J. theor. Biol. (1995) 176, 91–102

0022–5193/95/170091+12 $12.00/0 7 1995 Academic Press Limited

Evolutionary Cycling in Predator–Prey Interactions: Population Dynamics and

the Red Queen

U D†‡, P M‡> R L§

† Arbeitsgruppe Theoretische O� kologie, Forschungszentrum Julich GmbH, Postfach 1913,52425 Julich, F.R.G., ‡ Institute of Ecological and Evolutionary Sciences, Leiden University,

Kaiserstraat 63, 2311 GP Leiden, The Netherlands and § Department of Biology,University of York, York YO1 5DD, U.K.

(Received on 21 October 1994, Accepted on 26 April 1995)

This paper describes the coevolution of phenotypes in a community comprising a population of predatorsand of prey. It is shown that evolutionary cycling is a likely outcome of the process. The dynamical systemson which this description is based are constructed from microscopic stochastic birth and death events,together with a process of random mutation. Births and deaths are caused in part by phenotype-dependentinteractions between predator and prey individuals and therefore generate natural selection. Threeoutcomes of evolution are demonstrated. A community may evolve to a state at which the predatorbecomes extinct, or to one at which the species coexist with constant phenotypic values, or the speciesmay coexist with cyclic changes in phenotypic values. The last outcome corresponds to a Red Queendynamic, in which the selection pressures arising from the predator–prey interaction cause the species toevolve without ever reaching an equilibrium phenotypic state. The Red Queen dynamic requires anintermediate harvesting efficiency of the prey by the predator and sufficiently high evolutionary rateconstant of the prey, and is robust when the model is made stochastic and phenotypically polymorphic.A cyclic outcome lies outside the contemporary focus on evolutionary equilibria, and argues for anextension to a dynamical framework for describing the asymptotic states of evolution.

7 1995 Academic Press Limited

1. Introduction

Predator–prey interactions are ubiquitous in nature(Crawley, 1992). Sometimes the ecological interactionsbetween predator and prey species can be strongenough for the predator to be a major component ofthe environment in which the prey is evolving, and viceversa. Such interactions have therefore motivated avariety of theoretical models of phenotypic coevolu-tion in predator–prey communities (e.g. Rosenzweig,1973; Parker, 1985; Abrams, 1986; Brown & Vincent,1992). Of some interest has been the question ofwhether the phenotypes of the predator and preyevolve to an equilibrium asymptotic state, such as anevolutionarily stable strategy (Maynard Smith&Price,

1973). An alternative could be that their interactionprevents attainment of an equilibrium point and thatthere is continuous evolutionary change of theirphenotypes. Following Van Valen’s (1973) Red Queenhypothesis, the latter behaviour has become known asRed Queen dynamics (Stenseth & Maynard Smith,1984; Rosenzweig et al., 1987; Marrow et al., 1992). Tomake this notion precise, we refer here to a Red Queendynamic as any phenotypic dynamic that, in theabsence of external forcing, does not tend to anequilibrium state.

In the literature, it has been argued that a Red Queendynamic would require the set of feasible phenotypesto be unbounded, so that the phenotypes could evolveto ever more extreme states. Rosenzweig et al. (1987)concluded that ‘‘the Red Queen depends on theexistence of special phenotypic features, i.e. thosewhich are independent, boundless, and about which it

> Present address: Large Animal Research Group, Department ofZoology, University of Cambridge, Downing Street, CambridgeCB2 3EJ, U.K.

91

. E T A L .92

may be said, the larger (or smaller, or denser, or furrier,or...), the better’’. This requirement is unlikely to bemet in reality, and calls into question whether RedQueen dynamics could occur at all. To investigatewhether Red Queen dynamics are possible, we havedeveloped models of the evolutionary dynamics ofpredator and prey phenotypes (Marrow, 1992;Marrow et al., 1992; Marrow & Cannings, 1993).These models suggested that, over the course ofevolution, the phenotypes could either tend toequilibrium or to non-equilibrium asymptotic states.The models did not incorporate time explicitly, and forthis reason could give only qualitative information onthe direction of evolution. To determine theasymptotic states of coevolving systems, it is necessaryto build the time-dependent processes into theframework of a dynamical system (Dieckmann & Law,1995; Marrow et al., 1995).

In this paper, we utilize a hierarchy of threedynamical models to investigate the phenotypic statesto which coevolving predators and prey could tend.These models represent different balances betweendescriptive capacity and corresponding analytictractability. Mathematical details are given in theAppendix. Section 2 introduces the ecologicalinteractions which define the predator–prey commu-nity, and Section 3 briefly explains the distinctivefeatures of the three models used. In Section 4 wedemonstrate that the system eventually attains one ofthree different evolutionary states: (i) the predator goesextinct, (ii) coevolution leads to constant phenotypesin predator and prey, and (iii) the phenotypes in bothspecies undergo coupled and sustained oscillations ona limit cycle corresponding to Red Queen dynamics.Section 5 analyses the requirements for thisevolutionary cycling. The dependence of cycling on theinteraction and mutation structure of the predator andprey is revealed, and we show that the phenomenon isrobust under changes in the modelling approach. Toour knowledge this is the first example of a Red Queencoevolutionary process driven by population dynamicchange (corresponding to the second mechanismproposed by Stenseth & Maynard Smith, 1984). Weconclude that the conceptual framework of phenotypicevolution, with its current focus on fixed points (likeevolutionarily stable strategies) as the endpoints ofevolution, needs to be expanded to encompass morecomplex evolutionary attractors such as the limit cyclespresented here.

