Random evolutionarily stable strategies

Random evolutionarily stable strategies

THEORETICAL POPULATION BIOLOGY 13, 276-293 (1978) Random Evolutionarily Stable Strategies* D. AUSLANDER,+ J. GUCKENHEIMER,* AND G. OSTER+ + Un...

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13, 276-293 (1978)

Random Evolutionarily

Stable Strategies*



+ University of California, Berkeley, California 94720, and * University of Califo*nia, Santa Cruz, California 95060 Received March

23, 1977

The game-theoretic notion of competitive equilibrium has frequently been used to evaluate evolutionary trends. These discussions have centered mostly on the static situation, ignoring the constraints of Mendelian genetics. In this paper we illustrate by an example that, when population and genetic dynamics are included in a model, the outcome in a competitive situation can be quite different from that deduced from the corresponding static model. In particular, the nonlinearities due to density and frequency dependence can produce chaotic dynamics whose statistical properties may or may not fluctuate about the static “evolutionary stable strategy.”

1. INTRODUCTION 1.1. The notion that natural selection is an “optimizing” process has been a recurrent theme in evolutionary thought since Darwin. The idea that those genotypes better designed to cope with their environments will enjoy a selective advantage in a competitive world appears self-evident. In service to this conception of the evolutionary process a great deal of effort has been expended in constructing models of life history strategies using the formal machinery of mathematical optimization theory. Recently, however, several authors have voiced reservations concerning the validity of optimization arguments in evolutionary theory (Levins, 1976; Sahlins, 1976; Slatkin, 1976; Oster and Wilson, 1977). A close examination of most optimization models reveals a number of implicit assumptions which are not easily justified. The particular issue we wish to address here is the difficulty a prey population faces in achieving an “optimal” genetic configuration in the presence of a coevolving predator population. Such situations seem to fall naturally within the conceptual framework of game theory. Although there have been a few attempts to model evolutionary processes from the viewpoint of game theory, these efforts have not been too successful, partly because of the mathematical difficulties one encounters in all but the simplest situations (Lewontin, 1961; * This work was supported by NSF Grant DEB74 21240. 1 In game theory parlance an ESS is known as a Nash equilibrium.

276 OCMO-5809/78/0132-0276$02.00/0 Copyright Q 1978 by Academic Press, All


of rqxoduction

in my


Inc. reserved.




Stewart, 1971). The one contribution that game theory has made to evolutionary theory is the notion of stable adaptive strategies. That is, under situations where there is a conflict of interest between the two parties (individuals or populations) the notion of an “optimal” strategy must be discarded, since neither party can generally maximize their competing fitnesses. Instead of looking for optimal strategies, one looks for strategies capable of stable coexistence. These John Maynard Smith termed “evolutionarily stable strategies” (ESS) (Maynard Smith, 1974).1 An ESS is a strategy which is immune to small perturbations by a mutant strategy. Presumably, if both parties achieve a genetic configuration which implements an ESS the populations will coexist stably, since mutant types will be at a selective disadvantage. Maynard Smith and his colleagues have studied animal fighting behavior using the concept of ESS, and have succeeded in making some sense out of a heretofore confusing area of animal behavior (cf. Maynard Smith and Parker, 1976). However, one strong assumption has underlain most game theory models: they have not explicitly considered genetics. If an ESS exists, it is assumed that (a) each population has the genetic wherewithal to realize the stable strategy, and (b) the demographic and genetic dynamics will ultimately evolve so that the system will actually achieve the ESS. The purpose of this note is to point out that this need not be so; when genetic and demographic constraints are added to the game there may be no way for the populations to realize an ESS. In such cases a broader definition of stable coexistence must be devised. 1.2. In Section 2 we shall explore the consequences of combining density and frequency dependence into a model for the coevolution of a predator-prey system. Our investigations have been inspired by the extensive experiments of Huffaker and his associates on laboratory host-parasite populations (Hassell and HufIaker, 1969). Although raised in a controlled, deterministic environment, nevertheless, the populations exhibited highly irregular behavior over some 40 host generations. Other studies on single populations indicate that insect populations can develop significant genetic adaptations in as few as 8-10 generations. Thus one might expect that some “accommodation” of life histories might evolve between the host and parasite. What the nature of this accommodation should be is not clear; however, similar experiments by Pimentel (1968) and his co-workers suggested that the violent fluctuations which usually characterize such systems are attenuated somewhat after a number of generations. To some extent this occurred in Huffaker’s experiments. But, as shown in Fig. 1, strong irregularities persisted until the end of the experiments. The fluctuations in the host and parasite populations shown in Fig. 1 have two striking features: (a) Both populations exhibit pronounced periodic behavior; moreover, the host cycles maintain a fixed phase relationship with those of the parasite. (b) The cycles appear “noisy.” That is, the pattern of peak heights




