Patterns in spatial simulations—are they real?

Ecological Modelling 155 (2002) 19–30

Patterns in spatial simulations—are they real?

Kevin Anderson a, Claudia Neuhauser b,*a Institute for Mathematics and its Applications, Uni�ersity of Minnesota, Minneapolis, MN 55455, USA

b Department of Ecology, E�olution and Beha�ior, College of Biological Sciences, Uni�ersity of Minnesota, Twin Cities,100 Ecology Building, 1987 Upper Buford Circle, St. Paul, MN 55108, USA

Received 11 September 2001; received in revised form 6 February 2002; accepted 4 March 2002

Abstract

We discuss a class of spatial exploiter–victim models that exhibit pattern formation when exploiters disperse fartherthan their victims on average. The patterns are not Turing patterns; they are akin to patterns seen in models forneurological activities where spatial differences in excitation and inhibition can yield spatially inhomogeneouspatterns. The patterns in our study may be common in ecological simulation studies where often dispersal kernels arechosen that appear to be particularly prone to exhibiting such patterns. Since these patterns are very sensitive to thechoice of dispersal kernels, our study points to a potential pitfall in inferring processes from patterns in ecologicalstudies that are based on computer simulations. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Exploiter–victim models; Turing pattern; Pattern formation; Spatially explicit models; Interacting particle systems

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1. Introduction

Inferring process from pattern is a central themein community ecology. Conflicting views on howcommunities are organized existed in the earlierparts of the last century. Clements (Clements, 1916,1928) proposed that communities are highly orga-nized, whereas Gleason (Gleason, 1917) viewedthem as random assemblages. Whittaker’s seminalpaper (Whittaker, 1956) and many other observa-tional and experimental studies since then point tothe latter. This does not imply, however, thatcommunities are void of patterns. Both exogeneous

and endogeneous factors generate patterns, as hasbeen demonstrated in numerous empirical andtheoretical studies (Roberts, 1987; Schmitt, 1987;Pacala et al., 1990; Hassell and Wilson, 1997;Logan and Silander, 1998; Neuhauser and Pacala,1999).

This paper focuses on theoretical explanationsfor ‘patchiness,’ a ubiquitous spatial pattern. Therehave been many empirical studies that investigatecauses for patchy distributions in nature. Patchi-ness is an inherent property of many ecologicalsystems, not just caused by physical factors, suchas pH, moisture, or temperature, but also generatedby species interactions in a spatially homogeneousenvironment (Watt, 1947; Roberts, 1987; Schmitt,1987; Logan et al., 1998). It will be the latter causewe are concerned with here.

* Corresponding author. Tel.: +1-612-624-6790; fax: +1-612-624-6777.

E-mail address: [email protected] (C. Neuhauser).

0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0 304 -3800 (02 )00070 -4

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K. Anderson, C. Neuhauser / Ecological Modelling 155 (2002) 19–3020

Many empirical studies follow the hypothetical-deductive approach: a theoretical framework isestablished that allows the formulation of hy-potheses, which are subsequently tested in empiri-cal studies. Following this approach, findingexplanations for spatial patterns requires the for-mulation of a spatial model. Both deterministicand stochastic spatial models are employed. Ex-amples for deterministic spatial models are partialdifferential equations and integro-differentialequations; examples for stochastic spatial modelsare interacting particle systems (see below). Therelationship between the deterministic andstochastic approaches is well-understood (Durrettand Neuhauser, 1994; Neuhauser, 1994); see(Durrett, 1995) for additional examples.