2. The Coevolutionary Community

Our models of phenotypic evolution are under-pinnedby ecological processes describing the dynamics

of predator and prey populations. This ensures thatthe process of natural selection directing evolutionis driven explicitly by the ecology of predator–preyinteractions, rather than by an external ad hocnotion of relative fitness of different phenotypes.For simplicity, we focus on a single phenotypic traitin each species; in view of the importanceof body size in determining interactions betweenpredator andprey (Cohen et al., 1993), onemight thinkof these traits as body sizes, s1 and s2, of prey andpredator, respectively.

To describe the population dynamics in ourcommunity, it is necessary to define the ecologicalprocesses that affect the population sizes of the twospecies (Table 1). Table 1(a) describes the birth anddeath events that are dependent on phenotype, thesebeing the events that arise from encounters with otherindividuals, as opposed to the constant birth and deathevents given in Table 1(b).

Evolutionary processes in the community requirea mechanism for generating phenotypic variationon which natural selection caused by the interactionbetween predator and prey can operate. We assumethat variation is created by a simple mutation process;in order to keep the analysis tractable we envisage thatthe genetic systems of the species are clonal. Table 1(c)shows that each birth event gives rise with probabilitiesm1 and m2 to a mutant offspring in the phenotypic traits,s1 and s2, of prey and predator, respectively. The newphenotypes are chosen according to the mutationdistributions, M1 and M2, of prey and predator,respectively.

Natural selection arises from the dependence ofthe birth and death probabilities per unit time a, b

and g on the phenotypes of the interacting individuals.Various functions could be used; we use functionsas described in Fig. 1. Thus the function a, whichcharacterizes the ecological processes responsiblefor self-limitation in the prey’s population size, istaken to be parabolic such that intermediatephenotypes are favoured in the absence of the predator[Fig. 1(a)]. The function b describing the effect ofa predator on the probability of death of the prey istaken to be bivariate Gaussian [Fig. 1(b)], on thegrounds that the predator is likely to show some degreeof specialization in the size of prey it chooses relativeto its own size (Cohen et al., 1993). On the basisthat what is bad for the prey is good for the predator,the function g is related to b by a constant ofproportionality, g=h ·b. We call h the harvestingefficiency.

The ecological community presented here extendsthe model of Marrow et al. (1992) by providing afull dynamical description of the birth, death and

93

T 1Definition of birth, death and mutation processes for a prey individual of size s1 and

predator of size s2

(a) Birth and death processes affected by phenotype

Target Encountered Probability of eventindividual individual Birth/death event per encounter per unit time

1. prey s1 prey s1 death of prey s1 a(s1)†2. prey s1 predator s2 death of prey s1 b(s1, s2)3. predator s2 prey s1 birth of predator s2 g(s1, s2)

(b) Birth and death processes independent of phenotype

Probability of eventTarget individual Birth/death event per capita per unit time

1. prey s1 birth of prey s1 r1

2. predator s2 death of predator s2 r2

(c) Mutation processes

Birth event Mutation event Probability distribution‡

1. birth of prey s1 prey s14 s'1 (1−m1)d(s'1−s1)+m1M1(s'1−s1)2. birth of predator s2 predator s24 s'2 (1−m2)d(s'2−s2)+m2M2(s'2−s2)

† This death event is taken to be dependent only on the phenotype s1 of the target individual, noton that of the encountered individual s1.

‡ mi is the probability that the birth event in species i is a mutant; d is the Dirac d function;Mi is the mutation distribution.

mutation processes. It generalizes the former accountin the sense that (i) it allows stochastic populationdynamics arising from individual-based encounters,and (ii) it permits the populations to have polymorphicphenotypic distributions since multiple phenotypictrait values may be present simultaneously in each

species. From Table 1 we recover as a special case thewell-known Lotka–Volterra equations

n1=n1(r1−a(s1)n1−b(s1, s2)n2)

n2=n2(−r2+g(s1, s2)n1) (1)

F. 1. Specification of the coevolutionary community given in Table 1. The functions used to describe the effect of phenotype (s1, s2) onthe birth and death probabilities arising from encounters between individuals are: (a) prey self-limitation a(s1)/u=c1−c2s1+c3s2

1 , (b) effectof predator on prey b(s1, s2)/u=exp(−d2

1+2c4d1d2−d22 ), where d1=(s1−c5)/c6 and d2=(s2−c7)/c8, and u is a constant that scales population

sizes. Parameters take the values: c1=3.0, c2=10.0, c3=10.0, c4=0.6, c5=0.5, c6=0.22, c7=0.5, c8=0.25. The function g(s1, s2) is not shownsince it is related tob(s1, s2) by the constant of proportionality h. The constant birth anddeath terms are: r1=0.5, r2=0.05.Mutationparametersused in the paper are: m1=10−3, m2=10−3; M1 and M2 are normal distributions with mean 0 and zvarM1=2×10−3, zvarM2=2×10−3, exceptwhere otherwise stated. The quantity u=10−3 is constant throughout.

