Techniques of spatially explicit individual-based models:...

Ecological Modelling 150 (2002) 55–81

Techniques of spatially explicit individual-based models:construction, simulation, and mean-field analysis

Luděk Berec *Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic,

Faculty of Biological Sciences, Uni�ersity of South Bohemia, Branišo�ská 31, 370 05 C� eské Budĕjo�ice, Czech Republic

Received 23 February 2001; received in revised form 24 August 2001; accepted 28 October 2001

Abstract

The main focus of this article is to present and relate four different frameworks in which spatially explicitindividual-based models (IBMs) can be defined. These frameworks differ in the way space and time are modeled; eachcan be treated either discretely or continuously. The emphasis is put on constructing and simulating one of thesimplest single-species IBMs in each spatio-temporal framework, discussing some of their technical subtleties, andderiving corresponding mean-field models when the homogeneous mixing conditions are assumed to hold. The fourframeworks are more supplements than competitors. Since at almost every step of IBM construction severalalternatives are a priori plausible I discuss the most important ones in more details. This article seems to be the firstattempt to collect and synthesize information of this kind that is scattered over the literature. © 2002 Elsevier ScienceB.V. All rights reserved.

Keywords: Cellular automata; Mean-field model; Methodology; Population dynamics; Space; Time

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1. Introduction

Not only the ‘What to study?’ and ‘What hasbeen found?’ questions are vital for a good scien-tific work. Rather, the ‘How to solve the problemunder consideration?’ question forms the in-evitable ‘how’ bones for the ‘what’ flesh. Appro-priate methodology may reveal important insightsinto a seemingly banal problem. On the otherhand, techniques that are not adequate enoughmay drown topics of prime importance.

Individual-based approach to modeling popula-tion dynamics, one of the main streams of today’stheoretical ecology, has already been reviewedmany times (Huston et al., 1988; Hogeweg andHesper, 1990; Lomnicki, 1992; DeAngelis et al.,1994; Judson, 1994; Uchmanski and Grimm,1996; Grimm, 1999). The point of interest in allthese works has been either the role of individual-based models (IBMs) in ecology as a whole or anattempt to discuss potentials and summarize pre-dictions of the existing IBMs. The present articleaims at reviewing fundamental techniques under-lying construction, simulation, and mean-fieldanalysis of spatially explicit IBMs, including

* Tel.: +420-38777-2327; fax: +420-3853-00398.E-mail address: [email protected] (L. Berec).

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L. Berec / Ecological Modelling 150 (2002) 55–8156

means of summarizing and visualizing vastamount of model data generated. Non-spatialIBMs also exist (Botkin et al., 1972; DeAngelis etal., 1979; Grist and des Clers, 1999) but are notthe subject of this article.

Spatially explicit models of population dynam-ics can be classified according to whether popula-tion sizes, space, and time are treated as discreteor continuous entities (Table 1). For the purposesof this review, I define IBMs as such models inwhich a discrete individual that has at least onefeature unique is the fundamental modeling unit.This is the case of the last four groups enlisted inTable 1; the feature that makes each individualunique is simply at least its spatial location. I referto just these four model types as the spatiallyexplicit IBMs. There is some controversy in theliterature about what should be considered theIBM par excellence (Murdoch et al., 1992; Uch-manski and Grimm, 1996; Grimm, 1999). TheIBM definitions and underlying arguments pro-posed in these articles are philosophical; tech-

niques used to build and work with such modelsare not affected by them at all. I note here thatthe concept of ‘individual’ need not always coin-cide with that of ‘individual organism’. Modelswith more general population units such as localpopulations (Dytham, 1994), ant colonies (Brittonet al., 1996), and bird flocks (Fahse et al., 1998)do not differ technically from models treatingindividual organisms.

Time seems to be mature enough to collect andsynthesize chips of information scattered over amultitude of articles, and review techniques un-derlying spatially explicit IBMs. In all four modelgroups of interest the methodologies are relativelywell established though elaborated to various de-grees. They have been used to address both theo-retical and practical aspects of populationdynamics, working with both single-species andmulti-species populations of both motile and ses-sile organisms. I aim at extracting their commonconceptual and methodological background. Thismay help to reveal artifacts of particular modeling

Table 1Eight possibilities of how spatially explicit models can be classified according to whether population size (P), space (S), and time(T) are treated as discrete (D) or continuous (C) entities

Common label FormulationP S T References

System of difference eq.D Coupled mapDC Hassell et al. (1991), Kaneko, 1998latticesa

Reaction-dispersalCDC System of ODEs Levin, 1974; Takeuchi, 1996networks

C DC Reaction-dispersal Neubert et al., 1995; Kot et al. (1996), Veit and Lewis, 1996Integrodifference eq.models

C PDEs, integrodifferentialC Reaction-dispersal Okubo, 1980; Britton, 1986; Murray, 1990; Wilson, 1998Ceq.modelsb

Individual-basedD Set of rules de Roos, McCauley and Wilson; see the next sectionD Dmodelsc

Durrett and Levin; see the next sectionSet of rulesD Interacting particleD CsystemsNeighborhoodD Set of rules Pacala and Silander; see the next sectionDCmodels

C CD Spatial point Set of rules Bolker and Pacala; see the next sectionprocesses

Abbreviations, eq., equations; ODEs, ordinary differential equations; PDEs, partial differential equations.a Coupled map lattices can also be considered discrete-space IBMs under specific conditions, e.g. when the value at each site

represents mass of the plant individual present (Hendry et al., 1996); they are used to evolve individuals’ characteristics rather thantheir spatial distribution.

b Reaction–dispersal models defined in continuous time cover widely used reaction-diffusion models.c Some instances of individual-based models are sometimes referred to as (probabilistic) cellular automata or artificial ecologies.

L. Berec / Ecological Modelling 150 (2002) 55–81 57

approaches as well as areas of particular modelgroup applicability. Further on, unless specifiedotherwise, I refer to the spatially explicit individ-ual-based models as IBMs or simply models.

The comparison studies that have been pub-lished and involved IBMs (DeAngelis and Rose,1992; Durrett and Levin, 1994a; Wilson, 1998)exclusively concentrated on, say, an inter-ap-proach comparison (i-state distribution versus i-state configuration models in DeAngelis andRose, 1992; mean-field models, patch models, re-action–diffusion models, and interacting particlesystems in Durrett and Levin, 1994a; IBMs, de-terministic and stochastic reaction–dispersalmodels in Wilson, 1998). I propose an intra-ap-proach comparison study, contrasting differentmethodologies used to formulate IBMs, and wantto focus on technical means needed to run andanalyze them.

This article is intended as the first one in aseries of three. The next article that is currentlyunder preparation should review techniques (bothstatistical and analytical) used to analyze IBMswhen the homogeneous mixing conditions under-lying mean-field models do not hold. The finalarticle should present some of the software pack-ages available that were developed to simulatemyriads of interacting organisms of various spe-cies sharing the common environment, and dis-cuss some IBM representatives found in theliterature in view of the techniques reviewed inthe first two papers.

2. Simple single-species IBMs

All four spatio-temporal frameworks of IBMs,listed in the last four rows of Table 1, have moreor less established methodologies. In other words,for each framework there is a series of articles inwhich modeling aspects, that is, issues of modelconstruction, simulation, and analysis, are at leastas important as ecological problems studied.These series are due to de Roos, McCauley andWilson (D-space, D-time IBMs; de Roos et al.,1991; McCauley et al., 1993; Wilson et al., 1993,1995; McCauley et al., 1996; Wilson, 1996, 1998),Durrett and Levin (D-space, C-time IBMs; Dur-

rett, 1988, 1993; Durrett and Levin, 1994a,b,1998; Durrett, 1999), Pacala and Silander (C-space, D-time IBMs; Pacala and Silander, 1985;Pacala, 1986, 1987; Pacala and Silander, 1990),and Bolker and Pacala (C-space, C-time IBMs;Bolker and Pacala, 1999; Bolker et al., 2000). Therespective methodologies have been motivated byvarious ecological problems and elaborated tovarious degrees. To introduce them in a conciseand unified manner, I first construct and simulateprobably one of the simplest IBMs of single-spe-cies population growth in a two-dimensionalhabitat in each spatio-temporal framework, andthen attempt at making (some) general comments.I (subjectively) order the frameworks so that theirtechnical complexity increases. Sets of rules de-scribe performance and thus determine the fate ofevery single individual. The processes of mortal-ity, reproduction (including offspring dispersaland establishment), and movement (in case ofmotile organisms) drive evolution of populationsize (temporal pattern) and distribution (spatialpattern) in the environment.

2.1. D-space, D-time

The two-dimensional, physically homogeneousenvironment is modeled as a lattice of M×Nidentical square sites, with periodic boundaryconditions (BC; the left and right edges and thetop and bottom edges of the lattice are joinedtogether so that it forms the surface of a torus).Time runs in discrete steps. At any time step, atmost one individual is allowed to occupy eachsite. Initially, x0 (�M×N) individuals are scat-tered uniformly randomly over the lattice. Theprocesses that determine the fate of each individ-ual are as follows.

2.1.1. MortalityEvery time step, each individual dies with a

probability Pm (density-independent, uniformmortality).

2.1.2. ReproductionEvery time step, each individual gives birth to

one offspring with a probability Pr (asexualreproduction).