Interacting particle systems (IPS) were intro-duced in the late 1960s (Spitzer, 1969; Dobrushin,1971a,b). They have become an established fieldin probability theory (Liggett, 1985; Durrett,1988), and a popular modeling framework in thesciences. They provide an appealing class of mod-els to describe ecological systems. IPS are spa-tially explicit and grid based stochastic modelswith local interactions which are also known ascellular automata (CA) or individual based mod-els (IBM). They belong to the general class ofstochastic processes that are known as Markovprocesses. An IPS consists of a countable collec-tion of spatial locations, S, called sites; typically,S=Zd, the set of points in d dimensional spacewith integer coordinates. Each site can be in oneof a finite number of states. Sites change theirstates at rates that depend on the states of neigh-boring sites within prescribed neighborhoods, forinstance, all neighboring sites within a fixed dis-tance. General overviews of IPS are given in(Liggett, 1985) and (Durrett, 1995), while a reviewof applications of IPS to ecology is given in(Durrett and Levin, 1994).

While rigorous results for IPS with ecologicalapplications have been established (Durrett andNeuhauser, 1997; Neuhauser, 1998; Krone andNeuhauser, 2000), it is more common to findecological studies where IPS are merely investi-gated through simulations. This is partially due tothe fact that IPS are quite difficult to analyzerigorously. Some examples of this approach are

(Gassmann et al., 2000; Jeltsch et al., 1999;Keymer et al., 1998). Just relying on simulations,however, can be dangerous since a deeper under-standing of the mechanisms that cause observedpatterns can only be gained through a thoroughtheoretical analysis.

In this paper we will discuss a class of IPS thatmodel exploiter–victim (+/− ) interactions(sensu May (May, 1975)) of two (or more) typesthat compete for space. We think of the two typesas plants who reproduce in such a way that one ofthe plants (the ‘exploiter’) needs the other plant(the ‘victim’) for reproduction. Both plants dis-perse their seeds within fixed neighborhoods. Wefind that under certain conditions on the repro-duction and dispersal neighborhoods spatially in-homogeneous patterns arise.

Instabilities of spatially homogeneous steadystates under small spatial perturbations have beenthe topic of interest ever since Turing (Turing,1952) suggested their existence. These so-calledTuring patterns arise in systems of reaction–diffu-sion equations when diffusion is present. Theunderlying mechanism driving the instability isdiffusion, and the diffusion constants of the inter-acting species need to differ. They can be found,for instance, in predator-prey models with diffu-sion (Mimura and Murray, 1978; Levin and Segel,1985; Alonso et al., 2002). See (Murray, 1993) fora complete discussion. The patterns that we dis-cuss in this paper, however, are not of the Turingtype.

1.1. A biological example showing patternformation

A biological system that can be modeled by thisclass of IPS is gynodioecy. This is a breedingsystem exhibited by about 10% of angiosperms inwhich some individuals produce flowers withoutfunctional pollen, thus rendering these individualsmale sterile or female (Delannay, 1978). Ingynodioecious populations, both hermaphroditeand female individuals coexist. Examples ofgynodioecious plants are the wild thyme (Thymuspolytriclus), the buckhorn plantain (Plantagolanceolata), and the meadow saxifrage (Saxifragagranulata) (Proctor et al., 1996). In this system,

K. Anderson, C. Neuhauser / Ecological Modelling 155 (2002) 19–30 21

male sterile plants are thought of as the ‘exploiter’and hermaphrodite plants as the ‘victim’ since themale sterile plants gain from the presence of thehermaphrodite plants, and the hermaphroditeplants lose in the presence of the male sterileplants since the male sterile plants compete withthe hermaphrodite plants for space.

Gynodioecious populations often display inter-esting spatial patterns. The male sterile plants of agynodioecious population frequently cluster to-gether (Couvet et al. 1990; Mannicacci et al.,1996). For example, van Damme (van Damme,1986) showed that in a particular population of P.lanceolata the total fraction of male sterile plantswas small, only on the order of 5%. However, themale steriles clustered together. In the center ofthese clusters, the male sterile plants made up to60% of the present plants. Clustering of malesterile plants is also observed in Beta �ulgaris.