. E T A L .94

for the population sizes, n1 and n2, of prey andpredator, respectively, by assuming no mutations,random encounters, deterministic population dynam-ics (the population sizes of the species are large),and monomorphic phenotypic distributions (only onephenotype present within each species).

3. Three Dynamical Models Of Coevolution

Equations 1 illustrate how the general coevolution-ary process defined in Table 1 can be reduced bymaking appropriate simplifying assumptions. In asimilar spirit, three dynamical models are derived inthe Appendix for the change in phenotypic traits s1 ands2 of the prey and predator respectively.

1. Polymorphic stochastic model. This provides a fulldescription of the dynamics defined in Table 1.It can be given as a multivariate functional masterequation, and depends only on the assumptionof random encounters, thus allowing both forpolymorphism and for stochasticity.

2. Monomorphic stochastic model. This retains thestochasticity in the coevolutionary process, butassumes that variation in the phenotypic distri-butions is small enough for an assumption ofmonomorphism to provide a good approximation.The coevolutionary process can then be describedas a directed random walk in the phenotype spacespanned by s1 and s2. Stochastic steps occurwhen a resident phenotype is replaced by anadvantageous mutant, e.g. s14 s'1 ; a sequence ofsuch substitutions is called a trait substitutionsequence (Metz et al., 1992). The model is framedas a multivariate master equation.

3. Monomorphic deterministic model. This is adeterministic approximation to the monomorphicstochastic model above. It is given in termsof a system of ordinary differential equationsdescribing the expected evolutionary paths in thephenotype space.

Further information as to the relation between thethree models is given in the Appendix and inDreckmann (1994). The full derivation of themonomorphic models is given in Dieckmann & Law(1995), and a discussion of the third model can befound in Marrow et al. (1995).

4. Evolutionary Outcomes

In this section we describe the variety of possibleevolutionary outcomes in a predator–prey community,using the monomorphic deterministic model. Deter-ministic dynamics of this kind have been used

elsewhere in the literature (e.g. Hofbauer & Sigmund,1990; Vincent, 1991; Abrams et al., 1993), but have notpreviously been underpinned by a formal derivation.

In the case of the monomorphic phenotypicdynamics, we can immediately infer from eqns (1)that there is a region in the phenotype space whereboth species can coexist with positive populationdensities. The boundary of this region is depictedby the oval discontinuous curves in Figure 2. Onlywithin this region can the predator population harvestthe prey sufficiently to survive; given a pair ofphenotypes (s1, s2) outside this region, the predatorpopulation is driven to extinction by the populationdynamics (1). Accordingly, coevolution of thepredator and prey can only be observed within thisregion of coexistence.

For a coevolving predator–prey community startingwith phenotypes in the region of coexistence, there areeventually three possible outcomes.

1. Evolution to a fixed point. In Fig. 2(a) thephenotypic values tend to an equilibrium point;once this is reached, no further evolution occurs.There are in fact three fixed points at theintersection of the isoclines (i.e. at s1=0, s2=0, seeAppendix) in this example, as can be seen from theaccompanying phase portrait [Fig. 2(b)]; two ofthese are attractors and they are separated by thestable manifold of the third which is a saddle point.Notice that the coevolutionary process here ismultistable with two attractors having disjunctdomains of attraction; thus there may be no morereason for a particular observed asymptoticevolutionary state than the more or less arbitraryinitial conditions.

2. Evolution to extinction. In Fig. 2(c) the co-evolutionary process drives the phenotypic valuestowards the boundary of the region of coexistence[see Fig. 2(d)]. There the predator population goesextinct and the predator phenotype is no longerdefined. The phenotype space of the communitycollapses from (s1, s2) to the one dimensionalspace s1, where the prey phenotype continues toevolve to its own equilibrium point. Note here thatthe extinction of the predator is driven by theevolutionary dynamics in (s1, s2) and not merelyby the population dynamics in (n1, n2).

