PHYSICAL REVIEW E, VOLUME 63, 051909

Coexistence in a predator-prey system

Michel Droz1,* and Andrzej Pe¸kalski2,†

1Departement de Physique The´orique, Universite´ de Gene`ve, quai E. Ansermet 24, 1211 Gene`ve 4, Switzerland2Instytut Fizyki Teoretycznej, Uniwersytet Wrocławski, Wrocław, pl. M. Borna 9, 50-204 Wrocław, Poland

~Received 18 August 2000; revised manuscript received 27 November 2000; published 23 April 2001!

We propose a lattice model of two populations, predators and prey. The model is solved via Monte Carlosimulations. Each species moves randomly on the lattice and can live only a certain time without eating. Thelattice cells are either grass~eaten by prey! or tree~giving cover for prey!. Each animal has a reserve of foodthat is increased by eating~prey or grass! and decreased after each Monte Carlo step. To breed, a pair ofanimals must be adjacent and have a certain minimum of food supply. The number of offspring produceddepends on the number of available empty sites. We show that such a predator-prey system may finally reachone of the following three steady states: coexisting, with predators and prey; pure prey; or an empty one, inwhich both populations become extinct. We demonstrate that the probability of arriving at one of the abovestates depends on the initial densities of the prey and predator populations, the amount of cover, and the wayit is spatially distributed.

DOI: 10.1103/PhysRevE.63.051909 PACS number~s!: 87.23.Cc, 82.20.Wt, 87.10.1e

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I. INTRODUCTION

The problem of dynamical relations between two,more, interacting populations already has a long historystarted with the classic papers by Volterra@1# and Lotka@2#describing the fluctuations in the fish catch in the AdriatFor some time, investigation of the predator-prey systewas the domain of mathematicians and biologists@3#. For anextensive introduction see, e.g.,@4#. However, recently it at-tracted the attention of many physicists. First only homoneous populations~no spatial dependence! were consideredthen it was realized that new, interesting cooperative pnomena could occur in extended systems in which the polations live on a lattice.

Generally speaking, there are two goals in studyingdynamics of the predator-prey system. One is the explanaof the possible oscillations in the temporal evolution of tdensities of prey and predators, as well as of the correlatbetween them. This is the classic problem in the field. Tpapers by Lipowski and Lipowska@5#, Bradshaw and Moseley @6#, and Taitelbaumet al. @7# are examples of this approach showing different methods and starting points.modern attempt to explain the recorded correlations betwthe oscillations of prey and predators has been propose@8#.

The second problem is the derivation and discussionthe long-time steady states at which a predator-prey sysfinally arrives. Boccaraet al. @9# used a probabilistic cellulaautomata method to model a lattice system in which the ptries to avoid the predator and the predator moves indirection of the prey. The predator eats the prey with a givprobability and it may die with a given probability. Botspecies produce just one progeny.

This kind of approach has been continued by Rozenand Albano@10#, who added the condition that the speci

*Email address: Michel.Droz@physics.unige.ch†Email address: apekal@ift.uni.wroc.pl

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are aware of each other within a given neighborhood. Tauthors construct a phase diagram showing two phasessible in their model—coexisting and prey only—which thecall the absorbing one.

Gerami and Ejtehadi@11# studied a predator-prey systein which the predators~herbivores! move on a square latticeand eat out prey~plant!, which grows again after some timeA predator can survive without eating for a certain time athen breed, giving one offspring. The model is solved,mean-field approximation~MFA! -type differential equationsand Monte Carlo~MC! simulations, showing oscillations inthe time dependence of the densities of both speciesbifurcations in the phase space of the two densities. Notempt is made to discuss the final steady state.

The problem of the final state has been discussed at lein the paper by Maynard Smith and Slatkin@12#. They con-sider a model with breeding seasons separated by a wseason during which the predator must find prey at a cerminimum rate or starve to death. Young and adult predahave different hunting possibilities and the prey may ficover. The coexistence of prey and predators was possibthe density of predators was well below the carrying capaof the habitat.

In this paper, we present a lattice model that contaalmost all the features found in the papers cited above, psome that have not yet been considered but are neverthimportant. We use Monte Carlo simulations with sequenupdating.