To model gynodioecy, we consider an IPS withtwo states, or types of individuals. Space is repre-sented by the two-dimensional integer lattice Z2.We think of this as an exploiter–victim systemwhere type one, the hermaphroditic plant (thevictim), can exist in the absence of type two, the

male sterile plants (the exploiter), and the malesterile plants are dependent on the hermaphroditicplants for pollen. Victims give birth onto vacantsites within their seed dispersal neighborhood,R1,disp, at rate �1. Exploiters give birth onto va-cant sites within their seed dispersal neighbor-hood, R2,disp, at rate proportional to the numberof victims within whose exploitation neighbor-hood, Rexploit, they reside. The constant of propor-tionality is �2. For each type, death occurs at aconstant per capita rate, which we can set to oneby a suitable choice of time scales.

We simulated this system on the stochasticsimulator, S3. In Fig. 1, each point on the grid iseither vacant (white), or occupied by an exploiter(that is, male sterile plant) (black) or victim (thatis, hermaphroditic plant) (grey). Initial conditionsare the product measure with equal frequencies.On the left, seed dispersal is longer than pollendispersal, and we see a well-mixed population atequilibrium. On the right, pollen dispersal islonger than seed dispersal, and we see clusteringof male steriles.

We make two observations. First, in order forboth species to coexist, the exploiter’s birth rate

Fig. 1. �1=10, �1=2.5. �2=10. Rexploit=1, R1,disp=R2,disp=5 Rexploit=5 (left), R1,disp=R2,disp=1 (right). Interacting particlesystem simulation. Black represents exploiter, grey represents victims, and white represents empty space. On the left, the dispersalneighborhood is larger than the exploitation neighborhood, and we see a well mixed population. On the right, the exploitationneighborhood is larger than the dispersal neighborhood, and we see clustering of exploiters


Fig. 2. Numerical solutions of the integral differential equations. On the left, dispersal kernels are chosen so that the spatiallyconstant solution is stable, and on the right so that the spatially constant solution is unstable. In both cases, initial conditions areequal to the constant equilibrium plus noise.

must be significantly higher than the victim’s birthrate. Second, if the exploitation neighborhood,Rexploit, is much larger than the dispersal neigh-borhood, Ri,dispersal, then the exploiters tend tocluster together in groups.

It would be quite difficult to prove the phe-nomenon of clustering observed in Fig. 1 for longrange pollen and short range seed dispersal in theIPS. Instead, in (Anderson et al., 2001), we ap-proximated the IPS by a deterministic system thatexhibits the same phenomenon. This approxima-tion removes spatial correlations that are due tothe close vicinity of parents and offspring in theIPS framework. The deterministic system is a longrange approximation of this model (see below)that resulted in the following set of integro-differ-ential equations

�

�tu(x, t)=r1[1−u(x, t)−�(x, t)]

[k2×u(x, t)[k1×�(x, t)]]−u(x, t)

�

�t�(x, t)=r2[1−u(x, t)−�(x, t)][k2×�(x, t)]

−�(x, t)

where u represents the male sterile plants, �represents hermaphrodites, and k1(x) and k2(x)

represent pollen and seed dispersal, respectively.We showed that this system has a non-trivialequilibrium when r2�1 and r1�r2

2/(r2−1) andgave conditions under which the spatially con-stant solution is unstable under spatial perturba-tions. The conditions are phrased in terms of thedispersal kernels k1 and k2 and they amount tothat the seed dispersal of species �, thehermaphrodites, must be shorter range than therange at which the pollen is available to species u,the male-steriles, and that the dispersal of thepollen must be platykurtic, that is, has broadshoulders and thin tails. Fig. 2 shows two numer-ical solutions of the above integral–differentialequations. In the first, the dispersal kernels arechosen so that the spatially constant solution isstable, and in the second they are chosen so thatit is unstable, giving rise to spatial patterns. Therequirement that the pollen dispersal range isshorter than the seed dispersal range is as in thecorresponding IPS.

In the following, we will investigate a class ofinteracting particle systems together with their‘long range’ approximations and find conditionsunder which spatially heterogeneous patternsmight appear. Their biological significance andtheir implications for simulation studies are dis-cussed below.


