stability ofany potential feasible equilibrium that the systemmay have (15ndash17) More generally global stabilitylooks at the stability of any potential feasibleequilibrium point after a perturbation of anygiven amplitude (14ndash17) A technical definition ofthese different types of dynamical stability andtheir relationship is provided in (18)In most of these stability studies however a

feasible equilibrium point is always assumedwithout rigorously studying the set of conditionsallowing its existence (5 14 15 19) Yet in anygiven system we can find examples in which wesatisfy only one both or none of the feasibilityand stability conditions (3 16 17 19) This meansthat without a proper consideration of the fea-sibility conditions any conclusion for studyingthe stable coexistence of species is based on asystem that may or may not exist (3 5 19)To illustrate this point consider the following

textbook example of a two-species competitionsystem

dtfrac14 N1etha1 minus b11N1 minus b12N2THORN

8gtltgt eth1THORN

where N1 and N2 are the abundances of species1 and 2 b11 and b22 are their intraspecific com-petition strengths b12 and b21 are their inter-specific competition strengths and a1 and a2 aretheir intrinsic growth rates An equilibrium pointof the system is a pair of abundancesN

1 and N2

that makes the right side of the ordinary differ-ential equation system equal to zeroAlthough the only condition necessary to guar-

antee the global stability of any feasible equilib-rium point in this system is that the interspecific

competition strengths are lower than the in-traspecific ones (b12b21 lt b11b22) the feasibility

conditions are given byN1 frac14 b22a1 minus b12a2

b11b22 minus b12b21gt 0 and

N2 frac14 b11a2 minus b21a1

b11b22 minus b12b21gt 0 (3 4 19) This implies that

if we set for example b11 = b22 = 1 b12 = b21 = 05a1 = 1 and a2 = 2 we fulfill the stability conditionbut not the feasibility condition whereas if we setb11 = b22 = 05 b12 = b21 = 1 and a1 = a2 = 1 we cansatisfy the feasibility condition but not the sta-bility one To have a stable and feasible equi-librium point we need to set for instance b11 =b22 = 1 b12 = b21 = 05 and a1 = a2 = 1 (a graphicalillustration is provided in Fig 1)The example above confirms the importance

of verifying both the stability and the feasibilityconditions of the equilibrium point when analyz-ing the stable coexistence of species (3ndash5 19) Ofcourse we can always fine-tune the parametervalues of intrinsic growth rates so that the sys-tem is feasible (16 17) This strategy for examplehas been used when studying the success prob-ability of an invasive species (20) However whenfixing the parameter values of intrinsic growthrates we are not anymore studying the overalleffect of interspecific interactions on the stablecoexistence of species Rather we are answeringthe question of how interspecific interactions in-crease the persistence of species for a given param-eterization of intrinsic growth rates As we willshow below this is also the core of the problem instudies that are based on arbitrary numericalsimulations

Numerical simulations

Numerical simulations have provided an alter-native and useful tool with which to explore spe-cies coexistence in large ecological systems inwhich analytical solutions are precluded (3) Withthis approach one has as a prerequisite to param-eterize the dynamical model or a least to have agood estimate of the statistical distribution fromwhich these parameters should be sampled How-ever if one chooses an arbitrary parameterizationwithout an empirical justification any studyhas a high chance of being inconclusive for realecosystems because species persistence is stronglydependent on the chosen parameterizationTo illustrate this point we simulated the dy-

namics of an ecological model (6) with threedifferent parameterizations of intrinsic growthrates (21) Additionally these simulations wereperformed over an observedmutualistic networkof interactions between flowering plants and theirpollinators located in Hickling Norfolk UK (ta-ble S1) a randomized version of this observednetwork and the observed network without mu-tualistic interactions (we assume that there is onlycompetition among plants and among animals)As shown in Fig 2 it is possible to find a set ofintrinsic growth rates so that any network thatwe analyze is completely persistent and at thesame time the alternative networks are lesspersistentThis observation has two important implica-

tions First this means that by using differentparameterizations for the same dynamicalmodel

RESEARCH

SCIENCE sciencemagorg 25 JULY 2014 bull VOL 345 ISSUE 6195 1253497-1

1Integrative Ecology Group Estacioacuten Bioloacutegica de DontildeanandashConsejo Superior de Investigaciones Cientiacuteficas (EBD-CSIC)Calle Ameacuterico Vespucio sn E-41092 Sevilla Spain 2Unit ofEcology and Evolution Department of Biology University ofFribourg Chemin du Museacutee 10 CH-1700 FribourgSwitzerlandThese authors contributed equally to this work daggerCorrespondingauthor E-mail bascompteebdcsices

and network of interactions one can observe fromall to a few of the species surviving Second thismeans that each network has a limited range ofparameter values under which all species coexistThus by studying a specific parameterization forinstance one could wrongly conclude that a ran-dom network has a greater effect on communitypersistence than that of an observed network orvice versa (10ndash12) This sensitivity to parametervalues clearly illustrates that the conclusionsthat arise from studies that use arbitrary valuesin intrinsic growth rates are not about the ef-fects of network architecture on species coex-istence but about which network architecturemaximizes species persistence for that specificparameterizationTraditional studies focusing on either local

stability or numerical simulations can lead toapparently contradictory results Therefore weneed a different conceptual framework to unifyresults and seek for appropriate generalizations

Structural stability

Structural stability has been a general mathema-tical approach with which to study the behaviorof dynamical systems A system is considered tobe structurally stable if any smooth change in themodel itself or in the value of its parameters doesnot change its dynamical behavior (such as theexistence of equilibrium points limit cycles ordeterministic chaos) (22ndash25) In the context ofecology an interesting behavior is the stablecoexistence of speciesmdashthe existence of an equi-librium point that is feasible and dynamicallystable For instance in our previous two-speciescompetition system there is a restricted area inthe parameter space of intrinsic growth ratesthat leads to a globally stable and feasible solu-tion as long as r lt 1 (Fig 3 white area) Thehigher the competition strength r the larger thesize of this restricted area (Fig 3) (19 26) There-fore a relevant question here is not only whetheror not the system is structurally stable but howlarge is the domain in the parameter space lead-ing to the stable coexistence of speciesTo address the above question we recast the

mathematical definition of structural stability tothat in which a system is more structurally sta-ble the greater the area of parameter valuesleading to both a dynamically stable and feasibleequilibrium (27ndash29) This means that a highlystructurally stable ecological system is more like-ly to be stable and feasible by handling a widerrange of conditions before the first species be-comes extinct Previous studies have used thisapproach in low-dimensional ecological systems(3 19) Yet because of its complexity almost nostudy has fully developed this rigorous analysisfor a systemwith an arbitrary number of speciesAn exception has been the use of structural sta-bility to calculate an upper bound to the numberof species that can coexist in a given community(6 30)Here we introduce this extended concept of

structural stability into community ecology inorder to study the extent to which networkarchitecturemdashstrength and organization of inter-

specific interactionsmdashmodulates the range of con-ditions compatible with the stable coexistenceof species As an empirical application of ourframework we studied the structural stabilityof mutualistic systems and applied it on a dataset of 23 quantitative mutualistic networks(table S1) We surmise that observed networkarchitectures increase the structural stabilityand in turn the likelihood of species coex-istence as a function of the possible set of con-ditions in an ecological system We discuss theapplicability of our framework to other types ofinterspecific interactions in complex ecologicalsystems