2.1.2.1. Offspring dispersal and establishment. Re-specting BC, the conceived offspring is instanta-neously and equiprobably placed to a site in asquare neighborhood (excluding the original site)of side 2e+1 (e=1, 2, …) centered at its par-ent’s site, and is discarded provided that anadult individual occupies the selected site. If twoor more offspring attempt to recruit to the sameadult-free site at the same time step, only one ofthem is uniformly randomly chosen and allowedto do it, the rest discarded. Thus, reproductionis density dependent. Any established offspringbecomes the adult able of reproduction at thenext time step.

2.1.3. Mo�ementEvery time step, respecting BC, each individ-

ual moves equiprobably to a site in a squareneighborhood (including the original site) of side2d+1 (d=0, 1, 2, …) centered at its currentsite unless another individual occupies the se-lected site. In case the individual cannot move itremains in its original site. The order in whichindividuals attempt to move is modeled as uni-form random. For sessile species, one may for-mally set d=0.

I order the demographic processes concur-rently in the actual IBM implementation. Thatis, the mortality and reproduction rules are ap-plied independently to each adult individual: thereproduction rule takes place before the deadindividuals are removed from the lattice, andthe established offspring are not subject to themortality rule at the actual time step. Bookkeep-ing is then made by superposing the lattices ofnewborns and surviving adults. Finally, themovement rule is applied to each individual ofthe updated population. The reproduction ruleincludes the so-called ‘collision rule’ which de-termines what happens if two or more events tryto simultaneously influence a site.

2.2. D-space, C-time

The introductory assumptions are identical tothe above model, yet time is considered to runcontinuously now. The model rules differ intheir formulation and are as follows.

2.2.1. MortalityEach individual dies at a rate m (density-inde-

pendent, uniform mortality).

2.2.2. ReproductionEach individual gives birth to one offspring at

a rate r (asexual reproduction).

2.2.2.1. Offspring dispersal and establishment. Re-specting BC, the conceived offspring instanta-neously disperses equiprobably to a site in asquare neighborhood (excluding the original site)of side 2e+1 (e=1, 2, …) centered at its par-ent’s site, and is discarded provided that anotheradult individual occupies the selected site. Thus,reproduction is density dependent. Any estab-lished offspring immediately becomes the adultable to reproduce.

2.2.3. Mo�ementRespecting BC, each individual moves at a

rate w equiprobably to a site in a square neigh-borhood (including the original site) of side2d+1 (d=0, 1, 2, …) centered at its currentsite unless another individual occupies the se-lected site. In case the individual cannot move itremains in its original site. For sessile species,one may formally set w=0 and/or d=0.

There is no place for explicit ordering of mor-tality and reproduction rules in this model.Rather, the ordering is determined by the actualoccurrence of particular events. In this way,even the movement rule is mingled with the oth-ers. The rules are more concise now as at eachcontinuous-time instant at most one event takesplace (with probability one) and no ‘collisionrules’ are thus required.

I make a few notes concerning the actualcomputer implementation of the model. First,events are said to occur at a rate a if the occur-rence times are described by a Poisson processwith the parameter a ; see, for example, Mangeland Clark (1988) for details. The important con-sequence is that the time intervals � between theoccurrence of two successive events are indepen-dent and identically exponentially distributedrandom variables: P [�� t ]=1−exp(−at),


where P [A ] stands for a probability of an event A.Second, the above rules prescribe that three Pois-son processes should run for each individual(mortality, reproduction, and movement); threerunning processes vanish when an individual dies,and three new processes appear when an offspringis successfully established on the lattice. Althoughusing techniques of object-oriented programmingwould enable one to keep track of multiple Pois-son processes for each individual, it is unnecessaryand would introduce unnecessary overhead. In-stead, the so-called ‘thinning of Poisson processes’technique can be applied which keeps one back-ground Poisson process only (Durrett, 1995). Thissingle process generates time instants at whichevents may occur. Let S=MN be the number oflattice sites and let the background Poisson pro-cess generate time instants at a rate cS, wherec�m+r+w. Upon each time instant generateda lattice site is uniformly randomly chosen; ifoccupied the individual dies with the probabilitym/c, gives birth to one offspring with the proba-bility r/c, attempts to move with the probabilityw/c, and does nothing with the probability 1−(m+r+w)/c. If the chosen site is empty, nothinghappens to the lattice configuration at that timeinstant. Thus, each site is independently trying tochange at the rate c, as cS×1/S=c, 1/S being theprobability that a particular site is uniformly ran-domly selected.

In the literature, one can find at least some ofthe rules defined from different viewpoints. Forexample, in the above single-species model onemay use the viewpoint of either an individual(above) or a vacant site (Durrett, 1999) to specifythe reproduction rule. The latter gives the follow-ing rule: vacant sites become occupied at a rater× f, where f is the fraction of occupied neigh-bors. Note that when individuals give birth at arate r, vacant sites are filled at a rate at most r,since births onto occupied sites are discarded.Simulation of this rule works as follows. Upon atime instant generated by the background Poissonprocess a lattice site is uniformly randomly chosenand if vacant one of its neighbors is picked atrandom; if this neighbor is occupied (this eventhas the probability f ) the vacant site is madeoccupied with the probability r/c. Analogous rule

duality can be found in an example from epi-demics. Sites can be vacant or occupied by suscep-tibles or infecteds. The local disease transmissionrule can be defined as: (i) infecteds emit disease ata rate r equiprobably to a nearest neighbor and ifthat site is occupied by a susceptible, it becomesinfected, otherwise nothing happens, or (ii) sus-ceptible individuals become infected at a rate r×f, where f is the fraction of nearest neighborsoccupied by infecteds. Such rules are substitutesand use of one or another has no simulation oranalytical consequences. Rather, the choice ismore a matter of biological plausibility andinterpretation.

2.3. C-space, D-time

The two-dimensional, physically homogeneousenvironment is modeled as a continuous plot ofsize M×N, with periodic BC. Time runs in dis-crete steps. Initially, x0 individuals (now not lim-ited a priori) are scattered uniformly randomlyover the plot. The processes that determine thefate of each individual are as follows.

2.3.1. MortalityEvery time step, each individual dies with a

probability Pm.

2.3.2. ReproductionEvery time step, each individual gives birth to

one offspring with a probability Pr.

2.3.2.1. Offspring dispersal and establishment. Theconceived offspring instantaneously disperses to aposition generated by an offspring dispersal ker-nel, specified by a probability density function(pdf). In the physically homogeneous environ-ment, this pdf is typically radial and non-increas-ing with increasing distance between the newoffspring position (x �, y �) and its parent’s position(x, y). I assume here that this pdf is uniform on acircle of radius � centered at the parent’s position.That is, I generate a point u from the uniform pdfon (0, � ] and a point � from the uniform pdf on[0, 2 �), and set:

x �=x+u cos �, y �=y+u sin �, (1)


respecting BC. The offspring dispersal kernel issupplemented by an establishment probability de-termining whether the offspring is actually estab-lished in the selected position or is discarded. Iassume that this probability depends on the num-ber n of individuals located in a circle of radius ecentered at (x �, y �), decreases linearly from unityat n=0 to zero at some n=E, and stays zero forall n�E. The newborns are not counted to n.Any established offspring becomes the adult ableof reproduction at the next time step.

2.3.3. Mo�ementEvery time step, each individual moves to a

position generated by a movement kernel. I as-sume here that this kernel is specified by theuniform pdf on a circle of radius � centered at thecurrent individual’s position (x, y). That is, Igenerate a point z from the uniform pdf on [0, � ]and a point � from the uniform pdf on [0, 2 �),and set:

x �=x+z cos �, y �=y+z sin � (2)

respecting BC. The movement kernel is supple-mented by a movement probability determining,for z�0, whether the individual actually movesto the selected position or remains in the originalone. I assume that this probability depends on thenumber n of individuals in a circle of radius dcentered at (x �, y �), decreases linearly from unityat n=0 to zero at some n=D, and stays zero forall n�D. If an individual moves, only its newposition counts to n. The order in which individu-als attempt to move is implemented as uniformrandom. For sessile species, one may formally set�=0.

As time is considered discrete and all individu-als act at once, the ordering of mortality andreproduction processes has to be specified exter-nally. I assume they are ordered concurrently asin the ‘D-space, D-time’ case. The movement ruleis applied after mortality and reproduction rulesare accomplished. No ‘collision rules’ are requiredhere as at each time step each location is affected(with probability one) by at most one event.

The uniform random initial distribution of x0individuals in the plot implies that the number0�n�x0 of individuals located in a region of

area 0�A�S (=MN) is binomially distributedwith the event probability A/S,

P [there are n individuals in an area A ]

=�x0

n��A

S�n�

1−AS�x0−n

(3)

This probability distribution is valid no matterwhat the shape of the region is and where theregion is located within the plot. If S�� andx0�� so that x0/S is (or converges to) a con-stant, then A/S�0 for a fixed A, and the proba-bility that a region of area A�0 contains n�0individuals is Poisson distributed in this limit:

P [there are n individuals in an area A ]

=�n

n !exp(−�), �=�0A, (4)

where �0=x0/S has the meaning of initial popula-tion density.

An important technical problem in model im-plementation is to identify each individual’sneighbors in a relatively short time; Pacala andSilander (1985) suggested a very efficient datastructure to achieve this goal.

2.4. C-space, C-time

The introductory assumptions are identical tothe previous model, yet time is considered to runcontinuously now. The model rules differ in theirformulation and are as follows.