FIG. 1. Population census of adult hosts and parasites in a constant environment laboratory ecosystem; host: Anagasta kukniella, parasite: Venturia canescem. (From C. B. Huffaker, Univ. of California, private communication.)

is irregular, indicating either that there are strong stochastic forces influencing the dynamics or that the dynamics are very complicated. The synchronized cycles can be explained by the age-specific nature of the host-parasite intersection. A complete discussion can be found in Auslander et al. (1974). They constructed a model which included the age structure of both populations and which exhibited a stable periodic orbit. The behavior of the model reflects the ability of the populations to coexist with appropriately synchronized age structures. Here, however, we wish to focus attention not on the periodic behavior, but on the irregular features which one might naively ascribe to “noise,” or experimental error. The point we shall emphasize is that density and frequency dependent nonlinearities can reproduce the irregular features of the dynamics without random perturbations. Moreover, this same mechanism can prevent the populations from achieving an ESS which corresponds to steady, or even periodic population levels. Furthermore, we shall argue that this inability to achieve “steady” ESS may be a feature of many



ecological systems, and indeed may represent the best strategy for “tracking” an uncertain environment or avoiding a coevolving predator. 2. ESS IN HOST-PARASITE MODELS 2.1. An Experimental Prototype A complete description of Huffaker’s experiments can be found in Hassell and Huffaker (1969). The setup was roughly as follows. Cages containing a parasite wasp (Exidichthis) and its host, the moth Anagasta kiihnilla, were maintained in constant environment cages for roughly 75 parasite generations. The host larvae dwelt and pupated in a layer of rolled oats on the cage bottom. Adult parasites probed the oats with their ovipositors in search of vulnerable larvae. Parasitized larvae were doomed, but the conversion from parasitized larva to adult parasite was not always one-to-one. All other host life stages were invulnerable to parasitization. Figure 1 shows a running census of the host and parasite adult populations for one of the cages. 2.2. Static ESS In our treatment here we shall assume that the populations have already synchronized their age structure by the mechanism described by Auslander et al. (1974). In this periodic mode the host and parasite populations are virtually discrete, so that the generation-to-generation abundance of a particular age class can be described by difference equation models rather than differential equations. Hassell and Huffaker (1969) were able to match the generation-togeneration abundance by such models. The general forms of these equations are: (number of adult hosts in generation t + 1) = H’ = Hf(H, P), (number of adult parasites in generation t + I) = P’ = Hg(P), where f(H, P) is the fraction of hosts in generation t surviving to generation t + 1, and g(P) is the proportion of hosts parasitized. While the nature of the behavioral mechanisms governing the population interactions was not clear, they involved the parasites’ searching behavior and the hosts’ ability to avoid parasitization. The following hypothetical scenario, while surely incorrect in detail, illustrates the features of the strategic alternatives available to each population relevant to our model. The host larvae develop in the medium on the cage bottom. We shall assume that the depth at which they dwell is determined by some polygenic behavioral trait which we shall simplify to 1 locus with 2 alleles, denoted (A, a). (Alternatively, we could view the locus in question as one component of a more complex genetic configuration.) Furthermore, we shall assume simple dominance so that the host population can exercise but two phenotypic strategies: s1 = dwell shallow, s2 = dwell deep.







shallow (AA) r-I; Pu’ , ( e22 Deep (Aa, aa) lz?Fz! p12







shall.ow (BE)

E-P (Bb, bb) (a)








Similarly, the searching behavior of the parasites is also determined by 1 locus with 2 alleles (B, b) which permit but 2 strategies; T1 = search shallow, T, =

search deep. (This could correspond to ovipositor length or to a behavioral trait to probe shallowly

or deeply.)