2. Integro-differential equations

As mentioned above, IPS are difficult to ana-lyze; just proving that two species coexist in aspecific example is often challenging and coexis-tence proofs typically yield little insight into prop-erties, such as densities or correlations. Fewrigorous results about spatial patterns in IPS areavailable (Bramson and Griffeath, 1980; Neu-hauser, 1992; Neuhauser and Pacala, 1999).

An alternative approach that yields analyticallytractable models is to consider a related partial-differential or integro-differential equation. Theseequations are derived from IPS by either intro-ducing fast stirring (Durrett and Neuhausser,

1994), which results in a partial differential equa-tion, or by taking long range limits (Neuhauser,1994) (see below), which results in an integro-dif-ferential equation. The conceptual difference be-tween the two alternatives is that in the partialdifferential equation approach, individuals can bethought of as taking a large number of small stepsbetween reproduction events, whereas in the inte-gro-differential equation approach, individualstake one large step right after they were born butstay put afterwards. Which of these two ap-proaches is more appropriate depends on thespecific application. When modeling plant popula-tions, integro-differential equations are more ap-propriate. Formally, one can think of the longrange limit as letting the distance between sites inthe lattice (that is the spacing) become infinitesi-mal small while the other length scales involved(for example dispersal distances) remain constant.A consequence of this limit is that neighboringsites become independent, which greatly simplifiesthe analysis, but removes spatial effects that aredue to spatial correlations. It leaves, however,those effects that are due to space but do notdepend on spatial correlations.

2.1. General system

We are interested in two-species IPSs, whereindividuals of each type are recruited to a site at arate that may depend on the densities of bothspecies within a neighborhood of that site andthat is specified by a dispersal kernel, k1(·) forspecies 1 and k2(·) for species 2. The recruitmentrates for the two species may differ. Whether ornot the recruited individual can establish itselfdepends on the local density at that site.

Individuals of either species die at rate 1. Thelong range limit of this particle system yieldsintegro-differential equations of the form

The functions a(U, V), b(U, V), c(U, V), andd(U, V) are such that this system is well defined.We also assume that k1(·) and k2(·) are probabilitydispersal kernels, that is they are positive, even,and � k1(x)dx=� k2(x)dx=1. (This implies thattheir Fourier transforms are bounded between−1 and 1, a fact that we will use later.)

We will analyze these integro-differential equa-tions using linear stability analysis to determineconditions under which the spatially homoge-neous solution becomes unstable under spatialperturbations and one might expect patternformation.

The first step in the analysis is to consider thespatially homogenous system in which U(x, t) andV(x, t) do not depend on x. In this case, Eqs. (1)and (2) simplify to a set of ordinary differentialequations, known as mean field equations,

dUdt

=a(U, V)b(U, V)−U (3)

dVdt

=c(U, V)d(U, V)−V (4)

Assume that we have an interior equilibrium,i.e. (U*, V*) such that U*�0, V*�0, and

�

�tU(x, t)=a(U(x, t), V(x, t))

�k1(x−y)b(U(x−y, t), V(x−y, t))dy−U(x, t) (1)

�

�tV(x, t)=c(U(x, t), V(x, t))

�k2(x−y)d(U(x−y, t), V(x−y, t))dy−V(x, t) (2)


U*=a(U*, V*)b(U*, V*)

V*=c(U*, V*)d(U*, V*)

Using standard techniques, we linearize equa-tions Eqs. (3) and (4) about U* and V*. In matrixform this linearization can be written as

ddt�u

�

n=M

�u�

nwhere M is the Jacobian of the vector field in Eqs.(3) and (4) and u and � are small perturbationsabout U* and V*. For the solution (U*, V*) tobe stable, the eigenvalues of the matrix M mustboth be negative. This is equivalent to insistingthat the trace of M be negative and the determi-nant of M be positive.

If these conditions are met, the solution (U*,V*) will be stable. However, this mean field anal-ysis tells us nothing about the spatial distributionof U and V in the full-blown spatial model. Forthis, we need to analyze the spatial equations Eqs.(1) and (2).