2.4.1. MortalityEach individual dies at a rate m.

2.4.2. ReproductionEach individual gives birth to one offspring at a

rate r.

2.4.2.1. Offspring dispersal and establishment. Re-specting BC, the produced offspring instanta-neously disperses to a position generated by anoffspring dispersal kernel, and an establishmentprobability is used to decide whether the offspringis actually established in that position or is dis-carded. I assume both the kernel and the proba-bility to be the same as in the previous model.


Any established offspring immediately becomesthe adult able to reproduce.

2.4.3. Mo�ementRespecting BC, each individual moves at a rate

w into a position generated by a movement ker-nel, and stays there with a movement probability;otherwise, it remains in its original position. Boththe kernel and the probability are assumed to bethe same as in the previous model. Bolker et al.(2000) simulated plant population growth (nomovement) as follows. ‘Starting with a randominitial distribution of individuals, the simulatortakes small time steps to approximate a continu-ous-time process. In each time step, the simulatorpicks pseudo-random numbers to see if each planthas died or reproduced. If it has reproduced, thesimulator picks a random point from the dispersalkernel to see where its offspring disperses to. Itthen calculates the local density around that pointto determine the offspring’s establishment proba-bility, and picks another random number to see ifit establishes’.

There is no a priori need to approximate con-tinuous time by discrete steps. Last but not least,the model then looses its explicit ‘C-time’ charac-ter. One may rigorously simulate this IBM byrunning a background Poisson process with den-sity-dependent rate x(m+r+w), x being the ac-tual population size at the time an event isexpected to occur. Once a time instant is gener-ated an individual is uniformly randomly chosen:it dies with the probability m/(m+r+w), givesbirth to one offspring with the probability r/(m+r+w), and attempts to move with the remainingprobability w/(m+r+w). Reproduction/move-ment event is followed by generating a position inthe environment according to the respective ker-nel where the offspring disperses/adult moves.Finally, local density around the landing pointdetermines the offspring establishment/adultmovement probability.

2.5. Output �isualization

An appropriate summarization and visualiza-tion of the large amount of model data generatedby IBMs is important to a priori assess popula-

tion dynamics and, in turn, formulate and testhypotheses and/or select adequate tools for amore rigorous analysis. An obvious entity to lookat is temporal course of population size. Spatiallyexplicit models are appreciated for generating spa-tial patterns which evolve in time. These patternscan be seen by taking snapshots of the environ-ment at either all times population size changes orat specific times a distance apart; see Wilson et al.(1995), Berec et al. (2001). A kind of picture beingsomewhere in between these two extremes andcreated for ‘D-space, D-time’ systems, is the xt-plot of Wilson (1998); it is built up by appendinga fixed but uniformly randomly chosen row of thelattice to the bottom of the picture each time step.Analogous ‘spatio-temporal’ pictures can be de-veloped for the other frameworks, too. For ‘C-space, D-time’ systems, one may superimpose alattice on the continuous plot and follow thext-plot technique, with different numbers of indi-viduals in discrete sites corresponding to differentlevels of grey color. ‘C-time’ IBMs may be sam-pled at regular time instants, for example.

To show how model outcomes depend on vari-ous parameter values, one may exploit diagramsof Fryxell and Lundberg (1994). A grid of pointsis laid on a two-dimensional parameter space andthe model is run for each of the selected combina-tions. Different marking of the grid points can beused to distinguish different types of model be-havior (e.g. highly probable extinction, moder-ately probable extinction, lowly probableextinction (Berec et al., 2001), Grimm (1999) useda slight generalization of this type of picture inwhich the same marks were of different sizesdepending on the strength of the observed phe-nomenon (e.g. probability of extinction in the‘moderately probable extinction’ group).

2.6. A short discussion

At the qualitative level, all model formulationsand simulation outputs are essentially the same.The models describe behavior of each individualorganism in a spatially explicit habitat, and in-clude the fundamental processes. Fig. 1 showssigmoidal population growth under all four spa-tio-temporal frameworks. The most rapid growth


Fig. 1. Single-species population growth curves corresponding to the four spatio-temporal frameworks. (A). D-space, D-time;M=128, N=128, x0=10, Pm=0.05, Pr=0.5, e=1; simulation under the homogeneous mixing conditions (HM) virtuallycoincides with the solution of the mean-field model (5) (dotted line). (B). D-space, C-time; M=128, N=128, x0=10, m=0.05,r=0.5, w=1, c=1.6, e=1; HM virtually coincides with the solution of the mean-field model (8) (dotted line). (C). C-space, D-time;M=128, N=128, x0=10, Pm=0.05, Pr=0.5, �=3, e=2, E=11, d=2, D=11; HM virtually coincides with the solution of themean-field model (9) (dotted line). (D). C-space, C-time; M=128, N=128, x0=10, m=0.05, r=0.5, w=1, �=3, e=2, E=11,d=2, D=11; HM virtually coincides with the solution of the mean-field model (12) (dotted line).

is achieved under the homogeneous mixing condi-tions (see the next section), and the rate of ap-proach of the ‘stable’ population size decreaseswith decreasing the movement rate of individuals.The explanation for this behavior is as follows.The slower is the movement the closer offspringstay relative to their parents. As a consequence,spatial clustering evolves in the environmentwhich results in a decreased reproductive successwithin the clusters due to overcrowding. The re-

productive success on the cluster boundaries ishigher and the clusters grow. Eventually, the habi-tat is filled and the number of individuals ‘stabi-lized’ at a carrying capacity. Pacala (1986),Tilman et al. (1997), and others reported a de-crease in ‘equilibrium’ population size for smallmovement rates. Law and Dieckmann (2000)demonstrated that the carrying capacity may belarger or smaller than that obtained under thehomogeneous mixing conditions, and that it de-


pends on an intricate interplay between intraspe-cific competition and movement in single-speciesmodels. I observed an increase in the ‘equilibrium’population size with respect to the mean-fieldcarrying capacity with decreasing movement ratein both ‘C-space’ models.

At the technical level, no two models are thesame. Table 2 summarizes the fundamental tech-nical differences between ‘D-time’ and ‘C-time’modeling frameworks. I used the concurrent or-dering of demographic processes, followed bymovement, in the above ‘D-time’ IBMs. There arealternative possibilities. The processes of mortal-ity, reproduction, and movement may be orderedin any sequential manner, and the ordering evenmade independent for each individual. I discussthis issue in more details below.

Likewise, Table 3 summarizes the technical dif-ferences between the ‘D-space’ and ‘C-space’frameworks. The ‘interaction’ characteristic de-serves some explanation. In ‘D-space’ models cov-ering more individual types (such as predatorsand prey in de Roos et al. (1991) or males andfemales in Berec et al. (2001)) the interaction isoften direct: individuals of both types have tooccupy the same site for the interaction to takeplace. In analogous ‘C-space’ models an interac-tion neighborhood has to be defined (indirectinteraction) as two individuals occupy the sameposition with zero probability. In the above sin-gle-species IBMs, offspring establishment andadult movement demonstrate direct interactionbetween individuals in ‘D-space’ models and indi-rect interaction in ‘C-space’ models.

3. Mean-field analysis

Apart from their significant contribution to theissues of spatial pattern formation, IBMs providea way to determine population-level consequencesof specific individual-level behavior. In order tobe reliable, computer simulations must be runrepeatedly to provide information about average,or typical, population responses. Given often alarge dimensionality of model parameter space,computer simulations are more suitable to addressspecific questions about the model rather thanuncover complete model behavior. Mean-fieldmodels reside on the other side of the individuallyoriented modeling spectra: they are analyticalmodels that take no account at all of space andpossible individual variability. As such, construc-tion of mean-field models forces one to admitassumptions about individual biology that arerarely plausible in natural systems; I formulatethese ‘homogeneous mixing conditions’ in thissection. Mean-field models take the form of (asystem of) ordinary differential/difference equa-tions (ODEs). This brings at least one advantageof building them: an agreement between mean-field models and IBMs simulated under the homo-geneous mixing conditions provides an importantstarting point from which to explore complexitiesbrought about by assumptions destroying these

Table 2A technical comparison of ‘D-time’ and ‘C-time’ IBMs

D-time (t=0,1,2,Characteristic C-time (t�0)….)

Formulation Relatively simple More difficultRates‘Dynamic’ Probabilities

parametersAnalysis More difficult Simpler

(more complexdynamics)

OrdinaryMean-field Differenceequationsmodel differential

equationsUpdating Asynchronous (oneSynchronous (all

site updated at asites updated attime)once)

System state Changes abruptly Changes graduallyInternalProcess External

ordering

Table 3A technical comparison of ‘D-space’ and ‘C-space’ IBMs

D-spaceCharacteristic C-space (plot)(lattice)

InfinitesimalFiniteIndividual volumeInfinitesimalFiniteSmallest

heterogeneity levelIndirectPossibly directInteraction


conditions. Moreover, information obtained froma stability analysis of mean-field models allowsone to choose parameter values for IBMs thatyield stable or unstable population dynamics un-der the homogeneous mixing conditions, and tosee whether these features change when theseassumptions are not met. Still the mean-fieldmodels are rather regular companions to the pub-lished IBMs. That is why I now review rigoroustechniques of mean-field model construction ineach spatio-temporal framework.