Thus we have the game matrix shown in Fig. 2. The genetic payoff to each population for adopting a particular strategy is measured by the population levels of each in the subsequent generation (H’, P’): Payoff to hosts (parasites) for adopting strategy si while the parasites adopt strategy Tj = Hii( First, let us assume that the populations are steady at H, P (i.e., this amounts to assuming that the peaks in Fig. 1 are all congruent). For small deviations about the equilibrium (6H, 6P) we can linearize, and reduce the payoff matrix to a constant-sum game since each host escaping parasitization produces a proportional loss in to the parasite population: SH,, N -8Pij. By resealing the matrix

entries this can be converted

to a zero-sum

game whose entries

we denote 7rij . In general one could expect the relative payoffs to each player to obey the inequalities wll > n-z’22 > rIz > 7rzI . This follows if we assume that deep probing




is lower than that of shallow

probers. A moment’s reflection reveals that such a game has no ESS in pure strategies. If either population were monomorphic (say (AA) and (BB) as shown in Fig. 2), then a mutant allele a causing hosts to dwell deep would spread. When sufficient hosts had become deep dwellers, the allele, b, for deep probing would become advantageous. A mixed strategy ESS always

exists for zero-sum games. We regard the pure strategies as corner strategies on the unit square (0 < U, v < 1) as shown in Fig. 2b, and generate the payoff surface

qu>4 = (%1- 4 (Z: ?)(I ” .) A contour plot of L(u, v) = constant isoclines is shown in Fig. 2c. The ESS is located at2 u* = CT22 - ~,I)/4 (2.3) (2.4) (2.5) * The conditions for a mixed ESS (i.e., a game-theory saddle point) are computed from the relations:

aLlat = 0 = aLlav, a*Lja0* > 0. aLlau2 > 0,

(2.1) (2.2)

Since, by definition, L(u, v) is bilinear, the surface M is ruled (i.e., generated by straight lines, so that the level contours are hyperbolas), and so conditions (2.2) are always fulfilled. Note that an ESS corresponds to a (calculus) saddle point only when the additional condition a*L/aUav = 0 obtains, i.e., the saddle must “line up” with the coordinate axes. 653113/2-8





(b; FIG.







If we interpret u and ‘u as probabilities, then ESS (u*, w*) gives the equilibrium phenotype frequencies in the populations.3 The above zero-sum assumption breaks down for larger excursions from the equilibrium population levels since the gains to the host will generally not be proportional to the losses to the parasite. In the non-zero-sum case there are two payoff surfaces, L,(u, 7~)and L,(u, V) as shown in Fig. 3a. A mixed strategy ESS (II*, w*) must satisfy aL,/au Iv* = 0 = aLJaw lu* ,


ax,jav2 < 0,



aqat2 G 0, O

u* = [& - 4wH,


where the superscripts (H, P) refer to the host and parasites, respectively. Equations (2.9) and (2.10) will generally not have a solution in the unit square, a fact emphasized by Maynard Smith in analyzing animal conflicts (Maynard Smith, 1976). Therefore, he surmised that one should expect monomorphic strategies in most cases of asymmetric conflict. Furthermore, since

aqa22 lZlt= asqavaluI = 0, a polymorphic ESS, if it does exist, is only “neutrally stable” (i.e., a “weak Nash” solution). That is, small deviations from the ESS, while not advantageous, are not penalized either. We shall show that when the population and genetic dynamics are added to the model the stable coexistence conditions are polymorphic, but do not correspond to the static ESS. 2.2. Dynamic ESS We shall specify the general form for the equations in Section 2.1 by assuming (i) a logistic-type density dependence for the host population, and (ii) a random 8 Additional conditions are required to specify the gene frequencies of host (q, 1 - q) and parasite (p, 1 - p). If Hardy-Weinberg equilibrium is imposed, then the parasite gene frequencies are obtained from u = p2 and 1 - u = 2p(l - p) + (1 - p)$. The second equation follows from the first, so the ESS can always be realized by a H-W equilibrium.






encounter model for the parasite searching efficiency. The resulting equations are (Beddington et al., 1957).