Structural stability of mutualistic systems

Mutualistic networks are formed by themutuallybeneficial interactions between flowering plants

and their pollinators or seed dispersers (31)These mutualistic networks have been shown toshare a nested architectural pattern (32) Thisnested architecturemeans that typically themu-tualistic interactions of specialist species are pro-per subsets of the interactions of more generalistspecies (32) Although it has been repeatedlyshown that this nested architecture may arisefrom a combination of life history and comple-mentarity constraints among species (32ndash35) theeffect of this nested architecture on communitypersistence continues to be a matter of strongdebate On the one hand it has been shown thata nested architecture can facilitate the mainte-nance of species coexistence (6) exhibit a flexibleresponse to environmental disturbances (7 8 36)andmaximize total abundance (12) On the otherhand it has also been suggested that this nested

1253497-2 25 JULY 2014 bull VOL 345 ISSUE 6195 sciencemagorg SCIENCE

Fig 1 Stability and feasibility of a two-species competition system For the same parameters ofcompetition strength (which grant the global stability of any feasible equilibrium) (A to C) represent thetwo isoclines of the systemTheir intersection gives the equilibrium point of the system (3 19) Scenario(A) leads to a feasible equilibrium (both species have positive abundances at equilibrium) whereasin scenarios (B) and (C) the equilibrium is not feasible (one species has a negative abundance atequilibrium) (D) represents the area of feasibility in the parameter space of intrinsic growth rates underthe condition of global stability This means that when the intrinsic growth rates of species are chosenwithin the white area the equilibrium point is globally stable and feasible In contrast when the intrinsicgrowth rates of species are chosen within the green area the equilibrium point is not feasible Points ldquoArdquoldquoBrdquo and ldquoCrdquo indicate the parameter values corresponding to (A) to (C) respectively

RESEARCH | RESEARCH ARTICLE

architecture canminimize local stability (9) havea negative effect on community persistence (10)and have a low resilience to perturbations (12)Not surprisingly the majority of these studieshave been based on either local stability or nu-merical simulations with arbitrary parameter-izations [but see (6)]

Model of mutualism

To study the structural stability and explain theapparently contradictory results found in studies

ofmutualistic networks we first need to introducean appropriate model describing the dynamicsbetween and within plants and animals We usethe same set of differential equations as in (6)We chose these dynamics because they are sim-ple enough to provide analytical insights and yetcomplex enough to incorporate key elementsmdashsuch as saturating functional responses (37 38)and interspecific competitionwithin a guild (6)mdashrecently adduced as necessary ingredients for areasonable theoretical exploration of mutualistic

interactions Specifically the dynamical modelhas the following form

dtfrac14 Pi aethPTHORNi minus sum jb

ethPTHORNij Pj thorn

sum jgethPTHORNij Aj

1thorn hsum jgethPTHORNij Aj

dtfrac14 Ai aethATHORNi minus sum jb

ethATHORNij Aj thorn

sum jgethATHORNij Pj

1thorn hsum jgethATHORNij Pj

8gtgtgtgtgtltgtgtgtgtgt

eth2THORN

where the variables Pi and Ai denote the abun-dance of plant and animal species i respectively

Fig 2 Numerical analysis of species persistence as a function of modelparameterization This figure shows the simulated dynamics of speciesabundance and the fraction of surviving species (positive abundance at theend of the simulation) using the mutualistic model of (6) Simulations areperformed by using an empirical network located in Hickling Norfolk UK(table S1) a randomized version of this network using the probabilistic model

of (32) and the network without mutualism (only competition) Each rowcorresponds to a different set of growth rate values It is always possible tochoose the intrinsic growth rates so that all species are persistent in each ofthe three scenarios and at the same time the community persistencedefined as the fraction of surviving species is lower in the alternativescenarios

The parameters of this mutualistic system cor-respond to the values describing intrinsic growthrates (ai) intra-guild competition (bij) the bene-fit received via mutualistic interactions (gij) andthe saturating constant of the beneficial effect ofmutualism (h) commonly known as the hand-ling time Because our main focus is on mutual-istic interactions we keep as simple as possiblethe competitive interactions for the sake of ana-lytical tractability In the absence of empiricalinformation about interspecific competitionwe use a mean field approximation for thecompetition parameters (6) where we setbethPTHORNii frac14 bethATHORNii frac14 1 and bethPTHORNij frac14 bethATHORNij frac14 r lt 1 (i ne j)Following (39) the mutualistic benefit can be

further disentangled by gij frac14 ethg0yijTHORN=ethkdi THORN whereyij = 1 if species i and j interact and zero other-wise ki is the number of interactions of speciesi g0 represents the level of mutualistic strengthand d corresponds to the mutualistic trade-offThe mutualistic strength is the per capita effectof a certain species on the per capita growth rateof their mutualistic partners The mutualistictrade-off modulates the extent to which a speciesthat interactswith fewother species does it stronglywhereas a species that interacts with many part-ners does it weakly This trade-off has been jus-tified on empirical grounds (40 41) The degreeto which interspecific interactions yij are orga-nized into a nested way can be quantified by thevalue of nestedness N introduced in (42)We are interested in quantifying the extent to

which network architecture (the combination ofmutualistic strength mutualistic trade-off andnestedness)modulates the set of conditions com-patible with the stable coexistence of all speciesmdashthe structural stability In the next sections weexplain how this problem can be split into twoparts First we explain how the stability condi-tions can be disentangled from the feasibilityconditions as it has already been shown for thetwo-species competition system Specifically weshow that below a critical level of mutualisticstrength (g0 lt gr0) any feasible equilibrium pointis granted to be globally stable Second we ex-plain how network architecture modulates thedomain in the parameter space of intrinsic growthrates leading to a feasible equilibrium under theconstraints of being globally stable (given by thelevel of mutualistic strength)

Stability condition

We investigated the conditions in our dynamical sys-tem that any feasible equilibrium point needs to sat-isfy to be globally stable To derive these conditionswe started by studying the linear Lotka-Volterra ap-proximation (h = 0) of the dynamical model (Eq 2)In this linear approximation the model readsdP

dtdAdt

frac14 diag

aethPTHORN

aethATHORN

minus bethPTHORN minusgethPTHORN

minusgethATHORN bethATHORN

︸frac14B

eth3THORN

where the matrix B is a two-by-two block matrixembedding all the interaction strengthsConveniently the global stability of a feasible

equilibrium point in this linear Lotka-Volterramodel has already been studied (14ndash17 43) Par-ticularly relevant to this work is that an inter-action matrix that is Lyapunovndashdiagonally stablegrants the global stability of any potential fea-sible equilibrium (14ndash18)Although it ismathematically difficult to verify

the condition for Lyapunov diagonal stability itis known that for some classes of matricesLyapunov stability and Lyapunov diagonal sta-bility are equivalent conditions (44) Symmetricmatrices and Z-matrices (matrices whose off-diagonal elements are nonpositive) belong tothose classes of equivalent matrices Our interac-tion strength matrix B is either symmetric whenthe mutualistic trade-off is zero (d = 0) or is aZ-matrix when the interspecific competitionis zero (r = 0) This means that as long as thereal parts of all eigenvalues of B are positive(18) any feasible equilibrium point is globallystable For instance in the case of r lt 1 and g0 =