The predators and prey move on a lattice, and in ordesurvive each species must eat and breed. A certain percage of lattice sites serves as a cover for prey. Mean-field ttheories model the inhibition of population growth by crowing with an ad hoc ‘‘Verhulst factor,’’ but in our model thieffect emerges naturally~see below! and we do not need toimpose such a term. However, even in the absence of prtors, the population of prey stays below the carrying capaof the system.

We shall be interested mainly in discussing the long-ti

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steady state. In particular, we would like to find an answethe following questions.

~1! What is the role of the initial densities of prey anpredators on the chance of reaching a coexisting state?

~2! What is the role of cover and its spatial organizatio~3! How is the fate of the system affected by such deta

of the model as the habitat size, number of offspring, etcThe paper is organized as follows: in the next section

present the model, Sec. III contains the results of the MoCarlo simulations, and the conclusions are in the final stion.

II. THE MODEL

We consider a square lattice of linear sizeL with periodicboundary conditions. On a cell there may be a predator~e.g.,a wolf!, a prey ~e.g., a rabbit!, both of them, or neitherDouble occupancy by two animals of the same species isallowed. Each cell is either covered with grass or a treegrowing on it. Both types of animals move randomly on tlattice, following the blind ant rule, i.e., a direction formove is randomly chosen, but if the target cell is alreaoccupied by the same species, the move is not realizedthere is no second try during one Monte Carlo step~MCS!.On a grassy cell, a rabbit eats grass and a wolf eats rabthe two are on the same cell. The configuration of the tetory, i.e., grass and trees, does not change in time, wmeans also that the grass is always available to the rabbigrass sites.

Each animal has to feed at least once everyk MCSs. Thisis realized in our model by attributing to each animalcounter containing justk ‘‘food rations.’’ The counter is in-creased by 1 after each ‘‘meal’’~eating grass by a rabbit oeating rabbit by a wolf! and decreased after completing oMCS. For simplicity, we assume the same value ofk forpredators and prey. An animal that did not eat fork MCSsdies. Similar, although simpler, ideas of a finite reservefood characterizing predators has been considered in@11#and, in some sense, also in@12#.

If an animal has at least one nearest neighbor of the sspecies, the pair produces, at most,M offspring. M is thephysiological birth rate. In order to breed, apart from findia partner in its neighborhood, the animal must be stroenough, i.e., it must have at leastkmin food rations. Eachoffspring receives at birthkof food rations~which are notdeducted from the parents’ stores!. The offspring are locatedusing again the blind ant rule, within the Moore neighbohood @13#, which in the case of the square lattice contaeight cells. If the density is high, there is room for onlyfraction ofM, hence fewer progeny are born. This procedtakes care of the unrealistic unlimited growth of a populatfound in the classic Lotka-Volterra models~see@4#! and re-places, in a natural way, the phenomenological Verhulsttor.

When an animal moves into a cell with a tree, nothihappens. There is no grass for a rabbit to eat, but amakes a shelter against a wolf. Breeding is also imposson the tree-occupied cell. Hence, such sites make a coverabbits, but it is a potentially dangerous one, since the ra

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may die there of hunger. Of course, a wolf may also diehunger on a tree cell.

The parameters of the model are as follows: linear sizethe latticeL, initial densities of both species: prey~rabbits!cr(0) and predator~wolves! cw(0), concentration of treesct , maximum period of time~MCS! k an animal may survivewithout eating, the amount of food,kmin , needed to be fit forbreeding, the amount of food,kof , received at birth, andfinally M, the physiological birth rate. To make the modtractable, we have decided on fixing most of the parameat the following reasonable values:k56, kmin52, kof52,and M54. We shall therefore investigate mainly the roplayed by three parameters—cr(0),cw(0), and ct—on thetype of the steady state reached by the system, althoughrole of other parameters will be discussed briefly in the nsection. In general, we have usedL550.

The territory~distribution of tree and grass cells! is pre-pared before starting the simulations. The trees are eitherrandomly or they are organized into a compact cluster ming a forest. In the text below, we will refer to the two casas ‘‘trees’’ and ‘‘forest,’’ respectively.