2.1.1. Spatial analysisWe think of U* and V* as spatially constant

equilibrium solutions U*(x)=U*, V*(x)=V* ofthe spatial model given by Eqs. (1) and (2), whichare equal to the locally stable equilibrium solu-tions of the non-spatial system Eqs. (3) and (4).We now wish to know under what conditions thespatially constant solutions will be locally stableunder perturbations in space, u and �.

The system can be written as

�

�t(U*+u)

=a(U*+u, V*+�)[k1×b(U*+u, V+�)]

− (U*+u)

�

�t(V*+�)

=d(U*+u, V*+�)[k2×d(U*+u, V+�)]

− (V*+�)

Linearizing these equations about U* and V*and Fourier transforming the system, yields

�

�t�u(w, t)

�(w, t)n

=J�u(w, t)

�(w, t)n

where u and � are the Fourier transformed per-turbations and J is the matrix with entries

j11=�

k� 1(w)a(u, �)�

�ub(u, �)+b(u, �)

�

�ua(u, �)−1

nj12=

�k� 1(w)a(u, �)

�

�ub(u, �)+b(u, �)

�

�ua(u, �)

nj21=

�k� 2(w)c(u, �)

�

�ud(u, �)+d(u, �)

�

�uc(u, �)

nj22=

�k� 2(w)c(u, �)

�

�ud(u, �)+d(u, �)

�

�uc(u, �)−1

nwhere each term is evaluated at u=U* and �=V*

This is a system of differential equations for theFourier modes of the spatial perturbations. Thespatially constant solution will be stable to spatialperturbations if zero is a stable equilibrium ofthese equations for all values of the wave number�, that is, if both eigenvalues of J are negative. Asabove, this will be true if and only if the trace ofJ is negative and the determinant is positive.

Note that if the equations are solved on a finiteinterval, then only certain wave numbers are per-missible. However, since we are only interested inthe stability of the spatially constant solution, wewill ignore this complication in the following.

2.2. Specific examples

We will consider three related exploiter–victimexamples where two species compete for space.One of the species (the exploiter) requires thepresence of the second species (the victim) inorder to reproduce. We think of the second spe-cies as providing some resource for the first (forexample pollen for a male sterile member of agynodioecious population provided by ahermaphrodite). The second species is unaffectedby the first, except in competition for space (tocontinue the example of gynodiocy, this wouldimply that reproduction of hermaphrodites is notpollen limited).


The three examples differ in whether the re-source is available locally or at long range andwhether subsequent seed dispersal (that is afterthe resource has been made available) is local orlong range.

We are interested in conditions that would yieldspatially homogeneous solutions that are unstableunder spatial perturbations. We will see that forthis to occur, resource availability and dispersal ofthe two species have to occur at different spatialscales, one long range, and the other local. Fur-thermore, only long range kernels that areplatykurtic produce such unstable solutions.

In the first example, the resource provided bythe second species is available at long range andsubsequent seed dispersal is local. In the secondexample, the resource is available locally and thesubsequent dispersal is long range, and in thethird example, both the resource availability andthe subsequent dispersal are long range.

In the first two examples, the spatially homoge-neous solution becomes unstable provided thateither the resource or the dispersal of the speciesthat requires the resource is long range andplatykurtic (but not both) and the dispersal kernelof the species that provides the resource is shortrange. In the third example, where both the re-source and the dispersal kernel of the species thatrequires the resource are long range, all spatiallyconstant solutions are shown to be linearly stable.

2.3. Example 1a

In the first case, the integro-differential equa-tions that follow from taking the long range limitare

�

�tU(x, t)

=r1(1−U(x, t)−V(x, t))U(x, t)[k1×V(x, t)]

−U(x, t)

�

�tV(x, t)=r2(1−U(x, t)−V(x, t))[k2×V(x, t)]

−V(x, t)

In biological terms, the equations can be inter-preted in the following way. Plants of type V give

off seeds at rate r2. A seed given off by a plant oftype V at location y will land at site x withprobability equal to k2(x−y)dy. If x is vacant anew plant of type V is born at x ; if not, the seedperishes.

