Resolving Discrepancies between Deterministic Population Models and Individual‐Based Simulations

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Resolving Discrepancies between Deterministic Population Models and Individual‐BasedSimulationsAuthor(s): William G. WilsonSource: The American Naturalist, Vol. 151, No. 2 (February 1998), pp. 116-134Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/286106 .

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vol. 151, no. 2 the american naturalist february 1998

Resolving Discrepancies between Deterministic PopulationModels and Individual-Based Simulations

William G. Wilson*

Department of Zoology and Center for Nonlinear and Complex ics. Even the simplest model systems demonstrate impor-Systems, Duke University, Durham, North Carolina 27708-0325 tant consequences of the discreteness of interacting parti-

cles (Doi 1976; Kang and Redner 1985; Balding andSubmitted February 26, 1997; Accepted August 27, 1997Green 1989). For example, simulation models of discrete,interacting organisms often produce dynamic results thatcontrast with predictions of complementary mean-fieldpartial differential equations (Chesson 1978, 1981; Nisbet

abstract: This work ties together two distinct modeling frame-and Gurney 1982; de Roos et al. 1991; Wilson et al. 1993;works for population dynamics: an individual-based simulationDurrett and Levin 1994; Bascompte and Sole 1995; Ha-and a set of coupled integrodifferential equations involving popu-

lation densities. The simulation model represents an idealized rada et al. 1995). One important feature of systems of in-predator-prey system formulated at the scale of discrete individu- teracting individuals is stochasticity due to the randomals, explicitly incorporating their mutual interactions, whereas the choices individuals face. Previous examinations of preda-population-level framework is a generalized version of reaction- tor-prey models demonstrated enhanced predator persis-diffusion models that incorporate population densities coupled to

tence due to demographic or environmental stochasticityone another by interaction rates. Here I use various combinations(Leslie and Gower 1960; May 1973; Poole 1974; Cominsof long-range dispersal for both the offspring and adult stages ofand Hassell 1987).both prey and predator species, providing a broad range of spatial

and temporal dynamics, to compare and contrast the two model Acknowledging that individuals are distributed overframeworks. Taking the individual-based modeling results as space further complicates predictions regarding stabilitygiven, two examinations of the reaction-dispersal model are made: (Chesson 1978; Crowley 1981; Hastings 1992; Ruxtonlinear stability analysis of the deterministic equations and direct and Rohani 1996). For example, Roff (1974a, 1974b)numerical solution of the model equations. I also modify the nu-

demonstrates the stabilizing effect that space has on sto-merical solution in two ways to account for the stochastic nature

chastic models that tend to extinction without space, andof individual-based processes, which include independent, localChesson and Warner (1981) show that variability in birthperturbations in population density and a minimum population

density within integration cells, below which the population is set rates, a potential result of spatial complexity, is responsi-to zero. These modifications introduce new parameters into the ble for coexistence in lottery competitive systems. Like-population-level model, which I adjust to reproduce the individ- wise, unexpected spatial structuring occurs in simulationual-based model results. The individual-based model is then modi- systems whose deterministic mean-field formulations an-fied to minimize the effects of stochasticity, producing a match of

ticipate no such phenomena (McCauley et al. 1993).the predictions from the numerical integration of the population-Ecological theory now seems at the brink of con-level model without stochasticity.

structing ever more complicated individual-based modelsKeywords: predator-prey theory, individual-based models, stochas- that are divorced from the analytical population-levelticity, spatial dynamics, long-range dispersal.

models that seek generality across ecological systems.These simulations produce detailed ‘‘exact’’ results forsystems having complex and stochastic interactions

Thinking of interacting populations as collections of in-(Huston et al. 1988), but alone rarely provide the general

dividuals, rather than continuous densities, alters specificinsight that solutions to mathematical models do.

and general theoretical predictions of population dynam-Resolving the discrepancies between conceptually con-

gruent model systems—one formulated at the individualscale and the other at the population scale—clarifies im-* E-mail: [email protected] mechanisms affecting population stability inAm. Nat. 1998. Vol. 151, pp. 116–134. 1998 by The University of Chicago.

0003-0147/98/5102-0002$03.00. All rights reserved. these model systems (Wilson 1996; Wilson and Nisbet



Dynamics of Individuals and Populations 117

1997). In order to produce generalities that transcend theless, adjusting these stochasticity parameters allowgood reproduction of the IBM results, thereby sheddingspecific situations, it will be necessary to contrast simula-

tion results with solutions of approximate analytic mod- light on the important role stochasticity plays within in-dividual-based simulations.els (Brown and Hansell 1987; de Roos et al. 1991; Mat-

suda et al. 1992; Wilson et al. 1993; Durrett and Levin The remainder of the article is organized as follows.After first describing the three models examined in this1994; Harada et al. 1995). I address this contrast in the

following analysis. In particular, I use the rules embodied article, I demonstrate congruence between the IBM andthe deterministic RD model in the limit of large, local-by a spatially explicit individual-based model (IBM) to

define a predator-prey system. The IBM was carefully ized populations. Decreasing the size of local populationsshows the divergence of IBM results as stochastic pro-crafted to correspond, in specific limits, to a well-exam-

ined deterministic ordinary differential equation preda- cesses at the individual scale become important. Next,using the SRD model and its four adjustable parameterstor-prey model (de Roos et al. 1991; McCauley et al.

1993). The goal of this article is to determine an appro- that scale stochasticity, I reproduce the IBM results for avariety of dispersal scenarios. These comparisons demon-priate stochastic population-level model that captures

generality to gain insight beyond the specific simulation strate the important role stochasticity plays in spatialprocesses. Finally, reducing stochasticity within the SRDmodel studied here.