3.1. D-space, D-time

I show in Appendix A that if � and �� representpopulation densities at the beginning and at theend of a time step, respectively, the rigorousmean-field model corresponding to the above ‘D-space, D-time’ IBM is:

��=�(1−Pm)+ (1−�)�

1−�

1−�PrN�Nn

(5)

For small enough Pr or � the expression insquare brackets can be well approximated by theterm Pr �, giving:

��=�(1−Pm)+�Pr(1−�)=�+r��

1−�

K�

(6)

which is a discrete-time version of the Verhulstlogistic equation, with the intrinsic growth rater=Pr−Pm and the carrying capacity (with re-spect to the maximal admissible density �=1)K=1−Pm/Pr. For large enough N one may ap-proximately write

��=�(1−Pm)+ (1−�)(1−e−�Pr), (7)

as limN�� (1+z/N)N=ez for any real number z.To be precise, mean-field model (5) assumes

that individuals are,� identical in their parameters;� uniformly randomly distributed on the lattice

(each site has an equal probability of beingoccupied by an individual);

� members of an infinite population (thus livingon an infinite lattice).These assumptions are called the ‘homoge-

neous mixing conditions’ in the literature. Inother words, if individuals are more or less iden-

tical in their relevant characteristics, more or lessuniformly randomly distributed on the lattice ateach time instant, and forming a relatively largepopulation, difference Eq. (5), and possibly Eq.(6), is a good approximation of the populationdensity evolving according to the above definedIBM with discrete space and discrete time. Insimulations, the lattice size is always finite. Thefirst assumption is guaranteed here by the sim-plicity of IBM rules. For motile species, the mid-dle assumption can be technically approached bygiving individuals a high movement rate (large d)and/or letting the offspring disperse over largeneighborhoods (large e). The assumption is ex-actly fulfilled by taking up all individuals at theend of each time step and scattering them uni-formly randomly over the lattice, under the con-straint that at most one individual lands in asite. For sessile species, uniform random initialdistribution and uniform random offspring dis-persal guarantee the middle assumption at eachtime step, with large e approximating it quitewell.

3.2. D-space, C-time

I show in Appendix B that the rigorous mean-field model corresponding to the above ‘D-space,C-time’ IBM is:

�� =r�(1−�)−m�= r̃��1− �K�

(8)

This is the well-known Verhulst logistic equa-tion describing single-species population growth,with the intrinsic growth rate r̃=r−m and thecarrying capacity (with respect to the maximaladmissible density �=1) K=1−m/r.

To be precise, mean-field model (8) assumesthat individuals are subject to the homogeneousmixing conditions specified above. All the com-ments are valid here, too, with the high movementrate characterized by large d or w much greaterthan r. If the alternative rule for reproduction isapplied (see above), one arrives at the same mean-field model (8) since the fraction of occupiedneighbors f becomes the mean fraction of occu-pied neighbors x/S under the homogeneous mix-ing conditions.


3.3. C-space, D-time

I show in Appendix C that the rigorous mean-field model corresponding to the above ‘C-space,D-time’ IBM is:

��=�(1−Pm)

+�Pr�

PL �+�

n=0

(�A)n

n !exp(−�A)

�1−

n+1E

n+

+ (1−PL) �+�

n=0

(�A)n

n !exp(−�A)

�1−

nEn

+

n(9)

PL is the probability that if an offspring landsin a distance from its parent, the latter lies withinthe establishment neighborhood of the former(PL=e/� if e�� and PL=1 otherwise); for a realnumber z, [z ]+ =z if z�0 and zero otherwise.Provided that � is small in the course of timeand/or E is large, Eq. (9) will not stay far fromthat in which [z ]+ is replaced just by z (for a realnumber z). The latter sums to:

��=�+r��

1−�

K�

(10)

which is a discrete-time version of the Verhulstlogistic equation, with the intrinsic growth rater=Pr(1−PL/E)−Pm and the carrying capacity(with respect to the maximal admissible density�=1):

K=E−PL

A�

1−Pm

Pr(1−PL/E)�

(11)

To be precise, mean-field model (9) assumesthat individuals are subject to the homogeneousmixing conditions that are just ‘C-space’ ana-logues of those specified for the ‘D-space’ cases,� identical in their parameters;� uniformly randomly distributed on the plot;� members of an infinite population living in an

infinite plot.In other words, if individuals are more or less

identical in their relevant characteristics, more orless uniformly randomly distributed on the plot ateach time instant, and forming a relatively largepopulation, the ordinary difference Eq. (9) is agood approximation of the population densityevolving according to the above defined IBM with

continuous space and discrete time. In simula-tions, the plot size is always finite. The first as-sumption is guaranteed here by the simplicity ofIBM rules. For motile species, the middle assump-tion can be technically approached by giving indi-viduals a high movement rate (large �) and/orletting the offspring disperse over large neighbor-hoods (large �). The assumption is exactly fulfilledby taking up all individuals at the end of eachtime step and scattering them randomly over theplot. For sessile species, uniform random initialdistribution and uniform random landing of theoffspring guarantee the middle assumption at eachtime step, with large � approximating it quite well.

3.4. C-space, C-time

I show in Appendix D that the rigorous mean-field model corresponding to the above ‘C-space,C-time’ IBM is:

�� = −m�

+r��

PL �+�

n=0

(�A)n

n !exp(−�A)

�1−

n+1E

n+

+ (1−PL) �+�

n=0

(�A)n

n !exp(−�A)

�1−

nEn

+

n(12)

Analogously to the previous case, if � is small inthe course of time and/or E is large, Eq. (12) willnot stay far from that in which [z ]+ is replacedjust by z (for a real number z). The latter sums to:

�� = r̃��1− �K�

, (13)

which is the Verhulst logistic equation, with theintrinsic growth rate r̃=Pr(1−PL/E)−Pm andthe carrying capacity (with respect to the maximaladmissible density �=1):

K=E−PL

A�

1−Pm

Pr(1−PL/E)�

(14)

To be precise, mean-field model (12) assumesthat individuals are subject to the homogeneousmixing conditions specified for the ‘C-space, D-time’ case. All the comments are valid here, too,with the high movement rate characterized bylarge � or w much greater than r.


3.5. A short discussion

Formally, mean-field models are systems of or-dinary differential equations (C-time) or differ-ence equations (D-time) that describe temporaldynamics generated by the IBM rules under thehomogeneous mixing conditions. These condi-tions do not differ in principle for the four spatio-temporal model frameworks and can be generallyformulated as follows: (i) populations are to bedivided into a finite number of groups withinwhich individuals are supposed to be identical(averaging over individuals within respectivegroups), (ii) populations are well-mixed so thatany individual may equally interact with anyother (averaging over space), and (iii) populationsare sufficiently large so that individuals respondonly to means (averaging over systemrealizations).

Any deviation from the homogeneous mixingconditions causes that the mean-field (non-spatial,population-level) models have only an approxi-mate power. Pair and moment approximationsthat are going to be reviewed in the next articleare promising alternatives as better analyticalIBM counterparts (Bolker et al., 2000; Law andDieckmann, 2000; van Baalen, 2000).

In small populations (relaxation of the condi-tion (iii)) at least variance starts to play a role,too. IBM simulations then may demonstratestrong demographic stochasticity that alwaysgives a population a chance to go extinct at anytime instant. If interactions are only local (relax-ation of the condition (ii)) due to, for example,slow movement or small interaction neighbor-hoods, clusters of individuals are usually formedin the environment. Spatial pattern of individualsis then no more uniform random. Condition (i)may be relaxed by enabling model parameters tovary with individuals; then, parameter varianceand higher moments start to play the role inmean-field models. This influence can be dimin-ished by forming a larger number of groups andaveraging the parameters within them. The moregroups we form, however, the more we deviatefrom the condition (iii) within the individualgroups.

As for the simple IBMs formulated and ana-lyzed in this article, their rigorous mean-fieldmodels take forms of different complexity. Never-theless, under some mild assumptions, all theseforms are well approximated by the Verhulst lo-gistic model of single-species population growth,showing that even at this level the four spatio-temporal frameworks are more alternatives ratherthan competitors.

4. Alternatives and complexities

The use of spatially explicit IBMs demandsdescription of the environment and each individ-ual living in it, together with its individual– indi-vidual and individual–environment interactions.At (almost) every step of this description at leasta few alternatives are a priori plausible. Somemake the models possibly more realistic yet prob-ably more complex to formulate, simulate andanalyze. Others are just technical alternatives.Knowledge of whether and how a choice fromamong the alternatives changes spatio-temporalpatterns should help to separate effects of modelartifacts and biological processes. Some authorsclaimed that IBMs are rather robust against atleast some of these alternatives (Durrett andLevin, 1994b; McGlade, 1999; Wilson et al.,1999), causing no qualitative change in popula-tion-level behavior, but a few works only ad-dressed these questions explicitly (McCauley etal., 1993; Wilson et al., 1993).

One may ask whether a (discrete) lattice shouldbe modeled as regular or irregular, and in theformer case if it should be composed of squares,triangles or hexagons. Environment can be as-sumed homogeneous or heterogeneous in its phys-ical characteristics, movement may be directed asopposed to diffusive, initial population distribu-tion may be random or admit a specific spatialpattern, etc. Individuals need not be identical intheir behavior; IBM rules may (and usually do)depend on various individual characteristics suchas age, size, sex, genotype, etc. In particular, inevolutionary models different genotypes may havedifferent parameter values, leading to differencesin their respective reproductive success (Keeling


Fig. 2. (A) Projection of a hexagonal lattice and the nearest neighbors of its two sites onto a square lattice. (B) Projection of atriangular lattice and the nearest neighbors of its two sites onto a square lattice.

and Rand, 1995). This section discusses these andother model features in more details.