P’ = &H(l - e-“P),


The parameters {f, K, a, c?} have the following interpretation. f is the intrinsic growth rate of the host population at low densities, R is the host equilibrium population in the absence of parasites, a is the effective search area of the parasite and C is the proportion of parasitized host larvae that produce adult parasites. The overbars are intended to denote averages over the genetic variables to be introduced below. Beddington et al. (1975) studied the dynamic behavior of Eqs. (2.11) and (2.12) numerically for various values of the parameters. They found that, as the parameters r and ti were increased, the system passed through a sequence of bifurcations which gave rise to periodic orbits, invariant curves, and chaotic behavior. The periodic and almost periodic behavior can be understood in terms of conventional bifurcation theory; a discussion of bifurcation theory applied to a simple population model can be found in Guckenheimer et al. (1977). The chaotic behavior is not well understood, although numerous authors have studied the phenomenon in various settings (e.g., Smale, 1967; Bowen, 1975; Li and Yorke, 1976; Ruelle and Takens, 1971; May and Oster, 1976; May, 1974). Next, we model the genetic dynamics as follows: Denote by subscripts (i,j) the genotypic properties of the host and by (m, n) the parasite properties. We assume that the host locus affects the value of the parameters rif and Kij . Since both host and parasite loci influence the interaction coefficient, Cr, we shall denote by Uijmn the search area by mn-type parasites for G-type hosts. That is, aijmn measures the relative effectiveness of parasite genotype (tn, n) in parasitizing hosts of genotype (i,j). Note that we are assuming that the host locus affects both its net reproductive rate yij and its “visibility” to the parasite, aijmn . This is not essential to our subsequent discussion, but we retain it as a crude representation of possible epistatic trade-off effects. The genotype averaged parameters r; K, and B in Eqs. (2.11) and (2.12) are now replaced by the genotypic dependent parameters rij , kij , and aijmn. We denote by pi the gene frequency of the host and by p, the gene frequency of the parasite. In order to keep the number of parameters manageable we shall assume that the interaction coefficients aijmn are specified by the ruled “surface” shown in Fig. 4. To each host and parasite strategy pair we assign an effective search area, aijnln , which determines, through the equations of motion, the abundance of genotypes in the next generation. Note that the hetervzygote values are now intermediate to the homozygote strategies.




FIG. 4.



The equations relating the population levels (H, P) and gene frequencies (qi , pm) in generation t to those in generation t + 1 can be written as (H’, P’, q’, P’) = W,

P, q, 14.


The equations defining the map F: P + R4 are given in the Appendix. The resulting equations are quite complicated, though they appear to be the simplest set of equations embodying all of the relevent dynamical and genetic features for a host-parasite system.



We shall present numerical solutions projected onto the population (N, P) and frequency (q, p) pl anes. There are four basic types of attracting sets which give rise to different qualitative features of the dynamics; (1) attracting equilibria, i.e., stable fixed points of F; (2) attracting periodic orbits, i.e., stable fixed points of k iterates F’“’


(3) invariant curves, i.e., trajectories which lie on a closed curve and are quasiperiodic, and (4) complicated attracting sets within which the orbits wanders aperiodically. These “strange attractors” generate orbits which appear quite chaotic. Indeed, they have features which are indistinguishable from a sample path of a random process. One approach to their study involves examining the bifurcations of the map F as parameters in the model are varied. A discussion






and classification of bifurcation types can be found in Guckenheimer et al. (1977). In Figs. 5, 6, and 7 we present numerical simulations of the model which show what the attracting sets look like projected onto the density and frequency planes. In all cases, the heterozygote has properties intermediate 1.0



0.5 Host,


of ?. Allele

0 10 Host Population FIG. 5. Gene frequency (top) and population density planes for the model equations (5)-(14). The stable equilibrium shown corresponds to the following parameter values: /J = 0, c = 0.25, P = 0.3, I( = 10, c9 = 0.2.




between the homozygotes. (Explanations of the parameters used are given in the Appendix.) Several features of these solutions are worthy of comment.