This article extends previous work in two important model leads to new dynamic patterns, which I then re-produce in the IBM.ways. First, I examine localized individual interactions

coupled with long-range dispersal for all stages of preyand predator. McCauley et al. (1996) considered infinite-range predator offspring dispersal and found the com- Model Formulationsplete disappearance of the ordinary differential equation

In this section, I present three model formulations formodel’s limit cycle dynamics at both local and global

prey-predator systems: an individual-based model (IBM)scales. Nonoscillatory, slowly wandering prey-filled

that defines the system of interest; a complementary re-patches amid prey-barren regions and a constant, uni-

action-dispersal (RD) model of integrodifferential equa-form distribution of predators were observed instead.

tions that incorporates prey and predator interactionsDetailed analysis revealed that the combination of non-

and dispersal of various prey and predator stages; and alinearity in functional response coupled with the differ-

stochastic version of the RD model (SRD). It is impor-ences in spatial scales for predation and prey growth were

tant to note that the latter two are not derived mathe-responsible for this temporal stability and spatial pat-

matically from the stochastic rules defining the IBM.terning (de Roos et al., in press). As presented here, long-

Subtle terms can arise in detailed derivations of popula-range dispersal in other stages also produces a wide vari-

tion-level models of stochastic systems that have qualita-ety of interesting spatial dynamics, ranging from stable

tively important effects (Leslie 1958; Doi 1976; Chessonpattern formation to oscillatory dynamics (Levin and

1981; Durrett and Levin 1994). The IBM rules presentedSegel 1985; Kot and Schaffer 1986; Murray 1989; Bots-

here may indeed be amenable to such a derivation, yieldford et al. 1994).

these terms, and lead to greater mathematical insight.The second extension is to uncover mechanisms affect-

However, the philosophy of this work is that, even in theing spatial structure by comparing and contrasting IBM

absence of mathematical rigor, insights about importantresults with predictions from a set of integrodifferential

processes and mechanisms can be ascertained throughreaction-dispersal (RD) equations. Linear stability analy-

the comparison of a variety of modeling frameworkssis for the RD model qualitatively predicts many of the

(McCauley et al. 1993; Nisbet et al., in press; Wilson anddynamic features observed in the stochastic simulation.

Nisbet 1997).However, there are cases where the simulation showsspatial patterning not predicted by analytical methods.These disagreements prompt a stochastic version of the

Individual-Based Model (IBM)reaction-dispersal (SRD) model incorporating two fea-tures inherent within the stochastic IBM: ‘‘noise’’ from This work implements a well-studied individual-based

simulation model (de Roos et al. 1991; McCauley et al.the randomness of individual interactions, and extinc-tions due to the small size of local populations (Wilson 1993; Wilson 1996) that incorporates the movement and

interactions of thousands of individual prey and preda-et al. 1993; Wilson 1996). These features are incorporatedphenomenologically, which unfortunately introduces ad- tors over a spatial habitat. Given explicit rules, each indi-

vidual-based simulation run produces an ‘‘exact’’ in-justable parameters rather than parameters derived math-ematically from the given set of simulation rules. None- stance of the model system’s stochastic dynamics.



118 The American Naturalist

Discrete-time equations for the following rules are pre- Dispersal. Simulation results presented here incorporateboth local and long-range dispersal. Local dispersal is asented in appendix A.random step to one of four nearest neighbor cells, al-lowed if no other conspecific either is presently there orLattice Updating. The simulation makes space and time

discrete. A square lattice of cells approximates the homo- chooses to move to that cell during the same time step.This movement type is the default for all stages. Individ-geneous spatial habitat, with each cell containing at most

a single prey and predator. A ‘‘one-dimensional’’ 8 3 uals of at most one stage possess long-range dispersal de-fined by a function that gives the probability to disperse1,000 lattice is used. Previous results indicate that the

simulation results depend strongly on the width (Wilson from one point to another. The function used in thesesimulations is uniform dispersal within a cutoff distanceet al. 1995). Boundary conditions applied at the lattice’s

edges are periodic, connecting the top and bottom edges D about an individual’s present cell (i, j ), where i is therow number and j is the column number. The cutoff dis-and the left and right edges together like a long soda

straw joined at the ends. The simulation makes continu- tance D is larger than the lattice width for the cases ex-amined here, thus the uniform dispersal algorithm usesous time into discrete simulation steps and updates all

cells simultaneously while eliminating any conflicts that two random numbers, 1 # ρ1 # 8 (for a lattice of widtheight cells) and 2D # ρ2 # D to obtain the new spatialarise in the new states (McCauley et al. 1993; Wilson et

al. 1993). The simulation step defines the unit time, and coordinates (ρ1, j 1 ρ2).I use the terms interaction rate and event probabilitywithin a simulation step interchangeably.

Deterministic Reaction-Dispersal Model (RD)Prey Reproduction. During every time step, each prey has

Reactions. Nonspatial ordinary differential equation mod-a probability r of producing an offspring. Single prey andels of reaction kinetics define the base models for systemspredator occupancy requires placement of the offspringof interacting populations. This work’s baseline model isprey into a randomly selected nearest neighbor cell of itsa general predator-prey model, where g(V) representsparent’s cell. If a prey already occupies the selected cell,the prey growth in the absence of predators, and f (V) isthen the algorithm aborts the prey offspring. Thus, preythe functional response, or feeding rate per predator atreproduction is density dependent.prey density V. Consumed prey are converted into pred-ators with efficiency e, and µ is the predator mortality.Predation. A predator has a foraging status of eitherThese features outline the set of ordinary differentialsearching or sated. If a prey and a searching predator areequations,in the same cell during a time step, the predator captures

the prey with probability a. The predator then handles dV

dt5 g(V) 2 f (V)P , (1a)the prey for TH simulation steps (i.e., its satiation time)

during which the predator does not attack prey. There isno prey mortality other than predation. and

Predation Reproduction. A successful predator producesdP

dt5 e f (V)P 2 µP . (1b)

an offspring with probability e concurrent with enteringthe sated status. Starting from its parent’s cell, an off- As shown in appendix A, the well-mixed discrete-timespring takes a step to a randomly chosen nearest neigh- simulation model corresponds approximately to thebor cell. The offspring occupies that chosen cell if no functions,predator is present but takes another step if the cell con-tains a predator. Offspring remaining unplaced after 10

g(V) 5 rV(1 2 V) and f (V) 5aV

1 1 aTH V. (2)steps are aborted. This offspring placement algorithm in-

troduces a small degree of density dependence in thepredator population, which is much less severe than that McCauley et al. (1993) demonstrated the congruence be-

tween equations (1) and the simulation rules outlinedinvolved in prey reproduction, and I do not include it inthe analytic model. above under well-mixed conditions.