The dynamics of our model can be decomposed intophases: diffusion~motion! and reaction~eating and breed-ing!. The two are included in the following algorithm:~i! Acell is randomly chosen. If it is occupied by either animthen ~ii ! a direction for a move is randomly determined.the chosen cell is empty, the move is realized;~iii ! if there isgrass on the chosen cell then the following holds true:~a! ifthere are two animals~a wolf and a rabbit!, the wolf eats therabbit and the wolf’s counter is increased by 1,~b! if there isa rabbit but no wolf, the rabbit eats grass and the rabbcounter is increased by 1,~c! if there is a wolf and no rabbit,nothing happens, and~d! a check is made if the animal thaton the cell has enough resources to breed, i.e., whethecounter reads at leastkmin . If so, one search is made randomly in the nearest neighborhood for a partner. If itfound, the pair producesM offspring. For each of them, anindependent search is made for a free cell in the Moneighborhood. If found, the progeny is put there and itceiveskof food rations. If the cell is already occupied, noting happens and we search for a place for the next baby.~iv!If there is a tree on the chosen cell, nothing happens.~v!After completing as many picks as there are animals attime, one MCS has been accomplished and the food ratof all animals are decreased by 1.

Most of the simulations were performed on lattices50350 size, although to estimate the size effects wecreased it in some cases to 1003100 and 2563256. To findthe chances of a population with given initial parametreaching one of the three possible final states~coexistence,pure prey, or empty!, we have averaged over 100 indepedent runs, choosing a different territory each time, but wthe same amount of trees.

III. RESULTS

The main object of our study was the final state attainby a population. As could be expected, we have found thpossible states: the coexisting one with prey and predat

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COEXISTENCE IN A PREDATOR-PREY SYSTEM PHYSICAL REVIEW E63 051909

the absorbing one with prey only, and the empty one whno animal survived. Which one is the fate of a given poplation depends on the parameters characterizing the poption and the territory.

Consider first the ‘‘tree’’ case when the trees are distruted randomly over the lattice. One of the possible realitions is shown in Fig. 1. There are indeed very few and smtree clusters. In such a territory, the wolves and rabbits halso been put in a random way. After approximately 1MCSs, the system arrives at a quasisteady state. If it cosponds to a coexisting state, the densities of wolvesrabbits fluctuate around some ‘‘average’’ values, whichnot change in time and which are the same in all coexisstates, irrespective of the initial parameters~see Fig. 2!. Ifthat final state is that of prey alone, its density stabilizafter extinction of the predators, at a level below the carrycapacity of the habitat, which is equal to 1 in our model~seeFig. 3!. We have not found periodic oscillations in the timdependence of the prey density in the absence of predaSimilar kinds of curves have been found by Maynard Smand Slatkin@12# in their nonspatial model. The random ocillations of the two densities, as shown in Fig. 2, are veweakly intercorrelated. Fourier analysis performed ondata showed a wide uniform spectrum.

Initial densities

The chances of a population, starting with a given initdensity of wolves and rabbits, to arrive at each of the thpossible final states are shown in Fig. 4 for small concentions of trees (ct50.2). Since the initial configurations, evefor fixed ct ,cr ,cw parameters, are different, and the evotion is probabilistic, the system may reach different statioary states. It is clear that the prey and predator populatcan coexist only if the initial density of the wolves does n

FIG. 1. An example of the distribution of trees~black squares!and grass~white squares! for ct50.1.

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exceed that of the rabbits too much. However, if the initdensity of the wolves is greater than 0.4, then, as seen in4, the empty state is the most probable one. For a laenough initial concentration of predators, increasing thetial density of the rabbits may result in increasing the proability that the system will reach the pure prey state, butthe coexisting one. When increasingcr(0) for a fixed ratiocr(0)/cw(0)<1, one passes from a coexistence through ponly to an empty state. The probability to reach a givsteady state varies continuously as a function ofcr(0). Thefinal distribution of animals is shown in Fig. 5.

Cover

The existence of trees is beneficial for the prey sincepredators cannot eat them there, but for the very same rea tree does nothing for the predators. Moreover, the simtions show that in a territory without any trees, the probab

FIG. 2. Time dependence of the densities of predators~dottedline! and prey~solid line! when the final state is the coexisting oncr(0)50.1, cw(0)50.1, ct50.2, L5256.

FIG. 3. Time dependence of the densities of predators~dottedline! and prey~solid line! when the final state is prey only.cr(0)50.4, cw(0)50.5, ct50.1, L550.

