Evolutionary stable sets in mixed-strategist models

Evolutionary stable sets are used as an extension of the concept of an evolutionarily stable strategy (ESS). They have, as sets, essentially the same properties as ordinary ESSs. Here, ES sets are applied to the characterization of what will happen in an asexual population of mixed-strategists under frequency-dependent selection. Such a population will tend to establish some state, usually not a unique one, that belongs to an ES set. For an important class of widely used mixed-strategist models, ES sets are found to comprise just those population states that allow the possible behavioural acts to be equally successful, or, to put it more precisely, that establish an evolutionarily stable population strategy.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

THE EVOLUTIONARILY STABLE PHENOTYPE DISTRIBUTION IN A RANDOM ENVIRONMENT

In an unpredictably changing environment, phenotypic variability may evolve as a “bet-hedging” strategy. We examine here two models for evolutionarily stable phenotype distributions resulting from stabilizing selection with a randomly fluctuating optimum. Both models include overlapping generations, either survival of adults or a dormant propagule pool. In the first model (mixed-strategies model) we assume that individuals can produce offspring with a distribution of phenotypes, in which case, the evolutionarily stable population always consists of a single genotype. We show that there is a unique evolutionarily stable strategy (ESS) distribution that does not depend on the amount of generational overlap, and that the ESS distribution generically is discrete rather than continuous; that is, there are distinct classes of offspring rather than a continuous distribution of offspring phenotypes. If the probability of extreme fluctuations in the optimum is sufficiently small, then the ESS distribution is monomorphic: a single type fitted to the mean environment. At higher levels of variability, the ESS distribution is polymorphic, and we find stability conditions for dimorphic distributions. For an exponential or similarly broad-tailed distribution of the optimum phenotype, the ESS consists of an infinite number of distinct phenotypes. In the second model we assume that an individual produces offspring with a single, genetically determined phenotype (pure-strategies model). The ESS population then contains multiple genotypes when the environmental variance is sufficiently high. However the phenotype distributions are similar to those in the mixed-strategies model: discrete, with an increasing number of distinct phenotypes as the environmental variance increases.     

When will evolution lead to deceptive signaling in the Sir Philip Sidney game?

Sir Philip Sidney games are a widely used model of simple signaling. Johnstone and Grafen [Johnstone, R.A., Grafen, A., 1993. Dishonesty and the handicap principle. Animal Behaviour 46, 759–764] present a version in which the Evolutionarily Stable Strategy (ESS) is for most signalers to “honestly” signal, with a small minority of signalers who “cheat”. This model is among the most frequently cited papers on the topic of “dishonest” signaling and supports the view that signals may be “dishonest” as long as they are “honest on average”. Using genetic algorithms, we demonstrate that another solution exists to the game, an evolutionarily stable set of Nash equilibria in which members of the set never signal and all donors give their resource. Payoffs to players using this set of strategies is greater those when playing the “dishonest” signaling ESS. We demonstrate that a random population is far more likely to evolve to this non-communicating strategy set than the “dishonest” signaling ESS. We also discuss the dynamics of biological game theory models and the advances of genetic algorithms as a heuristic solution method for these models.     

Population dynamics and harvesting of semelparous species with phenotypic and genotypic variability in reproductive age