Dispersal. Incorporating dispersal of offspring and adultPredator Mortality. Each predator has a probability µ ofdying each time step. Mortality is unaffected due to ei- stages of both prey and predators extends the ODE

model (eqq. [1]) into a set of integrodifferential equa-ther lack of prey (i.e., starvation) or high predator den-sity (i.e., interspecific competition). tions,




1992; Lewis 1994; Gourley and Britton 1996; Gourley et∂V(x, t)

∂t5 #

1∞

2∞κ OV(y 2 x)g(V(y, t))dy

al. 1996), which then influences local movement.

2 f(V(x, t))P(x, t) (3a) Stochastic Reaction-Dispersal Model (SRD)

Demographic stochasticity arises in the IBM from both1 δV#

1∞

2∞κ AV (y 2 x)(V(y, t) 2 V(x, t))dy ,

interactions and movement. For example, when the preyreproduction rate is r 5 1/4, an isolated prey individual

and reproduces, on average, once every four time steps, butduring any one time step there is either a reproductionevent or not. The time between events is also distributed∂P(x, t)

∂t5 e#

1∞

2∞κ OP (y 2 x) f(V(y, t))P(y, t)dy (3b)

probabilistically. Thus one can envision IBM interactionsas a random series of instantaneous events, whereas theRD model incorporates interactions spread uniformly2 µP(x, t)through time. Dispersal is also a source of stochasticity inthe IBM since an individual leaving one site can only re-

1 δP#1∞

2∞κ AP (y 2 x)(P(y, t) 2 P(x, t))dy .

appear at one other randomly chosen site on the lattice.In contrast the RD model spreads an individual (or frac-

The kernels, κ(z), represent probability distributions for tion thereof) across space according to the dispersalan individual to move a distance z. Here the functions kernel.κOV(z) and κOP(z) represent the kernels for prey and Here I extend the RD model to incorporate two fea-predator offspring dispersal, and the functions κAV(z) and tures that mimic demographic stochasticity and label theκAP(z) represent adult prey and predator dispersal. The resulting model the stochastic reaction-dispersal (SRD)parameters δV and δP represent the fraction of adults that model. The first feature accounts for within-cell fluctua-move per unit time. Offspring dispersal corresponds to tions. One of the solutions to the RD model examinedthe simulation’s one-time dispersal of all offspring imme- below is a numerical integration using equations that arediately after reproduction, for example, with a fraction temporally and spatially discretized. I follow each itera-κOV(y 2 x)dy of the prey reproduced in the region dy tion of the RD model’s numerical solution by a perturba-centered about the location y moving to the location x tion of the population density at each discrete integration(Weinberger 1978). Similarly, adult prey dispersal in- point along the axis:volves two terms, the first propogates individuals from

V(x, t) :5 V(x, t) 1 πVR√V(x, t) , (5a)location y to location x, while the second term representsdispersal away from location x (Nisbet and Gurney and1982). Whereas diffusion is a specific approximation, a

P(x, t) :5 P(x, t) 1 πPR√P(x, t) , (5b)dispersal kernel is a general description of organismalmovement (Skellam 1951; Okubo 1980; Murray 1989) where R is a uniformly distributed random number, 21over timescales that are short relative to interaction , R , 1, and the π’s scale the perturbations. Althoughtimescales (Karieva 1983; Murray 1989). ad hoc, the dependence on the square root of the popu-

The uniform dispersal IBM rules described above reas- luation has been rigorously derived for a variety of sys-sorts all individuals leaving one location uniformly tems (e.g., Leslie and Gower 1960; Poole 1974; Kang andwithin a dispersal distance D and corresponds to the dis- Redner 1985). Leslie (1958) used these repeated pertur-persal kernel bations in numerical calculations on ‘‘an ordinary hand-

machine’’ for nonspatial stochastic single-species andpredator-prey models (see also Leslie 1958; Leslie and

κ(z) 5 51

2D|z |# D,

0 |z |.D ,

(4) Gower 1960; Poole 1974). Repeated perturbations havealso been incorporated to mimic stochasticity in treegrowth models (Garcia 1983), and Hastings (1993) useda density-independent variation to examine stochasticity(e.g., Lewis 1994). The maximum dispersal distance D is

also subscripted with ‘‘OV,’’ ‘‘OP,’’ ‘‘AV,’’ and ‘‘AP,’’ de- within coupled logistic models. I use these repeated per-turbations within the SRD model to reflect the net sto-noting offspring prey and predators, and adult prey and

predators, respectively. Others have incorporated similar chasticity of the IBM’s interactions and dispersals.The second effect of discrete individuals accounts forkernels to represent an individual’s measure of local con-

ditions (Turchin 1989; McLaughlin and Roughgarden the change in model structure resulting from small pop-




Table 1: Parameters and default values used in the three models

Parameter Description Default value Relevant model(s)

r Prey growth rate .25 IBM, RD, SRDa Attack rate 1 IBM, RD, SRDTH Handling time 3 IBM, RD, SRDe Conversion efficiency .5 IBM, RD, SRDµ Predator death rate .06 IBM, RD, SRDDOV Offspring prey dispersal distance 1 IBM, RD, SRDDAV Adult prey dispersal distance 1 IBM, RD, SRDDOP Offspring predator dispersal distance 1 IBM, RD, SRDDAP Adult predator dispersal distance 1 IBM, RD, SRDπV Prey perturbation scale .2 SRDπP Predator preturbation scale .2 SRDMV Minimum prey density .01 SRDMP Minimum predator density .004 SRDV0 Initial prey density 1.0 IBM, RD, SRDP0 Initial predator density .25 IBM, RD, SRDLattice size ⋅ ⋅ ⋅ 8 3 1,000 IBMGrid points ⋅ ⋅ ⋅ 1,000 RD, SRD

ulations. For the default parameter set used in the runs play in determining its spatiotemporal dynamics. Thethird part uses a variety of prey and predator dispersalbelow (table 1), limit cycle dynamics are anticipated