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ity of arriving at the pure prey state never exceeds the prabilities of the remaining states. The pure prey state mbecome the most probable one, in a rather narrow rangthe parameters, if the concentration of trees isct50.1.

FIG. 4. Phase diagram showing probability of reaching onethe final states starting from a given initial density of predatorsprey. Size of the white area corresponds to the probability offinal state with no animals, light gray to the state with prey onand the dark gray area to the coexisting one~prey and predators!.ct50.2. The probabilities vary continuously between shosquares.

FIG. 5. Typical spatial arrangement of predators and prey incoexisting state. On white squares there are no animals, ongray ones there are prey only, on the dark gray ones therepredators, and on the black squares there are predators andcw(0)50.1, cr(0)50.1, ct50.1.

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Let us consider now the case of trees growing on neiboring cells and forming a cluster~the ‘‘forest’’ case!. For agiven concentration of trees, this kind of configuration favothe predators, leaving more open space for hunting andcover for the prey. In the case of equal, and rather larinitial densities of prey and predators,cr(0)5cw(0)50.3and ct50.2 the chances a population has of ending upeither empty, coexisting, or pure prey states are 0.01, 0and 0.97, respectively, in the case of dispersed trees,0.07, 0.81, and 0.12 in the forest. A predator populationa good chance~over 20%! of survival, and hence of realizingthe coexisting state, even for very high initial densitiespredators and prey@like cw(0)50.5 andcr(0)50.6# if theevolution takes place in the forest. In contrast, the predapopulation becomes extinct in the ‘‘tree’’ case ifcw(0).0.3. For compact clusters of trees~or no trees at all!, wehave found correlations between the time dependencethe two densities, predators~wolves! and prey~rabbits! ~seeFig. 6!. The role of the concentration of the trees and the wthey are arranged are shown in Fig. 7. Here again, the slarity between the situation of no cover and trees groupin a compact cluster is clearly seen. The forest, after inipenetration of the animals, remains empty, as shownFig. 8.

Finite size effects

From a biological point of view, the most interesting cais that of populations living in a small habitat. It is weknown @14,15# that populations of small size are very vunerable and may become extinct because of stochastictuations. This is exactly what we have found in our modPopulations of prey and predators with the initial densitiescr(0)50.1, cw(0)50.2, and the concentration of forect(0)50.1 were quite stable on a 50350 lattice, having an83% chance of coexistence and a 17% chance of becomipure prey state. The same system on a smaller, 20320, lat-tice showed quite different behavior: 30% of coexististates, 14% of the prey only, and 56% of an empty o

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FIG. 6. As in Fig. 2, except that the trees form a compact clusand the size of the lattice is smaller (L5100), hence the fluctuations are larger.

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COEXISTENCE IN A PREDATOR-PREY SYSTEM PHYSICAL REVIEW E63 051909

Increasing the number of predators, the probability of arring at an empty state grows dramatically, and atcw(0)50.5 it is above 90% for all initial densities of the prey, othe 20320 lattice. The important factor is the numberindividuals ~size of the population! rather than the concentration. Consider two populations with an equal initial number of prey and predators—one on a small and one on a llattice. The population living on a larger~e.g., 50350! latticereached higher densities fairly soon and the stochastic fltuations left no marked effect, while the one living onsmall lattice had no chance for growing~there is not muchroom for growth! and was easily affected by such fluctutions.

Taking lattices bigger than 50350 does not change thgeneral trends, although when the initial density of predais high and that of prey is low, coexistence becomes mlikely as the lattice grows. Analyzing the fluctuations arouaverage densities for different lattice sizes, fromL520 toL5256, convinced us that they decrease approximately1/L.

Other parameters

For simplicity, we have assumed the same numericalues for the parameters characterizing the prey and the pr

FIG. 7. Chances of reaching the coexisting state versus indensity of predators~a! and of prey~b!, for different concentrationsand arrangement of trees. If the trees are randomly located anct

>0.3, the chances of reaching the coexisting state do not ex5%. L ~case of no trees!, 3 (cf50.1), 1 (cf50.2), andh ~forestandcf50.2). Lines are guides to the eye only.