3. Evolutionary cycling. In Fig. 3(a) the coevolution-ary process in the predator–prey communitycontinues indefinitely; mutants replace residents ina cyclic manner such that the phenotypes eventuallyreturn to their original values and do not reach anequilibriumpoint.As can be seen fromFig. 3(b), theattractor is a limit cycle, confirming the conjecture

95

F. 2. Patterns of evolution of prey (s1) and predator (s2) phenotypes obtained from the monomorphic deterministic model. (a) Solutionthat tends to an equilibrium point over the course of time obtained using the parameter values in figure 1 with h=1. (b) Phase portrait ofthe phenotype space from which (a) is drawn with orbits shown as continuous lines in the direction indicated by the arrows: the starting pointof the orbit corresponding to the solution in (a) is shown as the diamond. The boundary of the region of coexistence of the predator andprey is given as the discontinuous oval line. Isoclines are shown as dotted lines (straight line: predator; curved line: prey); equilibrium pointsoccur at the intersection of the isoclines and are indicated by the filled circles. (c) Solution for a community that evolves to predator extinctionat time=1.30×106. After this time, the prey continues to evolve in the absence of the predator. Parameter values as in figure 1, except c1=1.0,c2=1.0, c3=15.0, and with h=1. (d) Phase portrait of the phenotype space from which solution (c) is drawn; the starting point of the orbitcorresponding to the solution in (c) is shown as the diamond. The prey isocline lies outside the region of coexistence and orbits touch theboundary of the region of coexistence at which point the predator goes extinct.

made by Marrow et al. (1992) that Red Queencoevolution can occur in this predator–preycommunity.

These three outcomes of coevolution correspond to theendpoints of evolutionary arms races discussedqualitatively by Dawkins & Krebs (1979), namely:(i) equilibrium endpoints, (ii) one side wins, and(iii) cyclic endings.

5. Requirements For Cycling

Here, we investigate the robustness of thephenomenon of evolutionary cycling. We do this intwo ways. First a bifurcation analysis of themonomorphic deterministic model is given; this allowsone to establish the range of parameters in the modelthat permit evolutionary cycling to occur. Second,we examine the monomorphic stochastic model and

. E T A L .96

F. 3. Example of evolutionary cycling using the monomorphic deterministic model in (a) and (b), the monomorphic stochastic modelin (c) and (d), and the polymorphic stochastic model in (e) and (f). Graphs (a), (c) and (e) show the values of the prey (s1) and predator (s2)phenotypes as functions of time (mean values in the case of the polymorphic model). The corresponding orbits are shown as continuous linesin the phase spaces given in graphs (b), (d) and (f). See Fig. 2 for an explanation of the phase portrait. Parameter values for these simulationsare identical and are set as given in figure 1, except m1=10−2, m2=10−2 and with h=0.14.

97

F. 4. Results of bifurcation analysis, showing the effect of theharvesting efficiency h, and the ratio of the evolutionary rateconstants r on the dynamics of the monomorphic deterministicmodel. Regions are: (1) predator absent, (2) one fixed point present,which is an attractor, (3) three fixed points, two of which areattractors, (4) limit cycle attractor.

of cases 2 and 3 with the switch occurring ath=12.6%.

The boundary of region 4 is in fact slightly morecomplicated than the description above suggestsbecause two further kinds of dynamics can occur here:(i) a limit-cycle attractor around each of the outer fixedpoints, and (ii) a limit-cycle attractor around all threefixed points with each of the outer fixed points alsobeing an attractor. But the parameter space permittingthese dynamics is very small compared to the othersand they are therefore of less biological interest. Weconclude that evolutionary cycling requires anintermediate harvesting efficiency plus prey evolutionto occur sufficiently fast compared to predatorevolution.

The results from the bifurcation analysis areintuitive in that evolutionary cycling requires: (i) theeffect of selection by the predator on the prey to begreat enough to drive the prey from the phenotypicequilibrium it would have in the absence of thepredator (h not too low); (ii) sufficient need for thepredator to track the prey’s phenotypic change (h nottoo high); and (iii) in the resulting evolutionary race,the prey must be fast enough not to be ‘‘caught up’’ bythe predator (r not too low). In view of the respiratorycosts that the predators have to meet fromconsumption of prey simply to stay alive, one wouldexpect h to be substantially less than 1 andevolutionary cycling to occur in a range of h likely tobe observed in reality.

A realization of the monomorphic stochasticdynamics is given in Figs 3(c) and (d). The parametervalues used are the same as those in Figs 3(a) and (b),and we see that the cyclic behaviour is still maintained.In addition, two major new effects should be noted.First, it can be seen that the oscillations in phenotypicvalues do not all have the same period. Thisphenomenon, which is well known in the theory ofstochastic processes as phase diffusion (Tomita et al.,1974), comes about because stochastic perturbationsalong the limit cycle are not balanced bya counteracting force, whereas those orthogonal tothe limit cycle are. Second, limit cycles whose extensionin phenotype space is small relative to the typi-cal mutational step sizes (given by zvarM1 andzvarM2) will be obscured by the stochastic noise.The boundaries of region 4 (Fig. 4) will then beless sharp than those in the monomorphic determinis-tic model. Thus, if the evolutionary cycling is tobe visible, the mutational steps must not be toolarge.

finally the polymorphic stochastic model to see howrobust the phenomenon of evolutionary cycling iswhen the simplifying assumptions of the monomor-phic deterministic model are removed.