0 the interaction matrix B is symmetric andLyapunovndashdiagonally stable because its eigen-values are 1 ndash r (SA ndash 1)r + 1 and (SP ndash 1)r + 1For r gt 0 and d gt 0 there are no analytical

results yet demonstrating that Lyapunov diago-nal stability is equivalent to Lyapunov stabilityHowever after intensive numerical simulationswe conjecture that the twomain consequences ofLyapunov diagonal stability hold (45) Specifi-cally we state the following conjectures Conjec-ture 1 If B is Lyapunov-stable then B isD-stableConjecture 2 If B is Lyapunov-stable then anyfeasible equilibrium is globally stableWe found that for any givenmutualistic trade-

off and interspecific competition the higher thelevel ofmutualistic strength the smaller themax-imum real part of the eigenvalues of B (45) Thismeans that there is a critical value of mutualisticstrength (g0

r) so that above this level thematrix Bis not any more Lyapunov-stable To compute gr0we need only to find the critical value of g0 atwhich the real part of one of the eigenvalues ofthe interaction-strength matrix reaches zero (45)This implies that at least below this critical value

Fig 3 Structural stability in a two-species competition systemThe figure shows how the range ofintrinsic growth rates leading to the stable coexistence of the two species (white region) changes as afunction of the competition strength (A to D) Decreasing interspecific competition increases the area offeasibility and in turn the structural stability of the system Here b11 = b22 = 1 and b12 = b21 = r Our goal isextending this analysis to realistic networks of species interactions

g0 lt g0r any feasible equilibrium is granted to be

locally and globally stable according to conjec-tures 1 and 2 respectively We can also grant theglobal stability of matrix B by the condition ofbeing positive definite which is even strongerthan Lyapunov diagonal stability (14) Howeverthis condition imposes stronger constraints onthe critical value ofmutualistic strength than doesLyapunov stability (39)Last we studied the stability conditions for the

nonlinear Lotka-Volterra system (Eq 2) Althoughthe theory has been developed for the linearLotka-Volterra system it can be extended to thenonlinear dynamical system To grant the stabil-ity of any feasible equilibrium (Pi gt 0 and Ai gt 0for all i) in the nonlinear system we need toshow that the above stability conditions hold onthe following two-by-two block matrix (14 43)

Bnl frac14bethPTHORNij minus

gethPTHORNij

1thorn hsumkgethPTHORNik Ak

minusgethATHORNij

1thorn hsumkgethATHORNik Pk

bethATHORNij

266664377775eth4THORN

Bnl differs from B only in the off-diagonal blockwith a decreased mutualistic strength This im-plies that the critical value of mutualistic strengthfor the nonlinear Lotka-Volterra system is largerthan or equal to the critical value for the linearsystem (45) Therefore the critical value g0

r derivedfrom the linear Lotka-Volterra system (from thematrix B) is already a sufficient condition to grantthe global stability of any feasible equilibrium inthe nonlinear case However this does not implythat above this critical value of mutualistic

strength a feasible equilibrium is unstable Infact when the mutualistic-interaction terms aresaturated (h gt 0) it is possible to have feasibleand locally stable equilibria for any level of mu-tualistic strength (39 45)

Feasibility condition

We highlight that for any interaction strengthmatrix B whether it is stable or not it is alwayspossible to find a set of intrinsic growth ratesso that the system is feasible (Fig 2) To findthis set of values we need only to choose afeasible equilibrium point so that the abun-dance of all species is greater than zero (Ai gt 0and Pj gt 0) and find the vector of intrinsicgrowth rates so that the right side of Eq 2 is

equal to zero aethPTHORNi frac14 sum jbethPTHORNij Pj minus

1 thorn hsum jgethPTHORNij Aj

and aethATHORNi frac14 sum jbethATHORNij Aj minus

1 thorn hsum jgethATHORNij Pj

This recon-

firms that the stability and feasibility conditionsare different and that they need to be rigorouslyverified when studying the stable coexistence ofspecies (3 16 17 19) This also highlights that therelevant question is not whether we can find afeasible equilibrium point but how large is thedomain of intrinsic growth rates leading to afeasible and stable equilibrium point We callthis domain the feasibility domainBecause the parameter space of intrinsic growth

rates is substantially large (RS where S is thetotal number of species) an exhaustive numeri-cal search of the feasibility domain is impossibleHowever we can analytically estimate the centerof this domain with what we call the structuralvector of intrinsic growth rates For example in

the two-species competition system of Fig 4Athe structural vector is the vector (in red) whichis in the center of the domain leading to fea-sibility of the equilibrium point (white region)Any vector of intrinsic growth rates collinear tothe structural vector guarantees the feasibility ofthe equilibrium pointmdashthat is guarantees spe-cies coexistence Because the structural vector isthe center of the feasibility domain then it is alsothe vector that can tolerate the strongest devia-tion before leaving the feasibility domainmdashthatis before having at least one species going extinctIn mutualistic systems we need to find one

structural vector for animals and another for plantsThese structural vectors are the set of intrinsicgrowth rates that allow the strongest perturba-tions before leaving the feasibility domain Tofind these structural vectors we had to transformthe interaction-strength matrix B to an effectivecompetition framework (45) This results in aneffective competitionmatrix for plants and a dif-ferent one for animals (6) in which thesematricesrepresent respectively the apparent competitionamongplants and among animals once taking intoaccount the indirect effect via theirmutualistic part-ners With a nonzero mutualistic trade-off (d gt 0)the effective competition matrices are nonsymmet-ric and in order to find the structural vectors wehave to use the singular decomposition approachmdasha generalization of the eigenvalue decompositionThis results in a left and a right structural vectorfor plants and for animals in the effective compe-tition framework Last we need tomove back fromthe effective competition framework in order toobtain a left and right vector for plants (aethPTHORNL andaethPTHORNR ) and animals (aethATHORNL and aethATHORNR ) in the observedmutualistic framework The full derivation isprovided in the supplementary materials (45)Once we locate the center of the feasibility

domain with the structural vectors we can ap-proximate the boundaries of this domain byquantifying the amount of variation from thestructural vectors allowed by the system beforehaving any of the species going extinctmdashthat isbefore losing the feasibility of the system Toquantify this amount we introduced propor-tional random perturbations to the structuralvectors numerically generated the new equilib-rium points (21) and measured the angle or thedeviation between the structural vectors and theperturbed vectors (a graphical example is providedin Fig 4A) The deviation from the structural vec-tors is quantified for the plants by hP (a

(P)) = [1 minuscos(y(P)L)cos(y

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

y(P)R are respectively the angles between a(P)

and a(P)L and between a(P) and a(P)R The pa-rameter a(P) is any perturbed vector of intrinsicgrowth rates of plants The deviation from thestructural vector of animals is computed similarlyAs shown in Fig 4B the greater the deviation

of the perturbed intrinsic growth rates from thestructural vectors the lower is the fraction ofsurviving species This confirms that there is arestricted domain of intrinsic growth rates cen-tered on the structural vectors compatible withthe stable coexistence of species The greater thetolerated deviation from the structural vectors