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tors, although this is clearly not very realistic. The phyological birth rate, i.e., the maximum number of offspringpair may have, is an important parameter. Lowering it froM54, taken in the simulations, toM53 resulted in thealmost total elimination of the predators; the only availabfinal states were prey or empty. IncreasingM to 5 did notchange anything at lower initial densities, but greatlycreased~by a factor of 4 at least! the chances for coexistenceA higher reproduction rate for both populations favors tpredators, while lowering it below a certain threshold leato extinction of the predators or of both species.

The same effect of elimination of the predators hasincrease of the food reserves needed to breed. A decrfrom k56 to k55 favored the prey and resulted in a smalprobability for an empty state and a larger one for a stwith prey only.

Since the food rations are decreased after each MCS,ering the amount of food received at birth fromkof52 tokof51 kills most of the progeny and it results in the extintion of both populations. Raisingkof to a higher value per-mits the newborn to survive without eating for several MCand produces offspring. This may lead to quite an unrealisituation when the predators are still alive after eating outprey.

IV. CONCLUSIONS

We have presented a lattice model of prey and predathat move randomly on a territory containing either grasstree. On a grass cell, prey eat grass and predators eatwhile on a tree cell, the prey is safe from the predatorfinds no food. To survive, each animal has to eat at leonce everyk MCS. An animal may breed, provided it findspartner in the neighborhood and it has enough food reserThe offspring are located in the neighborhood of the pare

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MICHEL DROZ AND ANDRZEJ PEKALSKI PHYSICAL REVIEW E 63 051909

which do not die after giving birth, hence we have a systwith overlapping generations.

A real ecological system is of course very complex acontains many important factors, out of which the presmodel considers only a few, namely spatial inhomogeneof the habitat, limited ability of the animals to survive without eating, and their mobility. The inhomogeneity is realizhere by dividing the territory into two, nonoverlapping, rgions, one providing food and a mating ground for prey apredators, and the second providing cover for prey. As inreal world, here also reproduction and finding food arelated. In order to mate, an animal must have energy resouabove a certain level. To maintain this state, it has to gafood and avoid~especially predators! ‘‘cover’’ cells. By im-posing a limited time of survival without eating, we force thanimals to move, otherwise an animal will die and liberathe cell on which another animal may move. In our model,in nature, animal mobility is necessary for catching food amating. The age of the animals is not considered here expitly, but denying the reproduction power to newborns hasimpact on the dynamics, as with the different hunting skin the model of Maynard Smith and Slatkin@12#.

The MC simulations show that there are three possfinal states into which the dynamics will lead the poputions: coexistence of prey~with density about 0.50! andpredator~with density about 0.32!, prey only ~with concen-tration about 0.90!, and an empty state in which both poplations are extinct. The coexisting state is realized only ifparameters characterizing the populations are within atain, rather restricted, range. A population may die out ifbirth rate is too low, getting food requires a very long timor there are too many enemies. The size of the predatoprey population may fall below a safe number. This is tsituation found in nature@12,16#.

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The initial density of predators cannot be too large,though for smaller densities it may be greater than the dsity of the prey. However, the knowledge of the initial desities is not enough to define completely the initial staIndeed, different initial states characterized by the sameues of cr(0) and cw(0) can correspond to quite differenmicroscopic configurations. Two such configurations cevolve in very different ways and approach different statioary states. Accordingly, the stationary phase diagramquires a probabilistic character, giving the probabilitiesreaching a particular type of stationary states, starting fran initial state defined in terms ofcr(0) andcw(0).

As Maynard Smith and Slatkin@12# found in their ownmodel, the existence of cover may lead to extinction ofpredators, but it does not increase the chance of coexisteThe way in which the cover~here trees! is organized alsoplays a role. If the trees are scattered around all the territthus offering many shelters for the prey, the predators hfewer chances than when all~or most! of the trees aregrouped in a compact cluster~a forest!. When the initialdensity of predators exceeds 0.4, the prevailing final staare either prey only or empty state.

ACKNOWLEDGMENTS

A.P. is grateful for the hospitality he received at the D´-partement de Physique The´orique de l’Universite´ de Gene`ve,where a large part of this work was done, and he acknoedges useful discussions with Marcel Ausloos and NicoVanderwalle. M.D. is grateful for the hospitality he receiveat the Institute of Theoretical Physics of the UniversityWrocław, where this work was completed. This work wpartly supported by the Swiss National Foundation. Wegrateful to Je´rome Magnin for his help.

e,

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