We focus on the effect of two quantities of particularinterest from an ecological viewpoint. These are,first, the predator’s efficiency in harvesting the preyas given by the ratio h=g/b, and, second, the ratioof the evolutionary rate constants r=(m1varM1)/(m2varM2) (see Appendix). The results of thebifurcation analysis are presented in Fig. 4. Fourdistinct regions within the parameter space can beseen:

1. For hQ5% the two species cannot coexist, andtherefore no coevolution can occur.

2. For 5%QhQ9.8% there exists only one fixed pointfor the monomorphic deterministic model. Thisfixed point is an attractor; the system evolves to thispoint and there is no further coevolution once it isreached.

3. For hq14.8% there exist three fixed points of thedynamics. The two outer points are stable, andwhich of these is reacheddepends on the phenotypesinitially present.

4. For 9.8%QhQ14.8% and sufficiently high valuesof r (Fig. 4), the attractor turns into a limit cycle,giving rise to Red Queen dynamics. On the otherhand, for low values of r, the limit cycle breaksdown and we recover the dynamical behaviour

. E T A L .98

A realization of the polymorphic stochastic model isshown in Fig. 3(e) and (f), using as before theparameter values of Figs 3(a) and (b). Thephenomenon of evolutionary cycling still persistsdespite the phenotypic distributions now beingpolymorphic. In addition, this model allows for theeffects of demographic stochasticity also of the residentphenotypes (see Appendix). Although this superim-poses more random variation to the solution, cyclingis maintained. Provided that phenotypic variance isnot too large and population sizes are not too small,we thus conclude that evolutionary cycling is robustto relaxation of the simplifying assumptions of themonomorphic deterministic model, and that it canactually occur in predator–prey communities such asthe one defined in Section 2.

6. Discussion

The main result of this analysis is that evolutionarylimit cycles, in which the predator and prey phenotypescontinue to change indefinitely, are a natural outcomein a coevolutionary community. The cyclic behaviouris not an artefact of determinism or monomorphism,because the phenomenon can be observed both in thestochastic monomorphic simulations and in thestochastic polymorphic ones. Clearly, there is nogeneral rule in nature to say that phenotypic evolutionwould lead to an equilibrium point in the absence ofexternal changes in the environment.

A simple classification of the outcomes ofphenotypic evolution can be constructed from twodichotomies. The first depends on whether an attractorexists, and the secondonwhether the attractor is a fixedpoint. This gives three classes of dynamics:

(i) evolution to a fixed-point attractor withstationary phenotypes,

(ii) evolution to an attractor that is not a fixedpoint on which the phenotypes continue to changeindefinitely, and

(iii) evolution without an attractor, such that thephenotypes take more and more extreme values.

According to the definition in the Introduction,Red Queen dynamics would encompass classes (ii)and (iii). Class (iii) is unrealistic for most kinds ofphenotypes and, if the Red Queen were to dependon the existence of such dynamics in nature, onecould reasonably conclude that Red Queen dynamicswould be very unusual (Rosenzweig et al., 1987).But this would be to miss class (ii), and dynamics ofthis kind we have shown here to be feasible. In fact,

the limit cycle is but one of a number ofnon-equilibrium attractors; for instance in systemswith more than two coevolving species, chaoticevolutionary attractors could be found.

Cyclic phenotype dynamics can occur in coevolutionas is well known from theoretical studies of geneticpolymorphisms under frequency-dependent selection(e.g. Akin, 1981; Seger, 1992), and research into thedynamics of strategy frequencies (e.g. Nowak &Sigmund, 1989). The system considered here isdifferent in two respects. First, the trait values arecontinuous, whereas cyclic dynamics have typicallybeen observed in polymorphic systems with largequalitative differences between a small number ofcoexisting phenotypes. Second, and more important,the underlying genetic process here would be asequence of gene substitutions in which mutantskeep replacing the resident types rather than one inwhich the genes always coexist and undergooscillations in frequency. Thus we are here looking ata process operating on an altogether larger evolu-tionary scale.

The monomorphic deterministic dynamic describedhere in fact turns out to be canonical (Dieckmann &Law, 1995), and can be derived from other startingpoints such as quantitative genetics (Abrams et al.,1993). It seems, therefore, that there is a large class ofmodels of phenotypic coevolution with the potentialfor non-equilibriumasymptotic states. This needs to beemphasized because the assumption that asymptoticstates of evolution are fixed points underliesmuch contemporary evolutionary thought. Thisassumption and the techniques that go with it (inparticular evolutionarily stable strategies) are clearlynot appropriate for dealing with non-equilibriumasymptotic states. The prevailing view amongevolutionary biologists, centred on equilibrium points,needs to be extended to a dynamical framework toassimilate the Red Queen.

This research was supported by the Royal Society(P.M.), the Evangelisches Studienwerk e. V. (U.D.), theForschungszentrum Julich GmbH F.R.G. (R.L. and U.D.)and a NERC equipment grant GR3/8205. We thank S. A. H.Geritz, V. A. A. Jansen, J. A. J. Metz and S. Mylius fordiscussions of the work.