Fig 4 Deviation from the structural vector and community persistence (A) The structural vectorof intrinsic growth rates (in red) for the two-species competition system of Fig 1 The structural vector isthe vector in the center of the domain leading to the feasibility of the equilibrium point (white region) andthus can tolerate the largest deviation before any of the species go extinct The deviation between thestructural vector and any other vector (blue) is quantified by the angle between them (B) The effect ofthe deviation from the structural vector on intrinsic growth rates on community persistence defined asthe fraction of model-generated surviving species The example corresponds to an observed networklocated in North Carolina USA (table S1) with a mutualistic trade-off d = 05 and a maximum level ofmutualistic strength g0 = 02402 Blue symbols represent the community persistence and the surfacerepresents the fit of a logistic regression (R2 = 088)

within which all species coexist the greater thefeasibility domain and in turn the greater thestructural stability of the system

Network architecture andstructural stability

To investigate the extent to which networkarchitecture modulates the structural stability ofmutualistic systems we explored the combina-tion of alternative network architectures (combi-nations of nestedness mutualistic strength andmutualistic trade-off) and their correspondingfeasibility domainsTo explore these combinations for each ob-

served mutualistic network (table S1) we ob-tained 250 different model-generated nestedarchitectures by using an exhaustive resamplingmodel (46) that preserves the number of speciesand the expected number of interactions (45)Theoretically nestedness ranges from 0 to 1 (42)However if one imposes architectural constraintssuch as preserving the number of species andinteractions the effective range of nestednessthat the network can exhibit may be smaller (45)Additionally each individual model-generatednested architecture is combined with differentlevels of mutualistic trade-off d andmutualisticstrength g0 For the mutualistic trade-off we ex-plored values d isin [0 15] with steps of 005 thatallow us to explore sublinear linear and super-linear trade-offs The case d = 0 is equivalent tothe soft mean field approximation studied in (6)For each combination of network of interactionsand mutualistic trade-off there is a specific crit-ical value gr0 in the level of mutualism strengthg0 up to which any feasible equilibrium is glob-ally stable This critical value gr0 is dependent onthe mutualistic trade-off and nestedness Howeverthe mean mutualistic strength g frac14 langgijrang shows nopattern as a function of mutualistic trade-off andnestedness (45) Therefore we explored values ofg0 isin frac120 gr0 with steps of 005 and calculatedthe new generated mean mutualistic strengthsThis produced a total of 250 times 589 different net-work architectures (nestedness mutualistic trade-off and mean mutualistic strength) for eachobserved mutualistic networkWe quantified how the structural stability (fea-

sibility domain) ismodulated by these alternativenetwork architectures in the following way Firstwe computed the structural vectors of intrinsicgrowth rates that grant the existence of a feasibleequilibrium of each alternative network archi-tecture Second we introduced proportional ran-dom perturbations to the structural vectors ofintrinsic growth rates andmeasured the angle ordeviation (h(A) h(P)) between the structural vectorsand the perturbed vectors Third we simulatedspecies abundance using the mutualistic modelof (6) and the perturbed growth rates as intrinsicgrowth rate parameter values (21) These devia-tions lead to parameter domains from all to a fewspecies surviving (Fig 4)Last we quantified the extent to which net-

work architecture modulates structural stabilityby looking at the association of community per-sistence with network architecture parameters

once taking into account the effect of intrinsicgrowth rates Specifically we studied this asso-ciation using the partial fitted values from abinomial regression (47) of the fraction of sur-viving species on nestedness (N) mean mutual-istic strength (g) and mutualistic trade-off (d)while controlling for the deviations from the struc-tural vectors of intrinsic growth rates (h(A) h(P))The full description of this binomial regression andthe calculation of partial fitted values are providedin (48) These partial fitted values are the contri-bution of network architecture to the logit of theprobability of species persistence and in turnthese values are positively proportional to the sizeof the feasibility domain

Results

We analyzed each observed mutualistic networkindependently because network architecture is

constrained to the properties of each mutualisticsystem (11) For a given pollination system lo-cated in the KwaZulu-Natal region of SouthAfrica the extent to which its network archi-tecturemodulates structural stability is shown inFig 5 Specifically the partial fitted values areplotted as a function of network architecture Asshown in Fig 5A not all architectural combina-tions have the same structural stability In par-ticular the architectures that maximize structuralstability (reddishdarker regions) correspond tothe following properties (i) a maximal level ofnestedness (ii) a small (sublinear) mutualistictrade-off and (iii) a high level of mutualisticstrength within the constraint of any feasiblesolution being globally stable (49)A similar pattern is present in all 23 observed

mutualistic networks (45) For instance usingthree different levels of interspecific competition

Fig 5 Structural stability in complexmutualistic systems For an observedmutualistic systemwith 9plants 56 animals and 103 mutualistic interactions located in the grassland asclepiads in South Africa(table S1) (58) (A) corresponds to the effectmdashcolored by partial fitted residualsmdashof the combination ofdifferent architectural values (nestedness mean mutualistic strength and mutualistic trade-off) on thedomain of structural stability The reddishdarker the color the larger the parameter space that iscompatible with the stable coexistence of all species and in turn the larger the domain of structuralstability (B) (C) and (D) correspond to different slices of (A) Slice (B) corresponds to ameanmutualisticstrength of 021 slice (C) corresponds to the observed mutualistic trade-off and slice (D) corresponds tothe observed nestedness Solid lines correspond to the observed values of nestedness and mutualistictrade-offs

(r = 02 04 06) we always find that structuralstability is positively associated with nestednessandmutualistic strength (45) Similarly structur-al stability is always associated with the mutual-istic trade-off by a quadratic function leadingquite often to an optimal value for maximizingstructural stability (45) These findings revealthat under the given characterization of inter-specific competition there is a general pattern of

network architecture that increases the struc-tural stability of mutualistic systemsYet one question remains to be answered Is

the network architecture that we observe innature close to the maximum feasibility domainof parameter space under which species coexistTo answer this question we compared the ob-served network architecture with theoreticalpredictions To extract the observed network

architecture we computed the observed nested-ness from the observed binary interactionmatrices(table S1) following (42) The observed mutual-istic trade-off d is estimated from the observednumber of visits of pollinators or fruits con-sumed by seed-dispersers to flowering plants(41 50 51) The full details on how to computethe observed trade-off is provided in (52) Be-cause there is no empirical data on the relation-ship between competition andmutualistic strengththat could allow us to extract the observed mu-tualistic strength g0 our results on nestednessand mutualistic trade-off are calculated acrossdifferent levels of mean mutualistic strengthAs shown in Fig 5 B to D the observed

network (blue solid lines) of the mutualistic sys-tem located in the grassland asclepiads of SouthAfrica actually appears to have an architectureclose to the one that maximizes the feasibilitydomain under which species coexist (reddishdarker region) To formally quantify the degreeto which each observed network architecture ismaximizing the set of conditions under whichspecies coexist we compared the net effect of theobserved network architecture on structural sta-bility against the maximum possible net effectThe maximum net effect is calculated in threestepsFirst as outlined in the previous section we