0 Host,


of 710 Allele

10 Host Population FIG. 6. An invariant circle traced out by a “nearly 4-point cycle” (i.e., a bifurcation near arg(A) M 2rr/4). The parameter values are /3 = 0, c = 0.25, 0 = 1.1, i? = 10, & = 0.342. For other parameter values the gene frequencies projection may or may not cycle along with the population densities depending on the orientation of the orbit in R’. The size of the invariant circle for the no-genetics case, c = 0, is generally much smaller than when c f 0.






I. For small values of the parameters ci and r^ (defined in the Appendix) the system asymptotically approaches a stable equilibrium point as shown in Fig. 5. As expected the parasite gene frequencies fix at the most efficient value of the search parameter. .O



of no Allele

FIG. 7. A typical stable chaotic polymorphism at /3 = 0, c = 0.25, P = 2.55, Z? = 10, d = 0.488. The size of the chaotic region is larger than for the corresponding no-genetic limit c + 0. The location and size of the chaotic region is also quite sensitive to the choice of parameters.



2. For larger values of d and r^ the system undergoes a bifurcation as a pair of eigenvalues (h@ of the linearized system pass out through the unit circle. If h = ea~~~l~,then the new attractor is expected to be an invariant curve when a//3 is irrational and a periodic orbit (with period close to 6) when a//3 is rational. The simulations of Beddington et al. (1975) show a typical example of such a bifurcation with a//? near a multiple of &. The gene frequencies in Fig. 6 do not tend to an equilibrium, but oscillate with the population densities, even though the heterozygote’s properties are intermediate between the homozygote properties. The time course of the orbit in this attracting set has the appearance of cycles with a long periodic “envelope” analogous to beats in a linear system. 3. At still larger values of ci and r^ the system passes into a region of chaotic behavior as shown in Fig. 7. The projections of this strange attractorlike set onto the frequency plane produces a stable, chaotic polymorphism. The gene frequencies in this parameter range are unpredictable. Moreover, the trajectories of the gene frequencies are extremely sensitive to changes in initial conditions. No matter how precisely the present gene frequencies are measured, small errors may be amplified exponentially and the predicted trajectory will diverge from the actual trajectory. Similarly, the population genotype at any one time is essentially history dependent. Small random perturbations are amplified and the trajectory diverges rapidly from the course it would have folIowed had the perturbation not occurred. This property of the strange attractor lends far greater significance to small, random selective events. In situations where density and frequency dependent selection is operating it may be quite impossible to characterize the system in a simple quantitative fashion; one must settle for a description of far less resolution than we have heretofore sought. One possible approach is to characterize the system trajectories in the strange attractor by a density function giving the asymptotic probability of finding the phase point in a given region. This approach is pursued in more detail in Guckenheimer et al. (1977). We note that in practice it is probably impossible to distinguish long cycles from truly chaotic orbits; population and gene frequency data of unprecedented resolution and duration would be required.


If we were to attempt to locate the ESS of the host-parasite “game” by ignoring the demographic and genetic constraints we would freeze the dynamics and examine the static payoff surfaces. However, if we start the system using this static ESS as initial conditions, and compute the genotype con&nation the next generation, we would find that the static ESS has shifted. Only with