(McCauley et al. 1993). However, I demonstrated (Wil- scenarios to examine the ability of the SRD model to re-produce the IBM spatiotemporal dynamics. Finally, theson 1996) that the individual-based simulation rules lead

to local dynamics dominated by extinction. It is well- last part examines the effect of reducing stochasticity, butnot entirely eliminating it, in both the SRD model andknown that in stochastic models, population trajectories

can be absorbed by the empty state, whereas their deter- the IBM.ministic counterparts have stable fixed points or limit cy-cles (e.g., Leslie 1958; May 1973, 1974). Here I account

The Importance of Discrete Individualsfor this extinction by defining lower bounds, MV and MP,for prey and predator populations, respectively. At the In the absence of a precise derivation of the RD model asend of each iteration, population densities of all cells are the deterministic limit of the given IBM rules, I presentchecked, and those that fall below these lower bounds are here a ‘‘numerical limit’’ to the deterministic RD model.set to zero. Adding these two features makes each nu- An implicit assumption of such model equations is thatmerical integration of the SRD model unique, dependent the populations contained in an infinitesimal area, or in-upon the stream of random perturbations. finitesimal length in one dimension, have large enough

populations to justify the deterministic approximation.One way to approach this limit in the IBM while not

Resultschanging the interaction rules is simultaneously to in-crease the lattice width and to mix the populations alongIn this four-part section I compare the IBM, RD, and

SRD models. All parameters take the values listed in table this dimension. The IBM and RD model should becomeidentical in the limit of an infinite IBM lattice width.1 unless otherwise stated. The first part extends the con-

gruence observed between the nonspatial RD model and Numerical solution to a straightforward temporallyand spatially discretized version of the deterministic RDthe homogeneously mixed IBM (McCauley et al. 1993)

into the one-dimensional spatial domain. It also demon- model reveals that the system’s long-term evolution de-pends on its initial condition (fig. 1A, B). This depen-strates the divergence of the IBM’s results from those of

the RD model as the discreteness of individuals becomes dence was described by de Roos et al. (in press) for thisRD model with infinite dispersal of predator offspringa dominating influence. The second part demonstrates

the roles that the SRD model’s stochasticity parameters (McCauley et al. 1996). Effectively, there are two struc-



Figure 1: A, B, Prey distributions resulting from a numerical integration of the deterministic RD model. The horizontal dimension(500 spatial units wide) plots high and low densities defined by a cutoff of V 5 0.125 as dark and light areas, respectively. Tempo-ral evolution (4,000 time units) goes from top to bottom. Dispersal is limited to nearest neighbor cells. Prey and predator densitiesin each cell are either 0 or 1.0 and 0.25, respectively, with probabilities 0.5 and 0.5 for the homogeneous runs (A) and 0.025 and0.1 for the heterogeneous runs (B). Two basins of attraction, one spatially homogeneous and the other spatially heterogeneous,are apparent. Subsequent images are simulation results using lattices of length 500 cells and width (C, D) 200 cells, (E, F) 32 cells,and (G, H) eight cells. Populations are homogeneously mixed along the width after each simulation step. Initial occupancies corre-spond to the RD model in the homogeneous and heterogeneous cases. Occupancy along a single row is shown every eight timesteps, with a dot indicating presence of an individual prey. The widest system demonstrates two basins of attraction in qualitativeagreement with A and B, but shrinking the width reduces local population numbers and enhances the importance of stochasticprocesses.




turally different basins of attraction (Hastings 1993; Jan- point the coupling between modes of large amplitudemay dominate. For example, the two possible scenariossen 1995), one with spatially homogeneous dynamics and

the other with spatially complex dynamics. Figure 1C, D for long-term dynamics in figure 1A, B depend on the setof initial large-amplitude modes.presents IBM results from a 200 3 500 lattice with mixed

populations along the narrow dimension and initialized The insufficiency of this comparison motivated theSRD model that includes that stochasticity features dis-analogously to figure 1A, B. Although specific runs using

either initial condition may land in one basin or the cussed above. In figure 3, I successively incorporate thesefeatures within the numerical integration of the SRDother, these results support the existence of multiple ba-

sins of attraction for this IBM limit, in qualitative agree- model. First, temporal and spatial scales are chosen iden-tical to those for the IBM runs. Temporal evolution inment with the deterministic RD model. Clear quantita-

tive differences between the RD and IBM results, such as prey dynamics for the deterministic equations (RD limitof the SRD model) is shown in figure 3A for initial con-the oscillation frequency, arise from IBM implementation

details such as the effort expended in placing predator ditions homogeneously resembling those of figure 2A.Figure 3B tests effects of initial randomness in the preyoffspring onto an almost filled lattice. This particular de-

tail introduces a degree of density dependence in preda- distributions seen in figure 2A. Neither of these two im-ages provides a good reproduction of the IBM results.tor reproduction that is ignored in the RD model.

Having established this connection between the deter- Next, repeated perturbations alone produce a much dis-torted image (fig. 3C) approaching the IBM results,ministic RD model and the IBM, I now demonstrate

their divergence by narrowing the lattice’s width, thereby whereas only including the minimum prey and predatordensities produces the large-scale prey extinction regionsreducing the total population at each point along the

one-dimensional space. Figure 1E–H shows that nar- observed in figure 3D. Finally, incorporating both the re-peated perturbations and the minimum densities pro-rowing the lattice ‘‘destablizes’’ spatially extended oscilla-

tions. The goal of this work is to ascertain the mecha- duces prey (fig. 3E) and predator (fig. 3F) images quali-tatively similar to the IBM model results (fig. 2A, B).nisms producing this observed spatial structure within

the individual-based simulation and to distinguish phe- It is important to note that I performed this parame-terization of the SRD model in parallel with the parame-nomena that arise from mechanisms contained within

the deterministic, population-level reaction-dispersal terization of the various dispersal scenarios presented inthe next section. Hence neither section represents inde-model from phenomena produced by having inherently

noisy individuals. pendent tests of the model.