computed the partial fitted values of the effectof alternative network architectures on speciespersistence (48) Second we extracted the rangeof nestedness allowed by the network given thenumber of species and interactions in the sys-tem (45) Third we computed themaximumneteffect of network architecture on structuralstability by finding the difference between themaximum and minimum partial fitted valueswithin the allowed range of nestedness andmutualistic trade-off between d isin [0 15] Allthe observed mutualistic trade-offs have valuesbetween d isin [0 15] Last the net effect of theobserved network architecture on structuralstability corresponds to the difference betweenthe partial fitted values for the observed archi-tecture and the minimum partial fitted valuesextracted in the third step described aboveLooking across different levels of mean mutu-

alistic strength in themajority of cases (18 out of23 P = 0004 binomial test) the observed net-work architectures induce more than half thevalue of the maximum net effect on structuralstability (Fig 6 red solid line) These findingsreveal that observed network architectures tendto maximize the range of parameter spacemdashstructural stabilitymdashfor species coexistence

Structural stability of systems withother interaction types

In this section we explain how our structuralstability framework can be applied to other typesof interspecific interactions in complex ecologi-cal systems We first explain how structural sta-bility can be applied to competitive interactionsWe proceed by discussing how this competitiveapproach can be used to study trophic inter-actions in food webs

Fig 6 Net effect of network architecture on structural stability For each of the 23 observednetworks (table S1) we show how close the observed feasibility domain (partial fitted residuals) is as afunction of the network architecture to the theoretical maximal feasibility domain The network ar-chitecture is given by the combination of nestedness and mutualistic trade-off (x axis) across differentvalues of mean mutualistic strength (y axis) The solid red and dashed black lines correspond to themaximum net effect and observed net effect respectively In 18 out of 23 networks (indicated byasterisks) the observed architecture exhibits more than half the value of the maximum net effect (grayregions)The net effect of each network architecture is system-dependent and cannot be used to compareacross networks

For a competition systemwith an arbitrary num-ber of species we can assume a standard set of dy-namical equations givenby dNi

dt frac14 Niethai minus sumjbijN jTHORNwhere ai gt 0 is the intrinsic growth rate bij gt 0is the competition interaction strength and Ni

is the abundance of species i Recall that theLyapunov diagonal stability of the interactionmatrix bwould imply the global stability of anyfeasible equilibrium point However in nonsym-metric competition matrices Lyapunov stabilitydoes not always imply Lyapunov diagonal sta-bility (53) This establishes that we should workwith a restricted class of competition matricessuch as the ones derived from the niche space of(54) Indeed it has been demonstrated that thisclass of competition matrices are Lyapunov diag-onally stable and that this stability is inde-pendent of the number of species (55) For acompetition systemwith a symmetric interaction-strength matrix the structural vector is equal toits leading eigenvector For other appropriateclasses of matrices we can compute the struc-tural vectors in the same way as we did with theeffective competition matrices of our mutualisticmodel and numerically simulate the feasibilitydomain of the competition system In generalfollowing this approach we can verify that thelower the average interspecific competition thehigher is the feasibility domain and in turnthe higher is the structural stability of the com-petition systemIn the case of predator-prey interactions in

food webs so far there is no analytical workdemonstrating the conditions for a Lyapunovndashdiagonally stable system and how this is linkedto its Lyapunov stability Moreover the compu-tation of the structural vector of an antagonisticsystem is not a straightforward task Howeverwe may have a first insight about how the net-work architecture of antagonistic systems mod-ulates their structural stability by transforming atwo-trophicndashlevel food web into a competitionsystem among predators Using this transforma-tion we are able to verify that the higher thecompartmentalization of a food web then thehigher is its structural stability There is no uni-versal rule to study the structural stability ofcomplex ecological systems Each type of inter-action poses their own challenges as a functionof their specific population dynamics

Discussion

We have investigated the extent to which differ-ent network architectures of mutualistic systemscan provide a wider range of conditions underwhich species coexist This research question iscompletely different from the question of whichnetwork architectures are aligned to a fixed set ofconditions Previous numerical analyses based onarbitrary parameterizations were indirectly ask-ing the latter and previous studies based on localstability were not rigorously verifying the actualcoexistence of species Of course if there is agood empirical or scientific reason to use a spe-cific parameterization then we should take ad-vantage of this However because the set ofconditions present in a community can be constant-

ly changing because of stochasticity adaptivemechanisms or global environmental changewe believe that understanding which networkarchitectures can increase the structural stabilityof a community becomes a relevant questionIndeed this is a question much more alignedwith the challenge of assessing the consequencesof global environmental changemdashby definitiondirectional and largemdashthan with the alternativeframework of linear stability which focuses onthe responses of a steady state to infinitesimallysmall perturbationsWe advocate structural stability as an integra-

tive approach to provide a general assessment ofthe implications of network architecture acrossecological systems Our findings show that manyof the observed mutualistic network architec-tures tend to maximize the domain of parameterspace under which species coexist This meansthat inmutualistic systems having both a nestednetwork architecture and a small mutualistictrade-off is one of the most favorable structuresfor community persistence Our predictions couldbe tested experimentally by exploring whethercommunities with an observed network archi-tecture that maximizes structural stability standhigher values of perturbation Similarly our re-sults open up new questions such as what thereported associations between network architec-ture and structural stability tell us about theevolutionary processes and pressures occurringin ecological systemsAlthough the framework of structural stability

has not been as dominant in theoretical ecologyas has the concept of local stability it has a longtradition in other fields of research (29) Forexample structural stability has been key in evo-lutionary developmental biology to articulate theview of evolution as the modification of a con-served developmental program (27 28) Thussome morphological structures are much morecommon than others because they are compatiblewith a wider range of developmental conditionsThis provided a more mechanistic understand-ing of the generation of form and shape throughevolution (56) than that provided by a historicalfunctionalist view We believe ecology can alsobenefit from this structuralist view The analo-gous question here would assess whether theinvariance of network architecture across diverseenvironmental and biotic conditions is due tothe fact that such a network structure is the oneincreasing the likelihood of species coexistencein an ever-changing world

REFERENCES AND NOTES

1 R M May Will a large complex system be stable Nature 238413ndash414 (1972) doi 101038238413a0 pmid 4559589

2 A R Ives S R Carpenter Stability and diversity ofecosystems Science 317 58ndash62 (2007) doi 101126science1133258 pmid 17615333

3 T J Case An Illustrated Guide to Theoretical Ecology (OxfordUniv Press Oxford UK 2000)

4 S L Pimm Food Webs (Univ Chicago Press Chicago 2002)5 A Roberts The stability of a feasible random ecosystem

Nature 251 607ndash608 (1974) doi 101038251607a06 U Bastolla et al The architecture of mutualistic networks

minimizes competition and increases biodiversity Nature 4581018ndash1020 (2009) doi 101038nature07950 pmid 19396144

7 T Okuyama J N Holland Network structural propertiesmediate the stability of mutualistic communities Ecol Lett 11208ndash216 (2008) doi 101111j1461-0248200701137xpmid 18070101