Finally, I note that most published IBMs areproblem-specific, and applied issues often drivethe choice of an appropriate alternative. For ex-ample, particular BC are used to study specifichabitable environments, specific initial conditionsallow for a study of species invasions, and partic-ular spatio-temporal variations in environmentalconditions can be used to investigate effects ofharvesting and habitat degradation.

4.1. Discrete space topology

Any habitat is characterized by its shape andsize (see below). Additionally, ‘D-space’ environ-ments require a topology, regular lattices made upof squares being by far the most common choice.The fundamental reason is no doubt their easycomputer implementation (the lattice sites aresimply represented as pairs of non-negative in-tegers). Nevertheless, the lattices composed ofhexagons (which probably capture better an ideaof circular individual neighborhood) or trianglescan be technically represented as square lattices,with properly transformed interaction neighbor-hoods. As an example, consider an individual thatinteracts with its nearest neighbors only (Fig. 2).For the hexagonal lattice with six nearest neigh-bors, an equivalent square lattice may be obtainedby shifting hexagon rows half the hexagon diame-ter (Fig. 2A). Note that there are two types ofneighborhoods, and no one is of the von Neu-mann type (i.e. containing four square-latticenearest neighbors). Triangular lattices may betransformed analogously (Fig. 2B). In fact, evenirregular space tessellation’s can be projected onto

square lattices, with interaction neighborhoodsgenerally differing for each site. To summarize,though the a priori lattice choice should alwaysrespect specific system properties, its subsequentcomputer coding and formal treatment can bedone via a square lattice with adequate (possiblysite-specific) interaction neighborhoods. In thisway, one may formally view the study of impactsof various space tessellation’s on population dy-namics as covered by the study of influences ofvarious interaction neighborhoods (see below).

4.2. Boundary conditions

The assumption of an infinite habitat simplifiesderivation of many mathematical results (Durrettand Levin, 1994b). However, finite habitats arewhat one encounters in applications and com-puter simulations. BC are then a necessary com-ponent of IBM formulation; they describe the fateof an individual that hits the habitat boundary.The individual ‘leaving’ the habitat may be ‘lost’and thus reduce the population size (a phe-nomenon termed the boundary effect) or may‘reappear’ elsewhere in the habitat. Also, an indi-vidual that is located near the boundary may haveincomplete interaction neighborhoods and thustend to have fewer neighbors than it would havein the middle of the habitat (a phenomenontermed the edge effect); this effect is often a modelartifact which may result in higher survivorshipand fecundity of annual plants (Pacala and Silan-der, 1985) and faster tree growth (Moravie et al.,1997) than actually observed.

Absorbing, reflecting, and periodic BC havebeen proposed in the literature. Under absorbingBC, individuals can step off the habitat and be


removed from the system. Absorbing BC do notdeal with the boundary and edge effects at all,and do not allow for immigration. To reduce theeffects one typically focuses on population dy-namics in a central portion of a model environ-ment (Pacala and Silander, 1985). Underreflecting BC, individuals that hit the boundaryricochet back into the habitat in a random ordeterminate direction. Keitt (1997) implementedthem in such a way that no movement occurredwhen an individual attempted to move off thelattice. Reflecting BC are realistic, for example,for ground animals living on an island or wateranimals living in a pond. They seem unrealistic,e.g. for plant seeds unless one interprets the‘reflected’ seed as another seed that ‘immigrates’into the studied area. Therefore, reflecting BCremove the boundary effect (do not change thepopulation size), but do not modify the edgeeffect. Pacala and Silander (1985) claimed that inthe examples they considered the dynamics onsmall habitats with reflecting BC are very similarto the dynamics on small portions that are embed-ded in large habitats with absorbing BC. In thisway one may considerably lessen computationalcosts as computer runs for small habitats arerelatively inexpensive. However, before this strat-egy is adopted one has to exemplify a congruencebetween results of these two approaches for thesystem under study. Under periodic BC, the op-posite edges of the habitat are connected together(the form of the habitat has to allow for such awrapping). There is no system with periodic BC innature. The only plausible argument for their useis just the same as for plant seeds and reflectingBC: an individual returning from the other side ofthe habitat is another individual; this may mimicdynamics in large habitats. Periodic BC have,however, nice technical properties: they removethe boundary effect, remove the edge effect (inter-action neighborhoods are always complete), andabove all they are easily implementable in com-puters. That is why they are by far the most usedBC in IBMs. Britton et al. (1996) got similarresults with periodic and reflecting BC. One couldprobably notice that the choice of BC (and in factof any other IBM attribute) for a given applica-tion is always a trade-off between realism andcomputational simplicity.

4.3. Scales

An important question in any modeling effort isthat of scales. In spatially explicit IBMs, size ofthe time step in ‘D-time’ models, size of the site in‘D-space’ models (with regular lattices), and sizeof the habitat (lattice/plot) are of a primary inter-est. All these scales are strongly influenced bybiological questions being addressed, modeledprocesses, and information available as modelinput. In many IBMs describing actual ecosystemsscales naturally follow from the interactions stud-ied and life histories of species involved; modelsof population dynamics of army ants (Britton etal., 1996), red grouse (Hendry et al., 1997), andwild daffodils (Durrett and Levin, 1994b) are justa few examples. In IBMs describing artificialecologies where the stress is put on interactions ingeneral, these scales should be considered modelparameters; in other words, it is important to viewthe system at all admissible scales.

Problem to be addressed, scale of spatial het-erogeneity in physical characteristics, area coveredby or under control of an individual, and compu-tational simplicity all play a role in choosing anadequate size of the lattice site (Keeling, 1999).Systems to be modeled are so diverse that it isvirtually impossible to formulate any general rule.Any lattice site should be physically homoge-neous; hence its size should be smaller than thesmallest scale of relevant heterogeneities. Itshould be large enough to contain at least oneindividual (often just a single individual) or anarea under its direct control (such as its territory);the latter is often related to the movement rate ofmotile organisms and the size of time step in‘D-time’ IBMs. A balance must be struck betweentoo fine a scale in which case the model will becomputationally intensive, and too coarse a scalein which relevant heterogeneities or interactionsmay be ignored. The choice is less clear and moreintricate if two or more individual types withdifferent spatial characteristics are considered,such as in many predator–prey and host–para-sitoid systems. Use of separate lattices for eachtype, with rules that relate them, is probably anappropriate way. In some systems, where oneindividual type is very small with respect to the


other, a continuous plot could even be consideredfor the former. Last but not least, the question ofthe size of each site is tightly linked to the rangeover which spatial interactions take place. Forsimplicity of model formulations and speed ofcomputations the best way is to only allow inter-actions in small neighborhoods, but this may bein a direct conflict with taking a small enoughscale, so these two factors need to be balanced(see below for more on interactionneighborhoods).

Once a ‘D-space’ topology and a site size arefixed, or when one decides to use ‘C-space’ mod-els, the question of habitat size arises. Smallerhabitats support smaller populations in which de-mographic stochasticity as well as individual mix-ing are stronger. As a result, spatiallyhomogeneous distributions and rapid extinctionsare often observed in IBMs with small habitats.On the other hand, if the habitat is too large,interesting dynamics may be averaged out (Keel-ing, 1999). Spatial patterns are commonly ob-served for larger (and even physicallyhomogeneous; see below) environments (Wilsonet al., 1995). Moreover, de Roos et al. (1991)demonstrated stabilization of temporal popula-tion dynamics in a two-dimensional, predator–prey system. Where lies the border between thesetypes of model behavior? Some authors have al-ready tried to address this question, with a char-acteristic length scale as an emergent,system-specific scale at which system dynamics aremost informative (de Roos et al., 1991; Keeling,1999).

The choice of a time step in ‘D-time’ models isnot so important in artificial ecologies. It is inti-mately related to life cycle details that are mod-eled, and respective model parameters. For manytime-step dependent parameters, such as the prob-ability of death or reproduction of an individualper time step, doubling the time step virtuallymeans doubling these probabilities. The care hasto be taken to keep these probabilities in betweenzero and one. Also, keeping these probabilitieslow may enable one to use simplifying assump-tions in mean-field model derivation (see ‘D-space, D-time’ framework above). Last but notleast, use of large time steps may lead to igno-

rance of important processes and interactions thatpossibly take place within them.

4.4. Initial conditions

Knowledge of initial population sizes is notsufficient to uniquely determine outcomes of spa-tially explicit IBMs. Rather, complete spatial dis-tribution of involved populations is required. Thisissue has a strong practical appeal and can beexemplified on a problem of species (re)intro-duction or invasion. What consequences has arelease of a number of individuals from a pointsource and how they differ if this number isdivided and multiple releases made from differentspatial points? Imagine, for example, a populationthat demonstrates the Allee effect, with a negativeper capita growth rate at low sizes (Stephens etal., 1999). Then, a threshold population size existssuch that below it the probability of populationextinction is inproportionately higher than abovethe threshold (Dennis, 1989). Population dynamicconsequences of such a system could be totallydifferent provided that the point source size isabove this threshold while all the multiple sourcelocal sizes fall below it; see Groom (1998) for areal example. Silvertown et al. (1992) exploredeffects of various initial distributions in five-spe-cies plant community on outcomes of interspecificcompetition. Although the ultimate fate of thecommunity did not depend on the order of plantsin the initial patterns with different species ar-ranged in monospecific bands, transient dynamicsdiffered significantly.