Plants of type U give off seeds at rate r1. Theseseeds are dispersed locally. That is, there must bea vacant site next to the plant producing the seed.In addition for its seeds to be viable, the plantneeds a resource (for instance, pollen) from aplant of type V. If there is a plant of type V at sitey, its resource will be dispersed to site x withprobability equal to k1(x−y)dy. Both plants dieat a constant per capita rate (scaled to 1).

The non-spatial (mean field) version of thismodel is

U� =r1(1−U−V)UV−U

V� =r2(1−U−V)V−V (5)

which has equilibrium solutions

U*=1−1r2

−r2

r1

, V*=r2

r1

Recall that in order for these equilibrium solu-tions to be stable both eigenvalues of

M=�−r2U* U*/V*−r2U*

r2V* −r2V*n

the Jacobian of Eq. (5), must be negative, which isequivalent to having the trace be negative and thedeterminant be positive.

Demanding that the trace is less then zero isequivalent to

r2�1

This is the same condition as for plants of typeV to survive in the absence of plants of type U ;the birth rate has to exceed the death rate. De-manding that the determinant is greater then zerois equivalent to U*�0, or

r1�r2

2

r2−1

These are the conditions for coexistence, that is,a stable, non-trivial equilibrium solution. The sec-ond condition is responsible for the observation


that the birth rate of the exploiter must exceedthat of the victim by a positive amount in orderfor coexistence to occur.

In order to analyze the spatial case, we lin-earize about the spatially constant solution

U*(x)=1−1r2

−r2

r1

V*(x)=r2

r1

The Jacobian of the spatial system (the matrix Jabove) is

J=�−r2U* −r2U*+ (U*/V*)k� 1(�)

−r2V* k� 2(�)−r2V*−1n

The trace of the matrix is

Tr=k� 2(�)−r2

which will be negative as long as the equi-librium solution of the mean field model is sta-ble. The determinant is

D=r2U*(1+k� 1(�)−k� 2(�))

If r1 and r2 are such that the equilibrium so-lution of the mean field model is stable, then

k� 2(�)�1+k� 1(�) (6)

is a necessary and sufficient condition for theinstability of the spatially constant solution.

We are now in a position to make two obser-vations. First, patterns will arise if the dispersalof offspring of V is of a shorter range than thatat which the resource is available to species U.

Second, because k� 2(�)�1, k� 1(�) must be neg-ative in order for k� 2(�)�1+k� 1(�) One broadclass of dispersal kernels with the property thatk� 1(�)�0 for some � is platykurtic kernels, thatis, kernels with ‘broad shoulders’ and ‘thin tails’.A prominent example of this class is the uni-form kernel over the interval (−R,R),k(x)=1/2R for −R�x�R and=0, otherwise.Dispersal kernels such as Gaussian or exponen-tial kernels have positive Fourier transforms andwill not lead to pattern formation in this sys-tem.

In computer simulations, dispersal kernels areoften taken to be uniform over a local neighbor-hood and so for better or for worse, this condi-tion is automatically satisfied and patterns could

be observed that are sensitive to the choice ofdispersal kernels.

2.4. Example 1b

Second, consider the case where the resourceneeds to be close to the parent that requires theresource and subsequent seed dispersal of theoffspring is long range:

�

�tU(x, t)

=r1(1−U(x, t)−V(x, t))[k1×U(x, t)V(x, t)]

−U(x, t)�


−V(x, t)

The mean field equations, the equilibrium so-lutions and the stability conditions are identicalto those in Example 1a.

For the spatial case,

J=�k� 1(�)−r2U*−1 r2U*+ (U*/V*)k� 1(�)

−r2V* −r2V*+k� 2(�)−1n

The trace of this matrix is

Tr= −r2(U*+V*)+k� 1(�)+k� 2(�)−2

which is negative. The determinant is

D= (1−k� 2(�))(1−k� 1(�))

+r2U* [1+k� 1(�)−k� 2(�)]+ (1−k� 1(�))r2V*

This time, Eq. (6) is a necessary (but notsufficient) condition for the determinant to benegative. Furthermore, we see that if

r1�3r2

2

r2−1

then the determinant will be negative for appro-priate kernels. Again, we observe that the dis-persal of offspring of species V must be shorterrange than the availability of the resourceneeded by U (i.e. k2(x) must be shorter rangethan k1(x)), and k1 must be a platykurtic disper-sal kernel.