very special parameter choices will the static ESS be dynamically stable. In general, we cannot expect the system state x(t) = (H, P,p, 4) to “track” the “static” ESS, and indeed, under most circumstances the system will achieve no equilibrium at all. How then are we to interpret the notion of ESS in the dynamic case? Clearly the static notion is not general enough. Perhaps the nonlinearities accompanying density/frequency dependent selection create statistical dynamics which are “optimal” in some sense. If so, it is not clear what is being optimized. Indeed, Slatkin (1976) and Rocklin and Oster (1976) have given examples of genetic systems wherein static optimization criteria need not correspond to dynamic equilibria. Our conclusions here are the same, with “equilibria” replaced by “stable attractor.” When the system is in a chaotic attractor then we can interpret the situation with regard to either the host or parasite population as if we were considering a single population facing a stochastic environment. In order to “track” the environment-in this case the other population-each must adopt a “mixed” strategy which somehow matches the statistical properties of the environmental fluctuations. The probabilistic processes accompanying sexual reproduction (e.g., segregation, recombination, mating patterns, etc.) contribute to adaptive polymorphisms and phenotypic variance. In addition, we see that density and frequency dependent nonlinearities can also engender statistical mixing and contribute to an adaptive demographic dynamics. For a host population being pursued by a coevolving predator, it is surely adaptive to maintain a demographic and genetic pattern as “untrackable” to the parasite as possible. Clearly, there is a very complicated dynamic interaction taking place. This interaction results in gene frequencies and population sizes that fluctuate by significant amounts on a time scale of a few generations. The whole system “coevolves” in a manner which is driven by its own internal dynamics. Rather than the system settling to a near equilibrium which is slowly evolving, the population dynamics and the genetics of the system have become inextricably entwined. The players of our game are following simple strategies which result in erratic behavior that is not evidently to the long range benefit of either party. Three features of the model interact to prevent realizing a stable ESS: (1) nonlinearity of the dynamic equations; (2) time lags intrinsic to nonoverlapping generations, and (3) discretized genetic strategies accompanying the I-lotus/2-allele model. The only essential feature is the nonlinearity of the model. Stable periodic and chaotic dynamics can be realized by models where (2) and (3) are relaxed (Oster et al., 1976). H owever, it would be interesting to investigate the role genetic flexibility plays in achieving stable coexistence; perhaps by adding more alleles and/or loci to the above model. One has the impression that more genetic degrees of freedom might enable a species to track its environment more closely (Slatkin, 1976). However, in the case of coevolving populations this is not at all clear. The relationship of these models to Huffaker’s experiments is an open



question. For the sake of concreteness, we have based our discussion of the models on these experiments and to lend to them biological plausibility. Whether the models are more than plausible is a difficult task, to be determined by future experiment and analysis. Our arguments are offered in the spirit that they may reflect upon a specific mechanism for the maintenance of genetic diversity in natural populations. We believe that the implications of these ideas for evolutionary biology warrant additional attention.


The calculation algorithm and equations used to update the populations and gene frequencies by one generation (one time unit) are shown in Fig. Al. The variables Wij and r introduced by boxes (1) and (2) can be interpreted as the fitness of genotype (i, j) and the mean fitness of the host population. There are already two simplifying assumptions in this system of equations. First, we assume that both populations mate randomly so that the genotype frequencies of host and parasite in each generation are determined by the gene frequencies in each generation. Second, in box (6) we assume that the genotype distribution of parasites is proportional to their parents’ searching ability. There are 24 parameters in the set of equations. We reduce this number by further hypotheses. First, we reduce the number to 15 by assuming that all heterozygote properties are symmetric: rij = rii , ajimn = aijnm , etc. (This amounts to assuming that it is irrelevant whether the genes in question are maternally or paternally inherited.) Second, we write the phenotypic property values in terms of the homozygote values and then in terms of nominal phenotypic values. Define G = $(i + j); i, j = 0, 1; then we compute any phenotppic property r by x = ~~(1 - G) + rlG + 2(1 - g) G(z-, - rO) /3.


Here 7r0and CT~ are the phenotypic values for the homozygotes, and /3 determines the relative values of the heterozygote. For example, /I = -1 or 1 represents dominance of the 0 or 1 allele, respectively, while 1/3 [ > 1 represents overdominance. 1j3 1 < 1 represents the case in which the heterozygous property is intermediate between the homozygous properties. Additionally, v,, and rI in (Al) can be expressed in terms of a nominal phenotypic value 73 and a parameter c measuring the maximum possible spread of phenotype properties between n,, and m1. Wesetrr,=(l -c)iiand vI = (1 + c) ii. Equation (Al) can be rewritten r = +[(l - c)(l - G) + (1 + c) G + 4(1 - G) Gcfl].







For each of the parameters, Y, k, and a, we specify a nominal phenotypic value i, k, and ci, a proportional spread in phenotypic values, c, and a relative value of the heterozygote, p. The search area, aijmn , which depends on both the host and parasite genotypes is computed from (A2) by first using G, = &(m + n) to locate ci and then using GH = -$,(i+ j) to compute a.