An important question is how to define initialconditions for established populations. Techni-cally, the easiest way is to assume that individualsare uniformly randomly distributed in their habi-tat, and this choice is by far the most common inthe literature. This case can easily be set up inlaboratory experiments; it may also be observedat the beginning of some ecological successionsand for trees in older forest stands (Szwagrzykand Czerwczak, 1993). When this assumptiondoes not hold, populations may be clustered oroverdispersed in space. Then, for ‘C-space’ IBMsinitial conditions may be generated from appro-priate spatial point process models. Cressie (1993)


made a review of such models and methods oftheir estimation from the observed patterns ofindividuals. These models also cover situations inwhich non-uniform characteristics such as age andsize have to be assigned to each individual at thestart of simulations. Lepš and Kindlmann (1987)suggested a flexible generator of clustered patternsin a ‘C-space, D-time’ IBM. If the number oflattice sites is sufficiently large these methods canbe used (as an approximation at least) to gain aninsight into ‘D-space’ IBMs, too, as exemplifiedby Wiegand et al. (1998).

4.5. En�ironmental heterogeneity

Spatial and/or temporal heterogeneity in envi-ronmental characteristics is one of the principaldeterminants of observed spatio-temporal popula-tion patterns. One may distinguish heterogeneityin physical characteristics of the environment asopposed to that in biological characteristics wherenon-random spatial patterns of individuals areformed in the environment that is physically ho-mogeneous (McCauley et al., 1993; Keeling, 1999;Herben et al., 2000). Patterns due to the formertype are imposed via extrinsic factors such astemperature, moisture, light, nutrients, topo-graphic heterogeneity, and habitat unsuitability ordestruction; spatial heterogeneity in physical char-acteristics of the environment can be describedeither phenomenologically (describing a cumula-tive effect of all potential sources of heterogene-ity) or mechanistically (separately describingindividual sources). Patterns due to the latter typeare emergent via intrinsic factors such as limiteddispersal and local interactions of the studiedpopulations; although these patterns are hard todetect in natural systems, Wilson et al. (1999)suggest its presence in a system composed ofwestern tussock moth feeding on perennial lupinesin coastal California.

Influence of individuals and the environment ismutual. The latter determines vital rates of popu-lations while the former contribute to the spatialheterogeneity by influencing, for example, dynam-ics of resource renewal and depletion. Britton etal. (1996) constructed a ‘D-space, D-time’ IBM inwhich army ant colonies exploited patches of

food, thus reducing their efficiency for and growthrate of other colonies; left exploited patches re-covered in a number of time steps. Doi et al.(1998) considered an IBM of a microorganicclosed ecosystem where each individual consumesa nutrient from the environment and excretes itsmetabolic products to the environment as de-tritus. Models of forest growth that are intendedto have a high predictive power describe spatialheterogeneity in a big detail (Busing, 1991; Pacalaet al., 1996).

In ‘D-space’ IBMs, spatial heterogeneity inphysical characteristics of the environment is eas-ily modeled by defining a variable giving each sitea value of its state. Britton et al. (1996) gave thisvariable a meaning of the state of recovery of thesite after a recent army ant raid. Dytham (1994)and Keitt (1997) distinguished destroyed and hab-itable sites in a two-species competition systemand a general food web, respectively. In ‘C-space’IBMs, a variable describing physical state of alocation is spatially continuous and hence moredifficult to define and handle. Pacala (1987) pre-sented two ways of dealing with it in one-dimen-sional environments of a finite length: theenvironment is divided into a finite number ofconnected intervals (patches), and physical char-acteristics of these patches are either fixed or varystochastically with time. In two-dimensional sys-tems a question arises how to handle possiblynon-rectangular patches. A possible solutionseems to be a use of a Geographic InformationSystem (GIS) model as a background descriptionof the physical environment. The statisticalmethod of kriging can also be exploited in ‘C-space’ models; it estimates a spatially continuousdistribution of an environmental variable from afinite number of measured samples (Cressie,1993). Interpolation of environmental variables in‘C-space’ IBMs can also be coupled with GIS(Briggs et al., 1997).

4.6. Process ordering

In ‘D-time’ models ordering of acting processesis not implied by them but has to be specifiedexternally. Yet it is not always obvious how theprocesses should be ordered in a time step. Move-


ment (for motile organisms) is mostly separatedfrom mortality and reproduction processes in theliterature, and put at the end (or equivalently atthe beginning) of the time step. I distinguish threecases concerning ordering of mortality and repro-duction in the above single-species, ‘D-time’IBMs, and give them the following labels, CC forconcurrent ordering (individuals reproduce anddie simultaneously, newborns cannot die nor es-tablish to the sites occupied by actually deceasedadults, individuals that are marked dead may stillreproduce in the current time step); RM for re-production preceding mortality (population firstreproduces, then all individuals (including new-borns) are exposed to death); and MR for repro-duction following mortality (individuals are firstexposed to death and those that die removed fromthe habitat, only surviving individuals are able toreproduce); see Fig. 3.

Which ordering to choose for a particular sys-tem? This issue is by no means a mere technicaldetail, since it can profoundly alter populationdynamics by introducing specific density-depen-dent relationships and/or by modifying parametervalues. McCauley et al. (1993) showed that for a‘D-space, D-time’, predator–prey IBM prey

growth is a function of prey density if the processordering is concurrent, but that it depends on thepredator density as well provided that the demo-graphic processes and predation are ordered in asequential way. Hence, an assumption made inthis regard is important. In some systems such asplants reproducing by seeds and animals withnon-overlapping generations or short reproduc-tive periods, mortality and reproduction are moreor less temporally separated processes. The RMand MR cases are equivalent if applied to sessilespecies. On the other hand, if movement is addedat the end of each time step it is placed inqualitatively different points of the RM and MRloops (Fig. 3); as movement may considerablymodify spatial patterns of individuals in the habi-tat, the RM and MR cases may now producedifferent results. Concurrent ordering is a way tomodel situations where mortality and reproduc-tion processes overlap. Some examples of processordering in more complex IBMs can be found inMcCauley et al. (1993) for predator-prey systemsand in Durrett and Levin (1994b) for competitionamong plants.

For a more quantitative insight, I now derivemean-field models for the CC, RM, and MR

Fig. 3. Ordering of mortality (M) and reproduction (R) processes in a time step. (A) Concurrent ordering. (B) Reproductionpreceding mortality. (C) Reproduction following mortality.


Fig. 4. Effects of ordering of mortality and reproductionprocesses in a time step on mean-field model dynamics in the‘D-space, D-time’ framework. Concurrent ordering ( Eq. (20);solid line), reproduction preceding mortality (Eq. (21); dashedline), reproduction following mortality (Eq. (22); dotted line).Parameter values: S=128×128, Pr=0.5, Pm=0.05, N=4,x0=10, �0=x0/S.

no more independent events. In particular, adultsthat currently die enable offspring to be estab-lished in the respective emptied sites. Conse-quently, two expectation equations have to becomposed:

E(X� �X=x)=x(1−Pm),E(X ��X� = x̃, X=x)

=x(1−Pm)+ (S− x̃)�

1−�

1−x̃S

PrN�Nn

(17)

where X� is a random variable giving the numberof surviving adults before reproduction starts.One may be tempted to combine these equationsto get:

E(X ��X=x)=x(1−Pm)+ (S−x(1−Pm))�

1−�

1−xS

(1−Pm)PrN�Nn

(18)

This is, however, not correct mathematically. Itcan be easily shown that:

E(X ��X=x)=E [E(X ��X� = x̃,X=x)�X=x ] (19)and unless one assumes that P(x̃ �x)=�(x̃−x(1−Pm)), where P(·�·) is a pdf and �(·) the Diracdelta function, that is, unless the mortality processis deterministic, the combined expectation in Eq.(18) does not hold. Yet it may be a good approx-imation if variance of X� is small, such as in largepopulations. Going through the same steps aswhen deriving the mean-field model within the‘D-space, D-time’ framework, one may get thefollowing equations for the evolution of popula-tions densities,

��=�(1−Pm)+ (1−�)�

1−�

1−�PrN�Nn

CC

(20)

��=�(1−Pm)+ (1−�)�

1−�

1−�PrN�Nn

(1−Pm) RM (21)

��=�(1−Pm)

+ (1−�(1−Pm))�

1−�

1−�(1−Pm)PrN�Nn

MR (22)

scenarios in the above ‘D-space, D-time’ IBM.Analogous, yet slightly more laborious calcula-tions can be made for the ‘C-space, D-time’ case.The CC case has been analyzed above; the mor-tality and reproduction processes are independentevents starting with the same number of individu-als. Under the homogeneous mixing conditions,this implies (see above):

E(X ��X=x)=x(1−Pm)

+ (S−x)�

1−�

1−xS

PrN�Nn

(15)

In the RM case, in order to be counted in thenext generation, offspring has to be conceived,placed successfully on the lattice, and survive tothe end of the time step. The adult survival andoffspring establishment processes are still inde-pendent events. Therefore:

E(X ��X=x)=x(1−Pm)

+ (S−x)�

1−�

1−xS

PrN�Nn

(1−Pm) (16)

The MR case is the most complex as the adultsurvival and offspring establishment processes are


Typical behavior of these three mean-field mod-els is shown in Fig. 4. Note that the MR modelattains the highest carrying capacity, the CCmodel demonstrates the most rapid growth, andthe RM and MR models coincide for low popula-tion densities.