8 E Theacutebault C Fontaine Stability of ecological communitiesand the architecture of mutualistic and trophic networksScience 329 853ndash856 (2010) doi 101126science1188321pmid 20705861

9 S Allesina S Tang Stability criteria for complex ecosystemsNature 483 205ndash208 (2012) doi 101038nature10832pmid 22343894

10 A James J W Pitchford M J Plank Disentanglingnestedness from models of ecological complexity Nature 487227ndash230 (2012) doi 101038nature11214 pmid 22722863

11 S Saavedra D B Stouffer ldquoDisentangling nestednessrdquodisentangled Nature 500 E1ndashE2 (2013) doi 101038nature12380 pmid 23969464

12 S Suweis F Simini J R Banavar A Maritan Emergence ofstructural and dynamical properties of ecological mutualisticnetworks Nature 500 449ndash452 (2013) doi 101038nature12438 pmid 23969462

13 A M Neutel J A Heesterbeek P C De Ruiter Stability in realfood webs Weak links in long loops Science 296 1120ndash1123(2002) doi 101126science1068326 pmid 12004131

14 B S Goh Global stability in many-species systems Am Nat111 135ndash143 (1977) doi 101086283144

15 D O Logofet Stronger-than-lyapunov notions of matrixstability or how flowers help solve problems in mathematicalecology Linear Algebra Appl 398 75ndash100 (2005)doi 101016jlaa200304001

16 D O Logofet Matrices and Graphs Stability Problems inMathematical Ecology (CRC Press Boca Raton FL 1992)

17 Y M Svirezhev D O Logofet Stability of BiologicalCommunities (Mir Publishers Moscow Russia 1982)

18 Any matrix B is called Lyapunov-stable if the real parts ofall its eigenvalues are positive meaning that a feasibleequilibrium at which all species have the same abundance isat least locally stable A matrix B is called D-stable if DB is aLyapunov-stable matrix for any strictly positive diagonalmatrix D D-stability is a stronger condition than Lyapunovstability in the sense that it grants the local stability of anyfeasible equilibrium (15) In addition a matrix B is calledLyapunov diagonally stable if there exists a strictly positivediagonal matrix D so that DB + BtD is a Lyapunov-stablematrix This notion of stability is even stronger than D-stabilityin the sense that it grants not only the local stability of anyfeasible equilibrium point but also its global stability (14)A Lyapunovndashdiagonally stable matrix has all of its principalminors positive Also its stable equilibrium (which may be onlypartially feasible some species may have an abundance ofzero) is specific (57) This means that there is no alternativestable state for a given parameterization of intrinsic growthrates (55) We have chosen the convention of having aminus sign in front of the interaction-strength matrix B Thisimplies that B has to be positive Lyapunov-stable (the realparts of all eigenvalues have to be strictly positive) so that theequilibrium point of species abundance is locally stable Thissame convention applies for D-stability and Lyapunovndashdiagonalstability

19 J H Vandermeer Interspecific competition A new approachto the classical theory Science 188 253ndash255 (1975)doi 101126science1118725 pmid 1118725

20 T J Case Invasion resistance arises in strongly interactingspecies-rich model competition communities Proc Natl AcadSci USA 87 9610ndash9614 (1990) doi 101073pnas87249610 pmid 11607132

21 All simulations were performed by integrating the system ofordinary differential equations by using the function ode45 ofMatlab Species are considered to have gone extinct when theirabundance is lower than 100 times the machine precisionAlthough we present the results of our simulations with a fixedvalue of r = 02 and h = 01 our results are robust to thischoice as long as r lt 1 (45) In our dynamical system the unitsof the parameters are abundance for N 1time for a and hand 1(time middot biomass) for b and g Abundance and time canbe any unit (such as biomass and years) as long as they areconstant for all species

22 V I Arnold Geometrical Methods in the Theory of OrdinaryDifferential Equations (Springer New York ed 2 1988)

23 R V Soleacute J Valls On structural stability and chaos inbiological systems J Theor Biol 155 87ndash102 (1992)doi 101016S0022-5193(05)80550-8

24 A N Kolmogorov Sulla teoria di volterra della lotta perlrsquoesistenza Giornale Istituto Ital Attuari 7 74ndash80 (1936)

25 Y A Kuznetsov Elements of Applied Bifurcation Theory(Springer New York ed 3 2004)

26 J H Vandermeer The community matrix and the number ofspecies in a community Am Nat 104 73ndash83 (1970)doi 101086282641

27 P Alberch E A Gale A developmental analysis of anevolutionary trend Digital reduction in amphibians Evolution39 8ndash23 (1985) doi 1023072408513

28 P Alberch The logic of monsters Evidence for internalconstraints in development and evolution Geobios 22 21ndash57(1989) doi 101016S0016-6995(89)80006-3

29 R Thom Structural Stability and Morphogenesis(Addison-Wesley Boston 1994)

30 U Bastolla M Laumlssig S C Manrubia A Valleriani Biodiversityin model ecosystems I Coexistence conditions for competingspecies J Theor Biol 235 521ndash530 (2005) doi 101016jjtbi200502005 pmid 15935170

31 J Bascompte Disentangling the web of life Science 325416ndash419 (2009) doi 101126science1170749pmid 19628856

32 J Bascompte P Jordano C J Meliaacuten J M Olesen Thenested assembly of plant-animal mutualistic networksProc Natl Acad Sci USA 100 9383ndash9387 (2003)doi 101073pnas1633576100 pmid 12881488

33 J N Thompson The Geographic Mosaic of Coevolution(Univ Chicago Press Chicago 2005)

34 E L Rezende J E Lavabre P R Guimaratildees P JordanoJ Bascompte Non-random coextinctions in phylogeneticallystructured mutualistic networks Nature 448 925ndash928 (2007)doi 101038nature05956 pmid 17713534

35 S Saavedra F Reed-Tsochas B Uzzi A simple model ofbipartite cooperation for ecological and organizationalnetworks Nature 457 463ndash466 (2009) doi 101038nature07532 pmid 19052545

36 L A Burkle J C Marlin T M Knight Plant-pollinatorinteractions over 120 years Loss of species co-occurrenceand function Science 339 1611ndash1615 (2013) doi 101126science1232728 pmid 23449999

37 D H Wright A simple stable model of mutualismincorporating handling time Am Nat 134 664ndash667 (1989)doi 101086285003

38 J N Holland T Okuyama D L DeAngelis Comment onldquoAsymmetric coevolutionary networks facilitate biodiversitymaintenancerdquo Science 313 1887b (2006) doi 101126science1129547 pmid 17008511

39 S Saavedra R P Rohr V Dakos J Bascompte Estimatingthe tolerance of species to the effects of global environmentalchange Nat Commun 4 2350 (2013) doi 101038ncomms3350 pmid 23945469

40 R Margalef Perspectives in Ecological Theory (Univ ChicagoPress Chicago 1968)

41 D P Vaacutezquez et al Species abundance and asymmetricinteraction strength in ecological networks Oikos 1161120ndash1127 (2007) doi 101111j0030-1299200715828x