REFERENCES

A, P. A. (1986). Adaptive responses of predators to prey andprey to predators: the failure of the arms-race analogy. Evolution40, 1229–1247.

A, P. A., M, H. & H, Y. (1993). Evolutionarilyunstable fitness maxima and stable fitness minima of continuoustraits. Evol. Ecol. 7, 465–487.

99

A, E. (1981). Cycling in simple genetic systems. J. math. Biol. 13,

305–324.B, J.S. & V,T.L. (1992). Organization of predator-prey

communities as an evolutionary game. Evolution 46, 1269–1283.C, J. E., P, S. L.,Y, P. & S, J. (1993). Body sizes

of animal predator and animal prey in food webs. J. Anim. Ecol.62, 67–78.

C, M. J. (ed.) (1992). Natural Enemies: The PopulationBiology of Predators, Parasites and Diseases. Oxford: BlackwellScientific.

D, R. & K, J. R. (1979). Arms races between and withinspecies. Proc. R. Soc. Lond. B 205, 489–511.

D, U. (1994) Coevolutionary dynamics of stochasticreplicator systems. Diplomarbeit RWTH Aachen, Berichte desForschungszentrums Julich Jul-3018, Julich Germany.

D, U. & L, R. (1995). The dynamical theory ofcoevolution: a derivation from stochastic ecological processes.J. math. Biol., in press.

F,R.A. (1958). The Genetical Theory of Natural Selection, 2ndedn. New York: Dover Publications.

G, D. T. (1976). A general method for numericallysimulating the stochastic time evolution of coupled chemicalreactions. J. comp. Phys. 22, 403–434.

H, J. & S, K. (1990). Adaptive dynamics andevolutionary stability. Appl. Math. Lett. 3, 75–79.

M, P. (1992) The evolution and dynamics of interactingpopulations. D. Phil. thesis, University of York, York, U.K.

M, P., L, R. & C, C. (1992). The coevolutionof predator–prey interactions: ESSs and Red Queen dynamics.Proc. Roy. Soc. Lond. B 250, 133–141.

M, P. & C, C. (1993) Evolutionary instability inpredator–prey systems. J. theor. Biol. 160, 135–150.

M, P., D, U. & L, R. (1995) Evolutionarydynamics of predator–prey systems: an ecological perspective.J. math Biol., in press.

M S, J. & P, G. R. (1973). The logic of animalconflict. Nature, Lond. 246, 15–18.

M, J.A. J.,N, R.M. & G, S.A.H. (1992). How shouldwe define ‘fitness’ for general ecological scenarios? Trends Ecol.Evol. 7, 198–202.

N, M. & S, K. (1989). Oscillations in the evolution ofreciprocity. J. theor. Biol. 137, 21–26.

P, G. A. (1985). Population consequences of evolutionarilystable strategies. In: Behavioural Ecology: Ecological Conse-quences of Adaptive Behaviour (Sibly, R. M. & Smith, R. H., eds)p. 35–58. Oxford: Blackwell Scientific.

R,D.A.,W,H.B.&MG, J.M. (1994)Dynamics andevolution: evolutionarily stable attractors, invasion exponentsand phenotype dynamics. Phil. Trans. R. Soc. B 343, 261–283.

R, M. L. (1973). Evolution of the predator isocline.Evolution 27, 84–94.

R, M. L., B, J. S. & V, T. L. (1987).Red Queens and ESS: the coevolution of evolutionary rates.Evol. Ecol. 1, 59–94.

S, J. (1992). Evolution of exploiter–victim relationships. In:Natural Enemies: The Population Biology of Predators, Parasitesand Diseases (Crawley, M. J., ed) pp. 3–25. Oxford: BlackwellScientific.

S, R., A, M., C, M. & Z, G.(1986). Introduction to the Physics of Complex Systems. Oxford:Pergamon Press.

S, N. C. & M S, J. (1984). Coevolutionin ecosystems: Red Queen evolution or stasis? Evolution 38,

870–880.T, T., O, T. & T, H. (1974). Irreversible circulation

and orbital revolution. Prog. theor. Phys. 52, 1744–1765. L, J. D. & H, P. (1995) Predator–prey

coevolution: interactions across different timescales. Proc. R. Soc.Lond. B 259, 35–42.

K, N. G. (1981). Stochastic Processes in Physics andChemistry. Amsterdam; North-Holland.

VV,L. (1973). A new evolutionary law. Evol. Theory 1, 1–30.

V, T.L. (1991). Strategy dynamics and the ESS. In: Dynamicsof Complex Interconnected Biological Systems (Vincent, T. L.,Mees, A. I. & Jennings, L. S., eds) pp. 236–262. Basel: Birkhauser.

Note added in proof: Since this paper was prepared, van der Laan &Hogeweg (1995) have also considered a model of Red Queendynamics incorporating population dynamic change.

APPENDIX

In this appendix we provide a brief derivation of thethree dynamical models describing the processof coevolution for the reader interested in the moretechnical details. The theory outlined here is general inso far as it applies to a large variety of N-speciescoevolutionary communities of which the predator–prey system analyzed in this paper is just a particularinstance.