All the above orderings can be referred to astemporal as they define an order of demographicprocesses in a time step. Ruxton (1996) and aboveall Ruxton and Saravia (1998) pointed out that anorder in which sites are processed in ‘D-space’models (spatial ordering) may also affect modeloutput. In particular, they considered random siteprocessing order and ‘top-left to bottom-right’ siteprocessing order in precisely the above ‘D-space,D-time’ IBM with e=1 and d=0 (no movement),together with a triplet of temporal process order-ings (including the above MR case). Moreover,they compared the ‘time type’ scenarios in whichchanges in the states of the lattice sites wereregistered immediately or at the end of a time step.Based on a couple of simulation experiments theycame with a hierarchical classification of theseordering alternatives: temporal ordering had themost pronounced effect, followed by time type andfinally spatial ordering. Although this ranking canbe model specific, the important consideration isthat the model output depends critically on as-sumptions made about process ordering. Ruxtonand Saravia (1998) ended up with two generalwarnings: first, spatio-temporal ordering must becarefully selected to match the biological charac-teristics of the system to be modeled rather thanled by programming expediency, and second, acomplete description of the details of this orderingshould be specified in the publications using IBMs.

4.7. Interaction neighborhoods

The interaction neighborhood is an area aboutan individual circumscribing a part of the habitat(possibly containing some other individuals) thatinfluences its current behavior. In the above single-species IBMs, I defined such neighborhoods forthe processes of offspring establishment and indi-vidual movement. Mortality and/or reproductionprobabilities could also be made neighborhooddependent (see below).

Interaction neighborhoods may be formulatedas regular or irregular. In the former case, onemay adopt ��z �� as a distance function and letN={z :��z ��r} be the set of sites or locationswithin a distance r from the origin. The interactionneighborhood for a process and an individuallocated at x is then the set {x+z :z�N}. Thefunctions ��z ��1= �z1�+ �z2� and ��z ��=max{�z1�,�z2�}, z= (z1,z2), probably most suit totwo-dimensional ‘D-space’ IBMs with latticescomposed of squares, giving diamond- andsquare-shaped neighborhoods, respectively (Dur-rett and Levin, 1994b); the most popular vonNeumann (or nearest neighbor) and Moore neigh-borhoods can be expressed as N={z :��z ��1=1}and N={z :��z ��=1}, respectively. For two-di-mensional ‘D-space’ IBMs with hexagonal andtriangular lattices one usually formulates regularinteraction neighborhoods in terms of rings ofnearest neighbors, second-nearest neighbors etc.(Tilman et al., 1997). When these lattices areprojected onto square lattices, these neighbor-hoods have to be transformed accordingly (seeabove). For two-dimensional ‘C-space’ IBMs, cir-cular neighborhoods due to the Euclidean norm��z ��2= (�z1�2+ �z2�2)1/2 are probably the most com-mon choice.

Durrett and Levin (1994a) claimed that ‘oneshould not worry too much about what neighbor-hood to choose. In most cases the qualitativebehavior of the model does not depend on theneighborhood used’. This statement concernsneighborhood shape but not its size. The observa-tions from many IBMs show and the above mean-field model derivations confirm that when theinteraction neighborhoods increase in size the sys-tem behaves more and more like a well-mixedsystem: spatial distribution of individuals resem-bles the uniform random distribution and tempo-ral evolution of density approaches mean-fieldmodel dynamics.

An important question concerns choice ofneighborhood sizes for the system under study.Generally, the adequate choice depends on theindividual life history and physiology. For ‘D-space’ IBMs this question is intimately related tothe site size. In ‘D-time’ IBMs the time step is alsoimportant. Britton et al. (1996) derived the move-


ment neighborhood size of the army ant coloniesfrom the actual observations of distances crossedper defined time step. Pacala and Silander (1985),Silander and Pacala (1985), Pacala and Silander(1990) touched this problem in a ‘C-space’ IBMof plant communities. Their seed set (or fecun-dity) predictor and survivorship predictor relatethe average number of seeds per plant and theprobability of reaching adulthood from seed, re-spectively, to a number of neighbor plants inrespective circular neighborhoods. The ‘best’neighborhood sizes were estimated by fitting thepredictors evaluated for a finite number of neigh-borhood radii to the observed data on seed setsand survivorship, and choosing such a radius foreach predictor that minimizes residual variance.

4.8. Life cycles

No two individuals are the same. Rather, theydiffer in age, size, stage, sex, genotype, and manyother physiological and behavioral traits. Theabove simple, single-species IBMs do not take thisuniqueness into account; various individuals differjust by their location in space. Some of these traitscan be technically considered as system states andthe model parameters (such as reproduction andmortality probabilities/rates) or even the wholemodel rules made dependent on them; in otherwords, the same local environment may havedifferent effects on the individual performancedepending on its state. One may also be interestedin the population-level distribution of these states(mostly age, size and sex) and its temporal dy-namics. Evolutionary models assume that differ-ent genotypes may have different parametervalues, which leads to differences in their respec-tive reproductive success (Keeling and Rand,1995). Temporal variation may be included inIBMs by altering the model rules from time totime (e.g. from generation to generation in annualplants).

Almost all published IBMs assume that move-ment is local but diffusive (equiprobable in alldirections). Yet often it may be directed: activelyto water, sexual partner, sward etc. passively bywind or river stream (Fahse et al., 1998; Berec etal., 2001). A predator may direct its movement to

areas with a higher local density of prey; likewise,a prey may direct it to areas with a lower predatordensity.

Although the vast majority of theoretical stud-ies considers the same mortality risk for eachindividual, non-uniform mortality scenarios aremore the rule than an exception in nature. Ad-verse environmental conditions, density depen-dence, and starvation are just a few forcesdetermining the unique mortality probability/ratefor each individual. As opposed to the aboveIBMs, Law and Dieckmann (2000) considereddensity-independent reproduction and density-de-pendent mortality in single-species populations inthe ‘C-time, C-space’ framework. Dieckmann andLaw (2000) modeled both processes as densitydependent. Starvation component of reproductionand mortality has been considered, for example,by Wilson and Keeling (2000).

Models may be constructed that allow for non-zero (‘C-time’) and non-unit (‘D-time’) develop-ment times for juveniles or production of two ormore offspring per reproductive event, that de-scribe ‘C-space’ individuals with a volume or atleast an area so that they could not overlap, andthat enable more than one ‘D-space’ individualoccupy a site. Such realistic features are even anecessity in many applied issues.

Last but not least, no population lives alone inits habitat, but rather interacts with other popula-tions. Spatially explicit IBMs can also be con-structed for multi-species systems such aspredator–prey systems (McCauley et al., 1993),competition systems (Durrett and Levin, 1998;Pacala, 1986), or one-predator two-prey optimalforaging systems (Berec, 2000; Berec and Křivan,2000). In such systems, a number of alternativescan also be considered to model inter-populationinteractions.

5. Conclusions

This article reviews spatially explicit IBMs froma methodological point of view. My focus is pri-marily on showing technical as well as conceptualdifferences between the four frameworks for suchmodels, differing by whether time and space are


each modeled discretely or continuously. Foreach of these frameworks I formulate one of thesimplest single-species IBMs and discuss its sim-ulation subtleties and mean-field analysis. Allfour mean-field models resemble the Verhulst lo-gistic model of single-species population growand are exact under the homogeneous mixingconditions. These conditions can be generallyformulated as follows: (i) populations are to bedivided into a finite number of groups withinwhich individuals are supposed to be identical(averaging over individuals within respectivegroups), (ii) populations are well-mixed so thatany individual may equally interact with anyother (averaging over space), and (iii) popula-tions are sufficiently large so that individuals re-spond only to means (averaging over systemrealizations). No one can probably claim thatone framework is better than another. This arti-cle shows that they are more supplements thancompetitors, and probably any system can bedescribed in any of the frameworks. Yet for agiven system, some frameworks may perhaps bemore suitable for implementing a given rule orfeature than the others.

I also discuss some alternatives in construc-tion of IBMs in more details. Some of them arejust technical but some make the model morecomplex yet possibly more realistic. One shouldnote at this point that finding an appropriatelevel of aggregation in a model is a decisive partof modeling procedure. This is exactly the rea-son why Murdoch et al. (1992) suggested build-ing a suite of models with increasing complexityand hence deciding on the adequate model reso-lution for a given purpose.

Finally, I cannot forbear one specific note.The only complete description of any IBM is itscomputer code. Its publication inside any articleis impossible, yet any conceptual description ofmodel rules is, due to an effort to save an ex-pensive space, in most cases incomplete, particu-larly in more complex systems. Hendry et al.(1997) published at least a flow chart of theirmodel, but even this practice is, in my opinion,unsatisfactory. Any modeller may implement theflow chart differently and when the system is atleast a bit more complex, the probability of

making an error increases. Moreover, everyreader should be allowed to check publishedsimulations. I am fully aware that there aremany possible computer languages available,some of them even specific to some model envi-ronments. But no reader should be a priori dis-allowed to view the code. Imagine how many ofus have to code the same or at least quite simi-lar IBMs simultaneously. Sharing our codeswould make the modeling much more effective,even though not always. Last but not least, thiswould force modelers to write clear, understand-able codes, and possibly create standardizedmodel outputs, which is useful by itself. That iswhy I suggest to describe models roughly bywords in the articles, putting stress on the mostimportant parameters and interactions, and en-able anyone downloading of IBM codes fromauthors’ web pages or at least sending the codeson request by e-mail. I do this just now. Any-one can download the codes of the above single-species IBMs, implemented in C-language, frommy personal web page http://www.entu.cas.cz/berec/ibm.html I hope this helps.