42 M Almeida-Neto P Guimaratildees P R Guimaratildees Jr R D LoyolaW Urlich A consistent metric for nestedness analysis in ecologicalsystems Reconciling concept and measurement Oikos 1171227ndash1239 (2008) doi 101111j0030-1299200816644x

43 B S Goh Stability in models of mutualism Am Nat 113261ndash275 (1979) doi 101086283384

44 Y Takeuchi N Adachi H Tokumaru Global stability ofecosystems of the generalized volterra type Math Biosci 42119ndash136 (1978) doi 1010160025-5564(78)90010-X

45 Materials and methods are available as supplementarymaterials on Science Online

46 R P Rohr R E Naisbit C Mazza L-F Bersier Matching-centrality decomposition and the forecasting of new links innetworks arXiv13104633 (2013)

47 P McCullagh J A Nelder Generalized Linear Models(Chapman and Hall London ed 2 1989) chap 4

48 For a given network we look at the association ofthe fraction of surviving species with the deviationfrom the structural vector of plants h(P) and animalsh(A) nestedness N mutualistic trade-off d and meanlevel of mutualistic strength g frac14 lang ranggij using thefollowing binomial generalized linear model (47)logitethprobability of survivingTHORN e logethhATHORN thorn logethhPTHORN thorn g thorng2 thorn g ˙ Nthorn g ˙ N2 thorn g ˙ dthorn gd2 Obviously at a level ofmutualism of zero (g = 0) nestedness and mutualistictrade-off cannot influence the probability of a species tosurvive We have to include an interaction between the meanlevel of mutualism and the nestedness and mututalistictrade-off We have also included a quadratic term in order totake into account potential nonlinear effects of nestednessmutualistic trade-off and mean level of mutualistic strengthThe effect of network architecture is confirmed by thesignificant likelihood ratio between the full model and anull model without such a network effect (P lt 0001) forall the observed empirical networks The effect ofnetwork architecture on structural stability can bequantified by the partial fitted values defined as followspartial fitted values frac14 b1gthorn b2g

2 thorn b3 g ˙ Nthorn b4g ˙ N2thornb5g ˙ dthorn b6g ˙ d2 where b1˙˙˙b6 are the fitted parameterscorresponding to the terms g to g ˙ d2 respectively

49 J Bascompte P Jordano J M Olesen Asymmetric coevolutionarynetworks facilitate biodiversity maintenance Science 312 431ndash433(2006) doi 101126science1123412 pmid 16627742

50 P Turchin I Hanski An empirically based model for latitudinalgradient in vole population dynamics Am Nat 149 842ndash874(1997) doi 101086286027 pmid 18811252

51 P Jordano J Bascompte J M Olesen Invariant propertiesin coevolutionary networks of plant-animal interactions EcolLett 6 69ndash81 (2003) doi 101046j1461-0248200300403x

52 To estimate the mutualistic trade-off d from an empiricalpoint of view we proceed as follows First there is availabledata on the frequency of interactions Thus qij is the observednumber of visits of animal species j on plant species i Thisquantity has been proven to be the best surrogate of percapita effects of one species on another (gij

P and gijA) (41)

Second the networks provide information on the number ofspecies one species interacts with (its degree or generalizationlevel ki

A and kiP) Following (41 50 51) the generalization

level of a species has been found to be proportional to itsabundance at equilibrium Thus the division of the totalnumber of visits by the product of the degree of plants andanimals can be assumed to be proportional to the interaction

strengths Mathematically we obtain two equations one forthe effect of the animals on the plants and vice versa(qij)(ki

PkjA) ordm gij

P and (qij)(kiAkj

P) ordm gijA Introducing the

explicit dependence between interaction strength andtrade-ofmdashfor example gij

P = g0(kiP)d and (qij)(ki

PkjA) ordm

(g0)[(kiP)d]mdashwe obtain (qij)(ki

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

A)d] In order to estimate the value ofd we can just take the logarithm on both sides of the previousequations for the data excluding the zeroes Then d is simplygiven by the slope of the following linear regressions

logqijkPi k

frac14 aP minus dlog kPi

and log

qijkAi k

frac14 aA minus dlog kAi

where aP and aA are the intercepts for plants and animalsrespectively These two regressions are performedsimultaneously by lumping together the data set The interceptfor the effect of animals on plants (aP) may not be the same asthe intercept for the effect of the plants on the animals (aA)

53 The competition matrix given by b frac143 31 15 5 64 1 3

is Lyapunov-stable but not D-stableBecause Lyapunov diagonal stability implies D-stability thismatrix is also not Lyapunovndashdiagonally stable

54 R MacArthur Species packing and competitive equilibria formany species Theor Pop Biol 1 1ndash11 (1970)

55 T J Case R Casten Global stability and multiple domains ofattraction in ecological systems Am Nat 113 705ndash714 (1979)doi 101086283427

56 D W Thompson On Growth and Form (Cambridge Univ PressCambridge UK 1992)

57 Y Takeuchi N Adachi The existence of globally stableequilibria of ecosystem of the generalised Volterra typeJ Math Biol 10 401ndash415 (1980) doi 101007BF00276098

58 J Ollerton S D Johnson L Cranmer S Kellie The pollinationecology of an assemblage of grassland asclepiads in SouthAfrica Ann Bot (Lond) 92 807ndash834 (2003) doi 101093aobmcg206 pmid 14612378

ACKNOWLEDGMENTS

We thank U Bastolla L-F Bersier V Dakos A Ferrera M A FortunaL J Gilarranz S Keacutefi J Lever B Luque A M Neutel A Pascual-GarciacuteaD B Stouffer and J Tylianakis for insightful discussions We thankS Baigent for pointing out the counter-example in (53) Funding wasprovided by the European Research Council through an AdvancedGrant (JB) and FP7-REGPOT-2010-1 program under project 264125EcoGenes (RPR) The data are publicly available at wwwweb-of-lifees

SUPPLEMENTARY MATERIALS

wwwsciencemagorgcontent34561951253497supplDC1Materials and MethodsFigs S1 to S14Table S1References (59ndash61)17 March 2014 accepted 3 June 2014101126science1253497

Science-2014-Rohr-pdf

RESEARCH ARTICLE

ECOLOGICAL NETWORKS

On the structural stability ofmutualistic systemsRudolf P Rohr12 Serguei Saavedra1 Jordi Bascompte1dagger

In theoretical ecology traditional studies based on dynamical stability and numericalsimulations have not found a unified answer to the effect of network architecture oncommunity persistence Here we introduce a mathematical framework based on theconcept of structural stability to explain such a disparity of results We investigated therange of conditions necessary for the stable coexistence of all species in mutualisticsystems We show that the apparently contradictory conclusions reached by previousstudies arise as a consequence of overseeing either the necessary conditions forpersistence or its dependence on model parameterizationWe show that observed networkarchitectures maximize the range of conditions for species coexistence We discuss theapplicability of structural stability to study other types of interspecific interactions

Aprevailing question in ecology [particularlysince Mayrsquos seminal work in the early 1970s(1)] is whether given an observed numberof species and their interactions there areways to organize those interactions that

lead to more persistent communities Conven-tionally studies addressing this question haveeither looked into local stability or used numer-ical simulations (2ndash4) However these studieshave not yet found a unified answer (1 5ndash12)Therefore the current challenge is to develop ageneral framework in order to provide a uni-fied assessment of the implications of the archi-tectural patterns of the networks we observe innature