2.5. Example 1c

Finally, consider the situation where the re-source that species U requires for reproductionmust be available long range and subsequentseed dispersal is close to the resource. This situ-ation is described by the integro-differentialequations

�

�tU(x, t)

=r1(1−U(x, t)−V(x, t))V(x, t)[k1×U(x, t)]

−U(x, t)

�


−V(x,t)

The mean field equations for this example arethe same as for example 1a. The equilibriumsolutions as well as the stability conditions forthose solutions will, of course, be identical aswell.

The Jacobian of the spatial system is

J=�k� 1(�)−r2U*−1 U*/V*−r2U*

−r2V* −r2V*+k� 2(�)−1n

The trace of this matrix is negative, and sothe spatially constant solutions will be locallystable to spatial perturbations if and only if thedeterminant is negative as well.

However, comparison with the mean fieldmodel above shows that the matrix can be writ-ten as

J=M+�k� 1(�)−1 0

0 k� 2(�)−1n

Therefore, it is impossible to have the equi-librium solutions of the mean-field model be sta-ble while the spatially constant equilibriumsolutions of the spatial model are unstable. Inother words, patterns do not develop in thismodel.

3. Conclusion

3.1. Summary and a warning about patternformation in IPS

Spatial simulations of ecological systems arewidespread. Common frameworks for such sim-ulation studies are interacting particle systems.A typical case involves a number of differentspecies that interact with each other in complexways. The dynamics are described by local inter-actions that are based on individuals. In manycases, the local interactions are chosen so thatthe neighbors involved are within a fixed rangeand contribute equally (uniform interactionneighborhood). This choice of neighborhood isnot driven by realism but rather by convenience.

Our analysis demonstrates that differences inthe interaction/dispersal neighborhood can resultin spatially heterogeneous solutions. To under-stand why these patterns arise, note that speciesU is resource limited. If species U were not re-source limited, it would outcompete species Vsince it has the higher birth rate. Without re-source limitation, the system would be a multi-type contact process (Neuhauser, 1992) forwhich we know that the species with the higherbirth rate outcompetes the other species (pro-vided both species have the same death rates).As long as species U is not resource limited, itcan outcompete species V locally. Conditionsthat favor this are provided by the first twoexamples.

In Example 1a, the species that requires theresource reproduces locally. As long as speciesV is within reach and abundant enough, speciesU can acquire the resource and outcompete spe-cies V locally. But there is a limit to how largea cluster of species U can grow since individualsof species U become resource limited.

In Example 1b, species U has a higher birthrate than species V provided species V is abun-dant enough and within reach of species U. Thismeans that locally, as long as both species arepresent, they behave like a multitype contactprocess in which one species outcompetes theother. Clusters of species U can form until spe-cies V reaches a low enough density and species


U becomes resource limited. This limits the size ofthe clusters that form.

In Example 1c, no such patterns form since theoffspring of the species that requires the resourcelands right next to the resource, which preventsformation of clusters.

The mechanism for pattern formation in ourexploiter–victim systems is akin to pattern forma-tion in models for neurological activities wherethe spatial dynamics are driven by activation andinhibition (this is discussed extensively in Chapter16 of (Murray, 1993). The interaction kernels inneurological models are such that they are posi-tive where activation occurs and negative whereinhibition occurs; this is a classical mechanism forgenerating spatially heterogenous patterns.

Our study has demonstrated that the choice ofuniform interaction within a finite neighborhoodis particularly prone to exhibiting spatial patternsthat are not robust with respect to the shape ofthe interaction neighborhood. This is cause forconcern since robustness cannot be simply testedby increasing the neighborhood, as is often done,or by changing the shape of the neighborhoodfrom a square to a circle (Molofsky et al., 2002).Instead, a theoretical analysis, such as the one weused in this paper is required to determine thecause of observed patterns.