Agreement between the IBM and SRD Model underRole of Stochasticity in the SRDVarious Dispersal Scenarios

Figure 2A, B shows the strong spatial structuring in preyand predator IBM dynamics (each dot indicates the pres- Long-Range Adult Prey Dispersal. Long-range uniform

adult prey dispersal, DAV 5 50, produces strong spatiallyence of an individual) that results when individuals moveslowly through the spatial arena by random walking (de homogeneous prey oscillations but more complex preda-

tor dynamics involving a mixture of global oscillationsRoos et al. 1991; McCauley et al. 1993). Figures 1A, Bprovides representative numerical integrations of the RD and transient patterning (fig. 4A, B). The IBM images are

reproduced by the SRD model in figure 4C, D. Eigen-model for these parameter values. Figure 2C plots thereal and imaginary components of the largest eigenvalue value plots (fig. 4E) clearly indicate the dominant global

oscillation seen in the prey and predators. In addition,λ arising from linear stability analysis of the RD modelagainst spatial mode k (see app. B). Small values for k this plot and the next two eigenvalue plots show peaks

near k < 0.09, or spatial modes of scale 2π/k < 70 cells.represent initial spatially oscillatory perturbations of largewavelength. The curves indicate that a range of modes The width of the IBM and SRD images correspond to

roughly 14 of these wavelengths. The patterning in thewill grow in amplitude (those modes for which Reλ .0;fig. 2C), and since the imaginary component of the ei- predator images presumably results from the ‘‘weakly’’

stable peak coupled with repeated perturbations thatgenvalue is nonzero, these modes will grow oscillatorilyin time. continually reintroduces this spatial mode.

Linear stability analysis only reveals the short-termevolution of small amplitude perturbations and is thus a Long-Range Adult Predator Dispersal. Long-range adult

predator dispersal, DAP 5 50, produces global oscillationslimited means of interpreting IBM simulation results. Itdoes not reveal the characteristics of long-term dynamics in predator density and spatiotemporal patterns combin-

ing global oscillations and stable patterning in the preyonce the perturbation amplitudes become large, at which




Figure 2: Low mobility results (no long-range dispersal) for (A) prey and (B) predators in the IBM simulation. Cells in the middlethird of the lattice are initially prey occupied with probability 1.0 and predator occupied with probability 0.25. Cells outside thisblock are initially empty. The images show an initial expansion of the prey into the empty region followed by a predator frontmoving at a slightly slower speed. C, Linear stability analysis for the deterministic RD model predicts that spatial structure withwavelengths larger than < 2π/0.28 < 22 cells will grow oscillatorially. Reλ, Imλ 5 real and imaginary components of the domi-nant eigenvalue.

distribution (fig. 5A, B). The SRD model, with increased is strictly due to long-range dispersal rather than purediffusion (Levin and Segel 1985). Approximately 14prey stochasticity πV 5 0.25, produces figure 5C, D. Lin-

ear stability analysis shows a positive real peak for non- roughly delineated bands are observed in the prey im-ages, thereby matching the value of the unstable spatialzero wave number (fig. 5E) along with the complex peak for

k 5 0, indicating either homogeneous oscillations or stable mode k < 0.09. Pattern formation observed in the ho-mogeneously unstable model (TH 5 3) with long-rangepattern formation, or some combination of the two.offspring predator dispersal is much more stable (notshown).Stable Model with Long-Range Offspring Predator Dis-

persal. A change in the handling time from TH 5 3 to TH

5 0 yields a stable equilibrium in the ODE (eqq. [1]). In Quantitative Comparisons. One way to quantify the abovespace-time images is through the traditional two-pointthis case long-range offspring predator dispersal (DOP 5

50) produces relatively fixed spatial patterns (fig. 6A, B). correlation functions, C(R, T) ; ,(ρ(x, t) 2 ρ) (ρ(x 1R, t 1 T) 2 ρ)., which measures the spatiotemporalNumerical results for the SRD model, using πV 5 0.25

and minimum values of MV 5 0.012 and MP 5 0.0, re- correlation between fluctuations in the density ρ(x, t)about the spatially and temporally averaged density ρ.produce the IBM images. Stable pattern formation is evi-

dent in the linear stability analysis (fig. 6E): the purely Figure 7 compares the correlation functions of the IBMand SRD model for selected images presented above. Allreal positive eigenvalue peak marks a Turing-like insta-

bility (Turing 1952), although in this case the instability plots are normalized by their autocorrelation C(0, 0).




homogeneous oscillations without spatial patterning (fig.8F). Finally, in the case of the stable model (TH 5 0) andhigh offspring predator dispersal, removing predator sto-chasticity leads to greatly stabilized pattern formation inprey density (fig. 8G vs. 8H). Reducing the minimumprey density (πP 5 0, MV 5 0.005) then produces an im-age that is a superposition of the k 5 0 stable state andstable pattern formation (fig. 8I).

Figure 1 demonstrated agreement between the deter-ministic RD model and the IBM by mixing the popula-tions along an increased lattice width. I now demonstratethat this change in the IBM appears analogous, in somecases, to the changes in stochasticity parameters for theRD model shown in figure 8. Figure 9 presents results forpredator distributions for high adult prey dispersal (fig.9A–C; cf. with fig. 8A–C), prey distribution for highadult predator dispersal (fig. 9D–F; cf. with fig. 8D–F),and prey distribution for offspring predator dispersal inthe stable model (fig. 9G–I; cf. with fig. 8G–I). Few dif-ferences are seen between the mixed and the unmixed re-sults for the narrow width, whereas results for the largestwidth demonstrate reasonable agreement with the re-duced noise and minimum density images for the SRDmodel (fig. 8).

Conclusions

Figure 3: Results for numerical integration of the RD and SRD This article’s specific goal was to connect an individual-models. In A and B, the middle-third populated block has cells based model (IBM) of predator-prey populations with ainitialized with prey (predators) with probability 0.01 (0.1), val-

population-level reaction-dispersal (RD) model (eq. [3])ues chosen to match qualitatively the initial eruption centers

based on generalizing standard reaction-diffusion modelsobserved in the IBM figures 2A, B. In C, random perturbationsto incorporate long-range dispersal kernels. Linear stabil-are applied continuously and in D minimum prey and predatority analysis and numerical integration of a stochastic ver-densities are imposed. Incorporating both features yields plotssion of the reaction-dispersal (SRD) equations demon-analogous to the IBM images of figure 2 for (E) prey and (F)strated good agreement with the simulation results.predators. In all images, dark regions indicate prey densities

.0.175 and predator densities .0.175. Because I was able to match the IBM and SRD results fora broad range of dynamics, I believe that I have success-fully captured the dynamics of the individual-based sim-ulation within a stochastic population-level framework.