Some more issues have to be addressed tomake IBM analysis more reliable, i.e. able tosatisfactorily describe systems even when the ho-mogeneous mixing conditions do not hold.These include,� statistical descriptions of spatial population

patterns;� non-spatial, population-level models that track

temporal population dynamics;� spatial, population-level models that track spa-

tio-temporal dynamics, etc.;These issues are going to be reviewed in the nextarticle.

Acknowledgements

The author acknowledges financial support bythe Grant Agency of the Czech Republic (Grant201/98/P202) and by the Ministry of Education ofthe Czech Republic (Grant MSM123100004). Thework was also supported from the projectZ5007907 hold by the Institute of Entomology,Academy of Sciences of the Czech Republic.

http://www.entu.cas.cz/berec/ibm.html


Appendix A. Mean-field model derivation in the‘D-space, D-time’ case

Let X � and X be random variables representingnumbers of occupied lattice sites at the beginningand at the end of a time step, respectively (notethat the model rules are stochastic). Let Xs and Xrbe two more random variables, representing num-bers of survived adult individuals and newly estab-lished offspring, respectively, at the end of the timestep. Obviously, X �=Xs+Xr. Hence:

E(X ��X=x)=E(Xs�X=x)+E(Xr�X=x), (23)where I denote by E (Y �X=x) the mean value ofa random variable Y conditioned on the eventX=x. Clearly:

E(Xs�X=x)=x(1−Pm) (24)as the number of individuals that die every timestep is binomially distributed. Similarly:

E(Xr�X=x)=xPrP [conceived offspring is estab-lished on the lattice] (25)

Yet I use an alternative expression for E(Xr�X=x) as the involved probability is easier to evaluate:

E(Xr�X=x)= (S−x)P [site that is vacant nowbecomes occupied at the next time step], (26)

where S−x is the current number of vacant sites.Provided that the individuals are uniformly ran-domly distributed on the lattice, it is:

P [site that is vacant now becomes occupied at

the next time step]=1−�

1−xS

PrN�N

, (27)

where N= (2e+1)2−1 is the number of sites inthe establishment neighborhood. Eq. (27) is due tothe fact that each neighbor of the vacant siteindependently sends an individual with the proba-bility that the neighbor site is occupied (x/S) timesthe probability that the individual present therereproduces (Pr) times the probability that it sendsoffspring to the focal vacant site (1/N). To sumup:

E(X ��X=x)=x(1−Pm)

+ (S−x)�

1−�

1−xS

PrN�Nn

(28)

Due to the concurrent ordering of mortality andreproduction rules, Xs and Xr are independentrandom variables. Hence, variance in the numberof individuals at the end of the time step is:

Var(X ��X=x)=Var(Xs�X=x)+Var(Xr�X=x)(29)

with (binomial distributions):

Var(Xs�X=x)=x(1−Pm)Pm (30)

and:

Var(Xr�X=x)= (S−x)�1−�1−xS PrN�Nn�1−

xS

PrN�N

(31)

Now, divide Eq. (29) by S2 and send x�� andS�� so that x/S remains (or converges to) aconstant �. Hence:

Var(V ��V=�)�0 (32)

where the random variable V=X/S stands for thepopulation density. This implies:

��=E(V ��V=�) (33)

in the limit, with �� being the actual populationdensity at the end of the time step. Consequently:

��=�(1−Pm)+ (1−�)�

1−�

1−�PrN�Nn

(34)

in the above limit. Eq. (34) can be alternativelyderived by means of the (strong) law of largenumbers.

Appendix B. Mean-field model derivation in the‘D-space, C-time’ case

Through the technique of thinning of Poissonprocesses, one background Poisson process ismade running at a rate cS, c�m+r+w (Durrett,1995). Lengths of intervals between two successiveevent times are thus independent and identically


exponentially distributed random variables withparameter cS. At every event time generated, asite is uniformly randomly chosen and if occupieda random number is picked uniformly from theinterval [0, 1] to decide on the particular event:death, birth, movement (for motile organisms) ornothing (see above).

Consider a small time interval h. By definition,probability that just one event time is generated inh is cSh+o(h) as h�0 (probability that two ormore event times are generated in h is o(h),probability that no event time is generated in h is1−cSh+o(h); the term o(h) has a precise mathe-matical meaning and for practical purposes can betreated as a quantity that is negligible in compari-son with h), and probability that the chosen site isoccupied is x/S, provided that individuals areuniformly randomly distributed on the lattice.The particular event may result in the increase ofpopulation by one individual (birth and successfuloffspring establishment), the decrease by one indi-vidual (death), or no change in abundance (noevent, reproduction but unsuccessful offspring es-tablishment, or movement [with uniform randompositioning of the individual]). Also, an empty sitemay be chosen or no event time generated in h.The event of birth and successful offspring estab-lishment takes place with the probability r/c(1−x/S), while the event of death with the probabilitym/c. To sum up, the mean change in the numberof individuals in a small time interval h is:

By computing variance in the number of indi-viduals and letting h�0, S��, and x�� sothat x/S is (or converges to) a constant �, onemay show that:

Var(V(t+h)�V(t)=�)�0 as h�0 (36)Time evolution of the population density � thus

converges to the solution of ODE:

�� =r�(1−�)−m�= r̃��1− �K�

(37)

Appendix C. Mean-field model derivation in the‘C-space, D-time’ case

With the notation established for the ‘D-space,D-time’ case,

E(Xs�X=x)=x(1−Pm), (38)and similarly:

E(Xr�X=x)=xPrP [conceived offspring is estab-lished on the plot] (39)

The probability that the conceived offspring isestablished on the plot depends on the place itlands and the number of neighbors around thatplace. The probability PE(n) of successful estab-lishment given n neighbors is:

PE(n)=�

1−nEn

+

��1−n/E if n�E,

0 if n�E.(40)

If the offspring lands in such a distance from itsparent so that the latter lies within the establish-ment neighborhood of the former (an event withthe probability PL, PL=e/� if e�� and PL=1otherwise), the parent influences the establishment

probability of its offspring. Therefore, one maywrite:

P [conceived offspring is established on the plot]

=PL �x−1

n=0

PN(n, x−1, e, �̃)PE(n+1)

+ (1−PL) �x

n=0

PN(n, x, e, �̃)PE(n) (41)

E(X(t+h)�X(t)=x)= x+ (−1)�cSh xS

mcn

+1�

cShxS

rc�

1−xS�n

+0

×�

cShxS�1− (m+r+w)

c+

rc

xS

+wcn

+cSh�

1−xS�

+ (1−cSh)n

+o(h)

=x+�

rx�

1−xS�

−mxn

h+o(h) as h�0 (35)


where PN(m, y, e, �̃) is the probability thatthe offspring located at �̃ has m adult neighborsout of y possible in its e-neighborhood, andequals:

PN(m, y, e, �̃)=�y

m��A

S�m�

1−AS�y−m

(42)

under the assumption of uniform random distri-bution of the individuals on the plot, withA=�e2. One may easily show that for x��and S�� (S=MN, the plot size) suchthat x/S is (or converges to) a constant �variance Var (V ��V=�)�0, which implies��=E(V ��V=�) in the limit. Moreover, thebinomial terms PN converge to the Poissoncounterparts with the parameter �A. Hence:

��=�(1−Pm)

+�Pr�

PL �+�

n=0

(�A)n

n !exp(−�A)

�1−

n+1E

n+

+ (1−PL) �+�

n=0

(�A)n

n !exp(−�A)

�1−

nEn

+

n(43)

Appendix D. Mean-field model derivation in the‘C-space, C-time’ case

The background Poisson process is maderunning at the rate x(m+r+w), x being thecurrent population size. Mean-field modelderivation is a combination of the ‘D-space,C-time’ and ‘C-space, D-time’ cases. In part-icular, consider a small time interval of length h,starting at time t at which X(t)=x. The pro-bability that just one event time is gener-ated within this interval is x(m+r+w)h+o(h)as h�0. Provided it is generated, an individ-ual is uniformly randomly selected; it dieswith the probability m/(m+r+w), reprodu-ces with the probability r/(m+r+w),and moves with the probability w/(m+r+w).The mean change in the number of individualsin h is:

E(X(t+h)�X(t)=x)=x+ (−1)

×�

x(m+r+w)hm

m+r+wn

+1×�

x(m+r+w)hr

m+r+wP [conceived off-

spring is established on the plot]n

+0×�

x(m+r+w)h� w

m+r+w+

rm+r+w

(1−P [conceived offspring is established on

the plot])�

+ (1−x(m+r+w)h)]+o(h)=x+ [rxP[conceived offspring is established on the plot]

−mx ]h+o(h) as h�0, (44)

where:

P [conceived offspring is established on the

plot PL= �x−1

n=0

PN(n, x−1, e, �̃)PE(n+1)

+ (1−PL) �x

n=0

PN(n, x, e, �̃)PE(n) (45)

Meaning of the involved probabilities in thelast exp

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