We study the evolution of polymorphic life histories in anadromous semelparous salmon and the effects of harvesting. We derive dynamic phenotypic and genetic ESS models for describing the evolutionary dynamics. We show in our deterministic analysis that polymorphisms are not possible in a panmictic random mating population. Instead, genetic or behavioral polymorphisms may be observed in populations with assortative mating systems. Positive assortative mating may be supported and generated by behavioral and phenotypic traits like male mate choice, spawning ground selection by phenotype, or within-river homing-migration-distance by size. In the case of an evolutionarily stable dimorphism, the ESS is characterized by a reproductive ideal free distribution such that at an equilibrium the individuals are indifferent from the fitness point of view between the two life histories of early and late reproduction. Different strategy models - that is, phenotypic and genetic ESS models - yield identical behavioral predictions and, consequently, genetics does not seem to play an important role in the present model. An evolutionary response to increased fishing mortality is obvious and may have resource management implications. High sea fishing mortalities drive the populations toward early spawning. Thus it is possible that unselective harvesting at sea may eliminate, depending on the biological system, behavioral polymorphisms or genetic heterozygozity and drive the population to a monomorphic one. If within-river homing migration distances depend on the size of fish, unselective harvesting at sea, or selective harvesting of spawning runs in rivers, may reduce local population sizes on spawning grounds high up rivers. Finally, harvesting in a population may cause a switch in a dominant life-history strategy in a population so that anticipated sustainable yields cannot be realized in practice.     

Frequency-dependent selection in sexual family-structured populations

In this paper, a two-phenotype, single-locus, n -allele matrix game diploid model incorporating interactions between full sibs influencing personal fitness is investigated. Necessary and sufficient conditions for an ESS are given. We show that if a strategy is an ESS for this model with the payoff matrix A, then it must be an ESS for the standard game formulation with payoff matrix A+(r/2) A(T) where r is the probability to interact with a sib, but it is also possible that no ESS exists. Moreover, under the assumption of weak selection, the partial change in phenotype frequencies brings the population closer to an ESS when it exists.     

Towards a genetic theory for the evolution of the sex ratio. III. Parental and sibling control of brood investment ratio under partial sib-mating

Models of sex ratio evolution under partial sib-mating are investigated in haplodiploids and diploids. In the cases of parental and sibling control of the brood investment ratio between the sexes in diploids, we find that the “unbeatable” investment ratio obtained by W. D. Hamilton (Science156, 477–488) for his local mate competition model corresponds in our inbreeding models to a weak form ESS (evolutionary stable strategy) fixation state and also to the population investment ratio at certain internal equilibria of our models. For haplodiploids, “strong form ESS” values exist under inbreeding in models involving father and sister control. Under brother and mother control, however, the ESS derived from local mate competition models is unstable in our inbreeding models to the introduction of any other investment ratio. We stress important qualitative differences between models involving local mate competition and inbreeding.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

Learning the evolutionarily stable strategy

The possibility that animals learn a “developmentally stable strategy” (DSS) (Dawkins, 1980) is an alternative in biological game theory to the idea that evolutionarily stable strategies (ESS) (Maynard Smith, 1972) are genetically determined. A learning rule is defined as a rule which assigns for every possible behaviour the probability of displaying that behaviour at each trial of a game as a function of previous payoffs. This report examines properties of the evolutionarily stable (ES) learning rule, i.e. the rule which, when adopted by a population, is uninvadable by a mutant with a different learning rule. The DSS is defined as the strategy used by individuals with the ES learning rule. With some simplifying assumptions, it is shown that the DSS is the ESS: the ES learning rule is a rule for learning ESSs. This and other properties of the ES learning rule suggested that an approximation to such a rule is the relative payoff sum (RPS) learning rule, which states that the probability of displaying a behaviour is equal to the cumulative payoff for that behaviour relative to the total sum of payoffs for the game. Residual payoffs and a memory factor are incorporated into the RPS learning rule to account for prior expectations of payoff and the decay of memory with time. Both features are adaptive. In simulations of several frequency dependent and frequency independent games using the RPS learning rule, the response of the simulated animals was consistent with the predictions of the ES learning rule. This analysis has shown how ESSs may be achieved by non-genetic means. The RPS learning rule is described in molecular terms utilizing synthesis, storage, and degradation of a substance which elicits the behavioural response. If the RPS learning rule is used by animals, it should be possible to identify within neurons substances whose synthesis is regulated by behavioural stimuli and which initiate alternative behaviours in proportion to their concentrations.     