Main approaches in theoretical ecology

Dynamical stability and feasibility

Studies based on the mathematical notions oflocal stability D-stability and global stabilityhave advanced our knowledge on what makesecological communities stable In particular thesestudies explore how interaction strengths needto be distributed across species so that an as-sumed feasible equilibrium point can be stable(1ndash4 13ndash17) By definition a feasible equilibriumpoint is that in which all species have a constantpositive abundance across time A negative abun-dance makes no sense biologically and an abun-dance of zero would correspond to an extinctspeciesThe dynamical stability of a feasible equilib-

rium point corresponds to the conditions underwhich the system returns to the equilibriumpoint

after a perturbation in species abundance Localstability for instance looks at whether a systemwill return to an assumed feasible equilibriumafter an infinitesimally small perturbation (1ndash3 13)D-stability in turn looks at the local stability ofany potential feasible equilibrium that the systemmay have (15ndash17) More generally global stabilitylooks at the stability of any potential feasibleequilibrium point after a perturbation of anygiven amplitude (14ndash17) A technical definition ofthese different types of dynamical stability andtheir relationship is provided in (18)In most of these stability studies however a

feasible equilibrium point is always assumedwithout rigorously studying the set of conditionsallowing its existence (5 14 15 19) Yet in anygiven system we can find examples in which wesatisfy only one both or none of the feasibilityand stability conditions (3 16 17 19) This meansthat without a proper consideration of the fea-sibility conditions any conclusion for studyingthe stable coexistence of species is based on asystem that may or may not exist (3 5 19)To illustrate this point consider the following

textbook example of a two-species competitionsystem

8gtltgt eth1THORN

where N1 and N2 are the abundances of species1 and 2 b11 and b22 are their intraspecific com-petition strengths b12 and b21 are their inter-specific competition strengths and a1 and a2 aretheir intrinsic growth rates An equilibrium pointof the system is a pair of abundancesN

1 and N2

that makes the right side of the ordinary differ-ential equation system equal to zeroAlthough the only condition necessary to guar-

antee the global stability of any feasible equilib-rium point in this system is that the interspecific

competition strengths are lower than the in-traspecific ones (b12b21 lt b11b22) the feasibility

conditions are given byN1 frac14 b22a1 minus b12a2

b11b22 minus b12b21gt 0 and

N2 frac14 b11a2 minus b21a1

b11b22 minus b12b21gt 0 (3 4 19) This implies that

if we set for example b11 = b22 = 1 b12 = b21 = 05a1 = 1 and a2 = 2 we fulfill the stability conditionbut not the feasibility condition whereas if we setb11 = b22 = 05 b12 = b21 = 1 and a1 = a2 = 1 we cansatisfy the feasibility condition but not the sta-bility one To have a stable and feasible equi-librium point we need to set for instance b11 =b22 = 1 b12 = b21 = 05 and a1 = a2 = 1 (a graphicalillustration is provided in Fig 1)The example above confirms the importance

of verifying both the stability and the feasibilityconditions of the equilibrium point when analyz-ing the stable coexistence of species (3ndash5 19) Ofcourse we can always fine-tune the parametervalues of intrinsic growth rates so that the sys-tem is feasible (16 17) This strategy for examplehas been used when studying the success prob-ability of an invasive species (20) However whenfixing the parameter values of intrinsic growthrates we are not anymore studying the overalleffect of interspecific interactions on the stablecoexistence of species Rather we are answeringthe question of how interspecific interactions in-crease the persistence of species for a given param-eterization of intrinsic growth rates As we willshow below this is also the core of the problem instudies that are based on arbitrary numericalsimulations

Numerical simulations

Numerical simulations have provided an alter-native and useful tool with which to explore spe-cies coexistence in large ecological systems inwhich analytical solutions are precluded (3) Withthis approach one has as a prerequisite to param-eterize the dynamical model or a least to have agood estimate of the statistical distribution fromwhich these parameters should be sampled How-ever if one chooses an arbitrary parameterizationwithout an empirical justification any studyhas a high chance of being inconclusive for realecosystems because species persistence is stronglydependent on the chosen parameterizationTo illustrate this point we simulated the dy-

namics of an ecological model (6) with threedifferent parameterizations of intrinsic growthrates (21) Additionally these simulations wereperformed over an observedmutualistic networkof interactions between flowering plants and theirpollinators located in Hickling Norfolk UK (ta-ble S1) a randomized version of this observednetwork and the observed network without mu-tualistic interactions (we assume that there is onlycompetition among plants and among animals)As shown in Fig 2 it is possible to find a set ofintrinsic growth rates so that any network thatwe analyze is completely persistent and at thesame time the alternative networks are lesspersistentThis observation has two important implica-

tions First this means that by using differentparameterizations for the same dynamicalmodel

RESEARCH

1Integrative Ecology Group Estacioacuten Bioloacutegica de DontildeanandashConsejo Superior de Investigaciones Cientiacuteficas (EBD-CSIC)Calle Ameacuterico Vespucio sn E-41092 Sevilla Spain 2Unit ofEcology and Evolution Department of Biology University ofFribourg Chemin du Museacutee 10 CH-1700 FribourgSwitzerlandThese authors contributed equally to this work daggerCorrespondingauthor E-mail bascompteebdcsices

Model of mutualism

eth2THORN

Stability condition

dtdAdt

frac14 diag

aethPTHORN

aethATHORN

︸frac14B

eth3THORN

gethPTHORNij

minusgethATHORNij

bethATHORNij

266664377775eth4THORN

This recon-

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Model of mutualism

eth2THORN

Stability condition

dtdAdt

frac14 diag

aethPTHORN

aethATHORN

︸frac14B

eth3THORN

gethPTHORNij

minusgethATHORNij

bethATHORNij

266664377775eth4THORN

This recon-

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Model of mutualism

eth2THORN

Stability condition

dtdAdt

frac14 diag

aethPTHORN

aethATHORN

︸frac14B

eth3THORN

gethPTHORNij

minusgethATHORNij

bethATHORNij

266664377775eth4THORN

This recon-

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Stability condition

dtdAdt

frac14 diag

aethPTHORN

aethATHORN

︸frac14B

eth3THORN

gethPTHORNij

minusgethATHORNij

bethATHORNij

266664377775eth4THORN

This recon-

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

gethPTHORNij

minusgethATHORNij

bethATHORNij

266664377775eth4THORN

This recon-

(P)R)][cos(y

(P)L)cos(y

(P)R)] where y

(P)L and

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Results

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Discussion

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

P and gijA) (41)

PkjA) ordm gij

P and (qij)(kiAkj

PkjA) ordm

PkjA) ordm (g0)[(ki

P)d] and(qij)(ki

AkjP) ordm (g0)[(ki

logqijkPi k

and log

qijkAi k

ACKNOWLEDGMENTS

Documents

ECOLOGICAL NETWORKS On the structural stability … · RESEARCH ARTICLE ECOLOGICAL NETWORKS On the structural stabilityof mutualistic systems Rudolf P. Rohr,1,2* Serguei Saavedra,1*