At time t there are ni individuals in species iwith i=1, . . . , N. These have phenotypes sij withj=1, . . . , ni such that the phenotypic distribution pi (si )in species i is given by

pi=sni

j=1

dsij (A.1)

with dx (y)=d(x−y) where d denotes Dirac’sd-function. These distributions change in time due tostochastic birth, death and mutation processes likethose specified in Table 1.

The dynamics of the phenotypic distributions can bedescribed by a functional master equation for P(p, t),the probability density of p=(p1, . . . , pN ) to berealized at time t (Dieckmann, 1994). The algorithmderived from this equation is the following.

1. Initialize the phenotypic distributions pi withi=1, . . . , N at time t=0 and specify the time Twhen to stop the dynamics.

2. Calculate the birth and death probabilities bi (sij , p)and di (sij , p) for each individual i=1, . . . , N,j=1, . . . , ni with phenotype sij in the environmentgiven by p.

Remark. According to Table 1 we have for thepredator–prey community

b1(s1, p)=r1,

d1(s1, p)=g a(s1)p1(s1)ds1+gb(s1, s2)p2(s2)ds2,

b2(s2, p)=g g(s1, s2)p1(s1)ds1, and d2(s2, p)=r2.

. E T A L .100

3. Construct the sums

wij=bi (sij , p)+di (sij , p),

wi=sni

j=1

wij

and

w=sN

i=1

wi

with i=1, . . . , N, j=1, . . . , ni .4. Choose the waiting time Dt for the next event to

occur according to

Dt=−1w

ln r

where 0QrE1 is a uniformly distributed randomnumber.

5. Choose species i with probability

1w

wi .

Choose individual j in species i with probability

1wi

wij .

Choose then a birth or death event with probability

1wij

bi (sij , p) and1wij

di (sij , p)

respectively.6. If a birth event occurs for an individual with

phenotype sij , choose a new phenotype s'ij withprobability density

(1−mi )d(s'ij−sij )+miMi (s'ij−sij ).

7. Update time and phenotypic distributionsaccording to t 4 t+Dt and pi 4 pi+ds'ij orpi 4 pi−dsij for a birth or death event in species irespectively.

8. Continue from Step 2 until teT.

The protocol above utilizes the minimal processmethod (Gillespie, 1976) to simulate the functionalmaster equation. Note in particular that according toStep 4 the waiting times follow an exponential distri-bution, the standard result for stochastic processesdescribed by homogeneous master equations.

If mi is sufficiently small for all i=1, . . . , N, thephenotypic distributions in each species will besharply concentrated around a single phenotype,the resident phenotype. The distributions thenare called monomorphic (precisely, one should referto them as quasi-monomorphic since still morethan one phenotype may be present in thepopulation) and can well be approximated by pi=nidsi

with resident phenotype si and population size ni .In this case phenotypic change occurs via asequence of‘ trait substitutions where a residentphenotype si is replaced by a mutant phenotype s'i(Dieckmann, 1994).

The resulting directed random walk in thephenotype space is described by the master equation

ddt

P(s, t)

=g [w(s =s')P(s', t)−w(s'=s)P(s, t)]ds', (A.2)

where P(s, t) denotes the probability density ofthe resident phenotypes to be given by s=(s1, . . . , sN )at time t. Equation (A.2) only holds if the stochasticdynamics of s are a Markov process; to guaranteethis it can be necessary to consider more than onetrait per species. This more general case is analysedin Dieckmann & Law (1995). In the infinitesimaltime interval dt, a trait substitution in only a singletrait can occur and thus the probability per unit timew(s'=s) for the transition s 4 s' can be decomposedaccording to

w(s'=s)=sn

i=1

wi (s'i , s) tn

j=1j$i

d(s'j −sj ). (A.3)

Here wi (s'i , s) denotes the probability per unit time fora trait substitution si 4 s'i to occur in species i given anenvironment of phenotypes s.

A trait substitution requires that, first, a specificmutant phenotype s'i enters the population of speciesi and, second, that it succeeds in replacing the residentphenotype si . Since these two processes are statisticallyindependent, their probabilities multiply and wi (s'i , s)is given by the product

a bZxxCxxVZxCxV

wi (s'i , s)=mibi (si , s)ni (s)Mi (s'i −si )b−1i (s'i , s)( fi (s'i , s))+. (A.4)

zxxxxcxxxxv zxxxcxxxvterm I term II

101

The functions bi (s'i , s), di (s'i , s) and fi (s'i , s)=bi (s'i , s)−di (s'i , s) denote the per capita birth, deathand growth probabilities per unit time (or rates) of aphenotype s'i in an environment given by thephenotypes s. They are defined in terms of theanalogous quantities of the polymorphic stochasticmodel by e.g. bi (s'i , s)=bi (s'i , p), withp=(n1(s)ds1, . . . , nN (s)dsN ). Here the population sizesni (s) are determined as the equilibrium solutions of theresident’s population dynamics

ddt

ni=nifi (si , s).