3.2. Biological rele�ance

Having made the point that spatial patterns insimulations might not be robust with respect tothe choice of interaction neighborhood, there re-mains the question whether these patterns havebiological significance.

Our analysis demonstrated that patterns appearonly if the dispersal kernel is sufficiently platykur-tic. Often, dispersal kernels are leptokurtic (Lewis,1997), that is they have narrow shoulders andthicker tails than platykurtic kernels, thus they arejust the opposite of the types of kernels for whichwe observed patterns.

However, there might be scenarios where ker-nels are effectively platykurtic. For instance, habi-tat destruction can truncate a dispersal kernelwhen the patches are small enough and suffi-ciently far spread apart so that dispersal is close

to uniform within a patch and few migrants reachneighboring patches.

We have demonstrated that these patterns arenot robust with respect to the shape of the disper-sal kernel. The question still remains whether ornot they are spurious artifacts. We believe that themechanism for pattern formation discussed in thispaper should be considered as a possibility whenpatterns of this type are observed in exploiter–victim interactions. However, it is important tokeep in mind that the presence of patterns de-pends on the shape of the dispersal kernels and,therefore, the modeling of the dispersal kernelsneeds to be done particularly carefully when mod-eling a real system in order to draw valid conclu-sions about the mechanism of pattern formation.

Besides the example of gynodioecy (Andersonet al., 2001) mentioned in the introduction, thereis one other model where this phenomenon hasreceived attention. In (Krone and Neuhauser,2000), epidemic models were studied that exhib-ited patchiness of healthy and diseased individualscaused by the same mechanism, namely differentdispersal ranges of the host and the disease andsufficiently platykurtic kernels. When the spatiallyconstant solution was unstable under spatial per-turbations in this epidemic model, epidemic out-breaks of the disease were observed. When thespatially constant solution was stable, the diseaseappeared to be endemic. If effectively platykurtickernels drive the dynamics in fragmented habitats,this would suggest that diseases might be moresevere in fragmented habitats than in continuoushabitats.

Acknowledgements

Supported by the National Science FoundationDMS-0072262.

Appendix A. These are not Turing patterns

Interacting particle systems are often analyzedusing reaction diffusion equations which arederived by a taking the fast stirring limit of theparticle system (Durrett and Neuhausser, 1994).


Intuitively, the fast stirring limit is thought of asadding an extra rule to the IPS dynamics in whichthe states of adjacent sites are exchanged at a fastrate.

One interesting aspect of the reaction diffusionequations that arise in this way is that they cangive rise to a type of spatial irihomogeneity calleda Turing pattern. An overview of these types ofspatial patterns is given in (Murray, 1993).

For our system, a reaction diffusion modelwould look like

U� =D1Uxx+a(U, V)b(U, V)−U

V� =D2Vxx+c(U, V)d(U, V)−V

Assume that the linearization of the mean fieldmodel about the interior solution (U*, V*) isgiven by

ddt�u

�

n=�m11 m12

m21 m22

n�u�

nand that this interior solution is stable, i.e. thatTr(M)�0, and that Det(M)�0. Furthermore,we will assume that m11�0 and m22�0 (May,1975).

If we linearize the reaction diffusion modelabout a the spatially constant solution, U(x)=U*, V(x)=V*, and then Fourier transform thesystem we have

ddt�u

�

n=�m11−D1�

2 m12

m21 m22−D2�2

n�u�

nFor this matrix

Trace=m11+m22− (D1�2+D2�

2)

Det=m11m22−m21m12−�2(D1m22+D2m11)

+�4D1D2

It must now be the case that Trace�0 and Det�0. Therefore, both eigenvalues are negative, andthe spatially constant solution is linearly stable forany values of D1, D2. This demonstrates thatTuring patterns would not appear in the modelsdiscussed here.

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Patterns in spatial simulations—are they real?