Reducing Effects of StochasticityOne crucial assumption to the deterministic analytic

model is its large number limit. Only for systems whereTurning off the stochasticity parameters in the SRDmodel serves as a prediction of which phenomena arise infinitesimal spatial regions hold very large numbers of

individuals can such models accurately describe dynam-from stochastic processes. Removing predator stochas-ticity (πP 5 0) eliminates the spatial structure in the ics. The simulation model violates this assumption, and

this violation is responsible for the discrepancies betweenpredator distributions for high adult prey dispersal (fig.8A vs. 8B), promoting homogeneous oscillations that the simulation and the deterministic analytic model. As a

result, there are three ingredients that determine the sys-lead to ultimate extinction. Further removing the mini-mum predator density (πP 5 0, MP 5 0) demonstrates its tem’s spatial dynamics: the inherent complex spatial dy-

namics of the deterministic RD model; the stochasticityrole in determining the predator’s wave speed (fig. 8C).Removing prey stochasticity (πV 5 0) in the case of high of individual interactions; and the tendency to extinction

of small populations.adult predator dispersal produces stable spatial patternsin the prey distribution (fig. 8D vs. 8E). Further removal A great amount of detailed structure in the simulation

results can be interpreted by considering the magnitudeof the minimum prey density (πV 5 0, MV 5 0) leads to




Figure 4: High adult prey dispersal (DAV 5 50) results for (A) prey and (B) predators in the IBM simulation. These images arereproduced in the SRD model for (C) prey and (D) predators. Dark regions indicate prey densities .0.225 and predator densities.0.2. E, Linear stability analysis of the RD model predicts purely homogeneous oscillations. Reλ, Imλ 5 real and imaginary com-ponents of the dominant eigenvalue.

of both stable and unstable eigenvalues in light of sto- knowledge that conclusions based only on linear stabilityanalysis of deterministic analytic formulations should bechastic processes (May 1973). In a spatial context, these

eigenvalues dictate the temporal development of pertur- qualified, yet these conclusions provide good qualitativeguidance.bations having specific spatial scales, that is, spatial sine

waves with a specific wavelength, and this development is Important biological conclusions arising from thiswork concern the influence of stochasticity and dispersalinfluenced by stochasticity. For example, one possibility

is that stochastic fluctuations frequently reintroduce per- on the dynamics of spatially averaged population mea-sures. Stochasticity promotes temporal stability (e.g., deturbations of a given wavelength that would otherwise

decay away over long timescales. Hence, spatial structure Roos et al. 1991), and its effects will be greatest for sys-tems that have small numbers of individuals, roughly lessat that wavelength is always present, whereas it is not an-

ticipated deterministically. The importance of stochastic than 100 or 200 (e.g., Leslie and Gower 1960), within lo-cal interaction regions, the size of which depends on bothfluctuations found here only emphasizes the common




Figure 5: High adult predator dispersal (DAP 5 50) results for (A) prey and (B) predators in the IBM simulation and (C) preyand (D) predators in the SRD model using πV 5 0.25. Dark regions indicate prey densities .0.125 and predator densities .0.15.E, Linear stability analysis results for the RD model admit solutions having stable patterns. Reλ, Imλ 5 real and imaginary compo-nents of the dominant eigenvalue.

species interactions and dispersal distances. In the simu- ics (fig. 1). However, even a system having a prey orpredator stage with long-range dispersal might have lowlation presented here, single-cell occupancy for both prey

and predator clearly enhances the effects of demographic population numbers for local interactions; for example,in figures 4–6 there are significant effects of stochasticitystochasticity (fig. 1), and one could well argue that this

limitation is an artifact of the simulation. But in defense despite a widely dispersing prey or predator stage.This work also demonstrates the important and well-of this occupancy limitation, it seems valid to consider

the cell size as the lower bound on an individual’s size or recognized role that dispersal plays in determining popu-lation dynamics. In the above IBM runs I was able tothe size of its well-defended territory (Wilson et al.

1993). If local population numbers increase, perhaps due produce a qualitatively broad range of temporal and spa-tial dynamics simply by altering dispersal parameters.to an increased carrying capacity or a proportionate in-

crease in all dispersal distances, then deterministic mod- One important caveat concerning the details of the re-sults presented here has to do with the specific dispersalels become valid approximations of the system’s dynam-




Figure 6: High offspring predator dispersal (DOP 5 50) results for the dynamically stable model (TH 5 0). Results are shown for(A) prey and (B) predators in the IBM simulation and (C) prey and (D) predators in the SRD model with πV 5 0.25, MV 50.012, and MP 5 0.0. Dark regions indicate prey densities .0.125 and predator densities .0.15. E, Linear stability analysis resultsfor the RD model admit solutions having stable patterns. Reλ, Imλ 5 real and imaginary components of the dominant eigenvalue.

kernel I used. Uniform dispersal within a fixed distance is interactions determine ecological dynamics at the popu-lation level (Levin 1992; Metz and de Roos 1992). Thebiologically implausible, having no apparent mechanistic

basis. Stability properties depend, in part, on the Fourier methodology and results of this article highlight two op-posing and important goals of theoretical ecology. Onetransform of the dispersal kernel, and this kernel’s trans-

form has changes in sign that enhance spatial pattern for- goal is to establish mathematical rigor behind the conclu-sions drawn from mathematical models encapsulatingmation. Less drastic spatial patterning occurs when, for

example, organisms disperse according to a normal dis- ecological processes. This goal’s objective is to solve apostulated model accurately and to prove that the statedtribution.