The general case of nonequilibrium populationdynamics is treated in Dieckmann & Law (1995).A more formal analysis is given in Rand et al.(1993).

Remark. For the predator–prey community,we obtain b1(s'1 , s)=r1, d1(s'1 , s)=a(s'1 )n1(s)+b(s'1 , s2)n2(s), b2(s'2 , s)=g(s1, s'2 )n1(s), and d2(s'2 , s)=r2.

We now explain the different terms in equation(A.4).

1. Term I represents the impact of the mutationprocess and is given by weighting (a) the probabilityper unit time for any mutant to occur inthe resident population of phenotype si by (b)the probability density for the mutant phenotypeto be given by s'i . Since mutations in distinctindividuals are statistically uncorrelated, term Iais composed of three factors. The per capitabirth rate bi (si , s) of the resident phenotype ismultiplied by mi , the mutation probability foreach birth event, giving the per capita mutationrate of the resident phenotype. This is multipliedby the resident’s population size ni to yield themutation rate of the resident population. Term Ibsimply is the mutation distribution Mi (s'i −si )for mutant phenotypes s'i given the residentphenotype si .

2. Term II stands for the process of selection. Inthe monomorphic stochastic model it is assumedthat the resident populations are sufficientlylarge not to be subject to accidental extinctiondue to stochastic fluctuations of their populationsizes. In contrast, the mutant population startswith population size 1 such that the impact ofdemographic stochasticity on its dynamics oughtto be considered. Since the function ( )+ returnsits argument if the argument is positive and 0otherwise, deleterious mutants, with fi (s'i , s)Q0,have no chance to survive in the residentpopulation. But even advantageous mutants,

with fi (s'i , s)q0, experience some risk ofaccidental extinction due to random sampling wheninitially rare (Fisher, 1958). Term II also shows thatfor large initial per capita growth rates of themutant, the probability for it to succeed in replacingthe resident saturates at 1. The exact form of termII is derived in Dieckmann & Law (1995).

Combining eqns (A.2), (A.3) and (A.4) yields acomplete description of the stochastic coevolutionarydynamics provided that phenotypic distributions aresufficiently concentrated and that population sizes aresufficiently large. The algorithm for this model againfollows the minimal process method.

To capture the representative features of themonomorphic coevolutionary dynamics directly,rather than having to consider a large number ofdifferent realizations of the monomorphic stochasticmodel, a deterministic approximation of the latter isdevised. The deterministic path �s� associated with astochastic process is described by the equation (Serraet al., 1986).

ddt

�s�=a(�s�), (A.5)

where the function a is the first jump moment of thestochastic process whose components in our case aregiven by

ai (s)=g (s'i −si )wi (s'i , s) ds'i , (A.6)

with i=1, . . . , N. If the different realizations of thestochastic process do not spread too far apart, i.e. if thevariance of the probability density P(s, t) in eqn (A.2)stays small, the deterministic path �s� provides a goodapproximation of the mean path

g sP(s, t) ds

(van Kampen, 1981).We obtain the deterministic monomorphic model by

introducing eqn (A.4) into (A.6). To simplify the result,we expand the functions fi (s'i , s) and b−1

i (s'i , s) in themutant phenotype s'i about the resident phenotype si .Here we only present the first order resultfor symmetricmutation distributions, the derivation ofhigher order correction terms and for arbitrarymutation distributions is given in Dieckmann & Law(1995). By introducing the result into eqn (A.5),we obtain the deterministic path of the monomorphicstochastic model

. E T A L .102

ddt

�si�=012 mivarMi1ni (�s�)1

1s'ifi (s'i ,�s�)bs'i=�si�

, (A.7)

zxxxcxxxv zxxxcxxxvterm I term II

for i=1, . . . , N. The deterministic path thus isdescribed by a simple, though typically nonlinear,dynamical system composed of N coupled first orderdifferential equations. The terms on the right-hand sideof eqn (A.7) have the following meanings.

1. Term I again captures the influence of mutationon the coevolutionary dynamics. The factor12 mivarMi , called the evolutionary rate constant,is affected by the proportion mi of births thatproduce mutants and by the variance varMi ofthe mutation distribution in the trait si . Togetherwith the population size ni , all these terms arenon-negative, so term I as a whole serves to scale therate of evolutionary change.

2. Term II accounts for the impact of selection as it

determines the direction of evolutionary change.When this derivative of the per capita growth rate fi

is positive (respectively negative), mutants withincreased (respectively decreased) phenotypic val-ues si will be advantageous in the environment givenby �s�. The lines on which the terms II are zero arethe isoclines of the monomorphic deterministicdynamics.

The Runge–Kutta method can be employed toconstruct an algorithm for the monomorphicdeterministic model. Equations (A.7) have features incommon with other models of adaptive dynamics(Hofbauer & Sigmund, 1990; Vincent, 1991; Abramset al., 1993), but are here explicitly derived from theunderlying stochastic ecological processes.