““iilfyqI~~ Fraction

of host genotypes


to t+l

i New parameter values rij

Host mean fitness


W = C Wij qi qi ,

, k.. 1J Total

a.. ljmn

Host population


H'=WH in next


generation A pJij q; = qi (L-

Host gene frequencies (4) in generation


qj w


1 Host genotypes

in Hij


= H' q; qj

gen. t+l










Note: Primed variables denote values at time t+l; onprimed denote values at


v = c mn ij


Jl I

-iL Yijmn

41-e Yijmn


z (l-e kL



) Hij

-a.. ijmn

P mn


'kL) 4

r Total


Population P'SC


in gen. t+l



in generation

for constructing

v mn mn

gene frequency

p;=xv (8)

FIG. Al.


(1 - e



/P' mn

the map F in Eq. (2.13).



REFERENCES AUSLANDER, D., OSTER, G., AND HUFFAKER, C. 1974. Dynamics of interacting populations, ]. Franklin Inst. 291, 345-376. BEDDINGTON, J., FREE, C., AND LAWTON, J. 1975. Dynamic complexity in predator-prey models framed in difference equations, Nature 255, 58-60. BOWEN, R. 1975. “Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,” Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin. GUCKENHEIMER, J., OSTER, G., AND IPAKTCHI, A. 1977. The dynamics of density dependent populations models, J. Math. Bid. 4, 101-147. HASSELL, M., AND HUFFAKER, C. 1969. Regulatory processes and population cyclicity in laboratory populations of Anagasta kiihniella (Zeller) (Lepidoptera: Phycitidae). III. The development of population models, Res. Pop. Ecol. 11, 186-210. LEVINS, R. 1975. “The Limits of Optimization,” preprint. LEWONTIN, R. 1961. Evolution and the theory of games, J. Theor. Biol. 1, 382-403. LI, T.-Y., AND YORKE, J. 1976. Period three implies chaos, Amer. Math. Monthly, 985-992. MAY, R. M. 1974. Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos, Science 186, 645-647. MAY, R., AND OSTER, G. 1976. Bifurcations and dynamic complexity in simple ecological models, Amer. Natur. 110, 573-599. MAYNARD S~TITH, J., AND PRICE, G. 1973. The logic of animal conflict, Nature 246, 15-18. MAYNARD SMITH, J. 1974. The theory of games and the evolution of animal conflicts, J. Theor. Biol. 47, 209-221. MAYNARD SMITH, J., AND PARKER, G. 1976. The logic of assymmetric contests, Anim. Behav. 24, 159-175. OSTER, G., IPAKTCHI, A., AND ROCHLIN, S. 1976. Phenotypic structure and bifurcation behavior of population models, Tkeor. Pop. Biol. 10, 365-382. OSTER, G., AND WILSON, E. 0. 1978. “Caste & Ecology in the Social Insects,” Princeton Univ. Press, Princeton, N. J., in press. PIMENTEL, D. 1968. Population regulation and genetic feedback, Science 159, 1432-1437. ROCKLIN, S., AND OSTER, G. 1976. Competition between phenotypes, J. Math. Biol. 3, 225-261. RUELLE, D., AND TAKENS, F. 1971. On the nature of turbulence, Commun. Math. Pkys. 20, 167-192. SAHLINS, M. 1976. “The Use and Abuse of Biology,” Univ. of Michigan Press, Ann Arbor. SLATKIN, M. 1976. “On the Equilibrium of Fittnesses by Natural Selection,” to appear. SMALE, S. 1967. Differentiable dynamical systems, Bull. Amer. Math. Sot. 23, 747-817. SMALE, S., AND WILLIAMS, R. 1976. The qualitative analysis of a difference equation of population growth, J. Math. Biol. 3, l-4. STEWART, F. 1971. Evolution of dimorphism in a predator-prey model, Tkeor. Pop. Biol. 2, 493-506. VARLEY, G., GRADWELL, G., AND HASSELL, M. 1974. “Insect Population Ecology,” Univ. of California Press, Berkeley.