Stepping back from the specific simulation model pre- conclusions are true. The present article does not furtherthis goal because the three examined models are notsented here, this work addresses the difficult theoretical

problem entailed by acknowledging that individual-level mathematically connected. Such a connection demands a



Figure 7: Two-point correlation functions, C(R, T) ; ,(ρ(x, t) 2 ρ) (ρ(x 1 R, t 1 T) 2 ρ)., for the population density ρcalculated over 100,000 time steps and normalized by C(0, 0). Each time step, a randomly chosen point serves as the origin for ameasurement set. A, B Predator correlation plots for low mobility in the IBM (fig. 2B) and SRD (fig. 3F). C, D, Predator correla-tion plots for high adult prey dispersal in the IBM (fig. 4B) and SRD (fig. 4D). E, F, Prey correlation plots for high offspringpredator dispersal in the IBM (fig. 6B) and SRD (fig. 6D).




Figure 8: Reducing stochasticity in the SRD model affects dynamics. A–C, Predator images for high adult prey dispersal with (B)πP 5 0 and (C) πP 5 MP 5 0. In C, the homogeneous oscillations lead to global extinction. D–F, Prey images for high adultpredator dispersal with (E) πV 5 0 and (F) πV 5 MV 5 0. G–I, Prey images for high offspring predator dispersal with TH 5 0. In(H) πP 5 0 and in (I) πP 5 0 and MV 5 0.005.

derivation of the stochastic population-level model from dressed by a population-level model that contains gener-alities. This goal’s objective is the formulation of generalthe specified individual-scale rules.

The second opposing goal is to understand inherently models that make falsifiable predictions regarding eco-logical systems. Given this objective, the use of sim-complicated and stochastic ecological systems. The role

of individual-based simulations is to bridge the funda- ulations can help refine general models by avoiding thelimitations of analytic tractability. As such this article ex-mental gap between real biological systems and general

population-level models. This biologically motivated goal amined the relevance of plausible population-level mod-els and their approximate solutions to a carefully posedstresses the importance of a two-tiered approach to un-

derstanding ecological systems, the tiers being the indi- simulation model.vidual and population scales (Wilson 1996; Wilson andNisbet 1997). Simulation inputs are reasonably wellknown but poorly parameterized individual-level behav-

Acknowledgmentsiors, while the outputs are averages that are as detailed asone desires. Correctly designed, a simulation represents I thank A. de Roos, S. Harrison, S. Levin, E. McCauley,

H. Metz, W. Morris, W. Murdoch, R. Nisbet, and D.an artificial example of the systems purportedly ad-




Figure 9: Sensitivity of IBM simulation to stochasticity. Sites perpendicular to the long dimension are mixed for three differentlattice widths: (A, D, G) 8; (B, E, H) 32; and (C, F, I) 200. A–C, Predator images for high adult prey dispersal. D–F, Prey imagesfor high adult predator dispersal. G–I, Prey images for high offspring predator dispersal in the stable model (TH 5 0). Severalfeatures are predicted qualitatively from figure 8, including changes in wave-front speed and pattern stabilization at intermediatestochasticity in (H).

Schaeffer for discussions and comments on early drafts. I tion dynamics (McCauley et al. 1993). In this case theindividual-based rules correspond to an unstructuredam forever grateful for support and encouragement by

W. Laidlaw and R. Nisbet. I would also like to thank P. prey population V(t) updated in discrete time,Kareiva and several anonymous reviewers. Part of thiswork was performed at the University of California, V(t 1 ∆ t) 5 V(t) 1 (r∆ t)V(t)(1 2 V(t))

(A1)Santa Barbara, supported by the U.S. Office of Naval Re-2 (a∆ t)V(t)Pf (t) ,

search (grant N00014-93-10952).

where ∆ t is the simulation time step, r is the prey repro-duction rate, a is the attack rate, and Pf (t) is the predatorpopulation actively foraging for prey items.APPENDIX A

The predator population is structured only accordingWell-Mixed System

to their foraging status. Each predator that successfullyfinds and attacks a prey item is placed in a series of statesMixing the individuals on the lattice breaks spatial corre-

lations and generates a nonspatial context for the popula- that ratchet the predator through its handling time. Av-




eraged over the entire population, this process corre- Taking the limit ∆ t → 0 yields the continuous-time dif-ferential equations,sponds to the discrete-time equations,

Pf (t 1 ∆ t) 5 Pf (t) 1 (1 2 µ∆ t)PTH/∆ t(t)

(A2a)

dV

dt5 rV(1 2 V) 2

aVP

1 1 aTh V, (A7)

1 ea∆ tV(t)Pf (t)and

2 (a∆ t)(1 2 µ∆ t)V(t)Pf (t)

2 (µ∆ t)Pf (t) ,dP

dt5

eaVP

1 1 aTh V2 µP . (A8)

P1(t 1 ∆ t) 5 a∆ t(1 2 µ∆t)V(t)Pf (t) , (A2b)The continuous-time approximation should be valid for

and small ∆ t, or equivalently, small interaction rates, and forshort handling times such that the time delay throughPi (t 1 ∆ t) 5 (1 2 µ∆ t)Pi21(t), (A2c)the handling states is unimportant.

with handling time per prey item TH, predator death rateµ, conversion efficiency e, and where i 5 2, . . . , TH/∆ t.These expressions ignore the details of placing a predator APPENDIX Boffspring onto the lattice. I define the total predator pop-

Stability Analysis of the RD Modelulation, P(t), as the sum of all predator states:

This appendix presents the linear spatial stability analysisP(t) 5 Pf (t) 1 ^

TH/∆ t

i51

Pi (t)(A3)

for the deterministic reaction-dispersal model (eq. [3]).Given equilibrium densities V * and P*,

< Pf (t) 1 (a∆ t)V(t)Pf (t)^TH/∆ t

i51

(1 2 µ∆ t)i , V * 5µ

a(e 2 µTH), (B1a)

where the approximation assumes the foraging popula- andtion is constant over a time equal to TH. Equation (A3)leads to an expression for the foraging predators in terms P* 5

r

a(1 2 V *)(1 1 aTH V *) , (B1b)

of P(t) and V(t),

the subsequent dynamics of small initial perturbationsPf (t) 5

P(t)

1 1 (a∆ t)V(t)^TH/∆ t

i51

(1 2 µ∆ t)i

. (A4) about the equilibrium,

V(x, t) 5 V * 1 ve ikx e λt (B2a)

andUsing equations (A3) and (A4), equation (A2a)–(A2c)

P(x, t) 5 P* 1 pe ikx e λ t (B2b)can be summed and rewritten for the dynamics of thetotal predator population. This replacement gives the determine the equilibrium’s stability. The spatial compo-discrete-time dynamic equations for the prey and preda- nent of such perturbations represents spatial variationstor populations, about the equilibrium densities with wave number k, that

is, wavelength 2π/k. Replacing equation (B2) into equa-V(t 1 ∆ t) 5 V(t) 1 (r∆ t)V(t)(1 2 V(t))tion (3) and assuming the perturbation amplitudes v andp are small enough to ignore second-order terms yields2

(a∆ t)V(t)P(t)

1 1 (a∆ t)V(t)^TH/∆ t

i51

(1 2 µ∆ t) i

, (A5)the characteristic equation for the eigenvalue λ as a func-tion of the wave number k,

and )(a11 2 λ) a12

a21 (a22 2 λ) ) 5 0 , (B3a)P(t 1 ∆ t) 5 P(t)

where1

e(a∆ t)V(t)P(t)

1 1 (a∆ t)V(t)^TH/∆ t

i51

(1 2 µ∆ t) i

(A6)a11 5 κOV(k)g ′ 2 P* f ′ 1 δV(κAV(k) 2 1) , (B3b)

a12 5 2µe

, (B3c)2 (µ∆ t)P(t) .




a21 5 eP* f ′ κOP(k) , (B3d) If a particular value of k satisfies either condition, thenperturbations of this scale grow in amplitude, and one

a22 5 µκOP(k) 2 µ 1 δP(κAP(k) 2 1) . (B3e)expects them to be important. A third condition dictatesoscillatory dynamics,Furthermore, f ′ 5 df/dV, g ′ 5 dg/dV, with both deriva-

tives evaluated at the equilibrium densities, and the κs[g ′κOV(k) 2 P*f ′ 2 δV(1 2 κAV(k))

are the Fourier transforms,1 µ(1 2 κOP(k)) 1 δP(1 2 κAP(k))]2

κ(k) 5 #1∞

2∞κ(z)e ikz dz , (B4)

, 4µP*f ′κOP(k) ,

(B7c)

of the dispersal kernels. If κ(z) is real and symmetric, which, if true, implies that perturbations undergo eitherthen the Fourier transform is also real and symmetric. oscillatory growth or decay. If condition (B7c) is false,Asymmetric dispersal kernels are relevant for organisms af- then perturbations do not show temporal oscillations.fected by air or water currents (Botsford et al. 1994). If the A basic feature of Fourier transforms is that short-transport distance is small compared with the dispersal range dispersal kernels transform into very broadlyrange, the asymmetry introduces a general drift in the di- peaked k-dependent functions. When all four stages, off-rection of the bias but otherwise has little qualitative effect spring and adult prey and predators, possess short-rangeon the patterns presented here; different dynamics may dispersal, then the first few terms of a Taylor’s seriesoccur, however, when the transport distance is large. This expansion,work only considers symmetric dispersal kernels.

κ(k) < 1 2 ek 2 , (B8)Fourier transform of equation (4) givesadequately approximates the transformed dispersal func-

κ(k) 5 1sin(kD)

kD 2 . (B5) tions for small k. Symmetry of the dispersal kernel de-mands that the linear and cubic terms are absent. Param-eter e is given subscripts similar to other dispersalReplacing all Fourier transforms by unity in the aboveparameters. Replacing (B8) into condition (B7a)–(B7c)equations recovers the k 5 0 results, representing the dy-yields the low-mobility conditionsnamics of a spatially homogeneous perturbation.

Dynamic properties are deduced from the properties (g ′ 2 P*f ′) . (g ′eOV 1 µeOP 1 δVeAV 1 δPeAP)k 2, (B9a)of the eigenvalues, determined from solving equation

µP*f ′ , (µg ′eOP 1 (g ′ 2 P*f ′)δPeAP)k 2, (B9b)(B3a):

andλ6 (k) 5

1

2 3(a11 1 a22) (B6)4µP*f ′ 2 (g ′ 2 P*f ′)2 . 2[(g ′ 1 P*f ′)µeOP

1 (g ′ 2 P*f ′)(δPeAP 2 g ′eOV 2 δVeAV)]k 2.(B9c)

6 √(a11 1 a22)2 2 4(a11a22 2 a12a21)4 .

Parameter combination δe . 0 is similar to the diffusionThe real and imaginary components of the eigenvalues coefficient in reaction-diffusion models, and g ′eOV anddetermine stability, and I present numerical solutions for µeOP act as effective diffusion rates for the offspring.the largest eigenvalue in the figures. Whenever Reλ(k) . When k 5 0, right-hand terms are zero and, for the0, the condition indicates that spatial perturbations of model and the numerical values for the parameters usedwave number k grow. This instability occurs whenever in the simulations (TH 5 3), condition (B9a) is true,one of the two inequalities, a11 1 a22 . 0 or a11a22 , condition (B9b) is false, and condition (B9c) is true,a12a21, are satisfied (Nisbet and Gurney 1982). Given the meaning that the homogeneous model’s equilibrium isabove expressions, we have the two sufficient conditions oscillatorily unstable. For nonzero k, increasing any offor instability: the four mobilities results in the violation of condition

(B9a). Increasing either the offspring or adult predatorg ′κOV(k) . P*f ′ 1 δV(1 2 κAV(k))

mobility results in condition (B9b) becoming true. Con-dition (B9c) is violated when either offspring or adult1 µ(1 2 κOP(k)) 1 δP(1 2 κAP(k)) ,

(B7a)

predator mobility is large but reinforced when either off-and

spring or adult prey mobility is increased.Hence, increasing any of the four mobilities enhancesµP*f ′κOP(k) , [g′κOV(k) 2 P*f ′

the relative importance of global oscillations as ascer-2 δV(1 2 κAV(k))] (B7b)

tained through the amplitude of Reλ. However, increas-ing either of the predator mobilities may admit nonoscil-3 [µ(1 2 κOP(k)) 1 δP(1 2 κAP(k))] .




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Resolving Discrepancies between Deterministic Population Models and Individual‐Based Simulations