Evolutionarily stable strategies in differential and difference equation models

Summary A definition for an evolutionarily stable strategy (ESS) is given which is applicable to a general differential equation population model and two difference equation analogs. With the introduction of a fitnessgenerating function, it is possible to develop necessary conditions for the determination of an ESS for each of these systems. In most situations, an ESS for one system will also be an ESS for the other. Necessary conditions for an ESS are obtained. Under certain restrictions, they are shown to be valid, even under an unstable equilibrium in population density. the results are illustrated with an example which has the same ESS solution whether a continuous or discrete model is used. The behavior of the ESS for the discrete model is then examined under unstable equilibrium conditions in population density.     

A generalized war of attrition

In the “War of Attrition” model of animal conflict, introduced by Maynard Smith, a reward is obtained by whichever of two opponents displays longer, each individual incurring a cost associated with the length of the contest. This model is generalized to allow more general reward and cost functions, and restrictions on the length of contest permitted. This permits unification of the “War of Attrition” model and the “Graduated Risks” model, and also the extension to models in which contests may end either due to injury, or to retreat.In each case it is demonstrated that either (i) there is no evolutionary stable strategy (ESS) or (ii) there is a unique ESS, which is fully specified. In the case where only a finite number of pure strategies are available, global convergence to the ESS is shown.A variety of interesting conclusions of biological relevance emerge, perhaps the most striking being the occurrence of a dichotomous behavioural pattern in an essentially continuous conflict.     

Epistatic selection opposed by immigration in multiple locus genetic systems

A model of “complete” epistatis is considered in which all “plus” alleles must be present in an individual before the adaptive phenotype is expressed. The conditions under which the plus alleles and hence the adaptive phenotype can increase and reach a stable equilibrium in the presence of immigration of gametes carrying minus alleles are found. In haploids and diploids in which the plus alleles are recessive, frequencies of the plus alleles are the same at all loci, regardless of the linkage relationships. Tight linkage favors the existence of a locally stable polymorphic equilibrium, but the equilibrium with only minus alleles is locally stable unless there is very tight linkage or very strong selection. Thus, this kind of epistasis, which provides a simple model for a character that requires several components to be present at the same time, is very sensitive to even a small amount of immigration. Hence, the evolution of such characters is likely only in completely rather than partially isolated populations.     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

Payoffs and strategies in territorial contests: ESS analyses of two ecotypes of the spiderAgelenopsis aperta

Summary Game-theoretic analyses were completed on the territorial contest behavior of two populations of a desert spider that exhibit markedly different levels of within-species competition. Numerical payoff matrices were constructed from field data collected on the behavior and demography of each population. Payoffs were expressed in terms of expected future egg production. Three behavior patterns that a spider might exhibit following assessment of its weight relative to that of its opponent and the value of the site were considered: withdraw, display, or escalate. The model predicts for the more harsh grassland habitat an evolutionarily stable strategy (ESS) that makes ownership decisive in settling contests between opponents with small weight differences, whereas it otherwise assigns victory to the heavier opponent. Whereas the empirical data collected for this grassland population closely approximates the predicted ESS, that for a population occupying a more favorable riparian habitat deviates significantly. The ESS prediction for this latter population is that an intruding spider will withdraw from a contest if it is similar in weight to the web-owner. Withdrawal is common in this population, but so are display and threat and these actions were not predicted. We hypothesize that gene flow from surrounding habitats is preventing the riparian population from completely adapting to its local environment.     

THE EVOLUTION OF STRATEGY VARIATION: WILL AN ESS EVOLVE?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.     

Ecological stability,evolutionary stability and the ESS maximum principle

Summary Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals in the community, the essential element of a game exists. This game is an evolutionary game where the individual organisms (players) inherit their strategies from continuous play of the game through time. Here, the strategies are assumed to be constants associated with certain adaptive parameters (such as sunlight conversion efficiency for plants or body length in animals) in a set of differential equations which describe the population dynamics of the community. By means of natural selection, these parameters will evolve to a set of strategy values that natural selection, by itself, can no longer modify, i.e. an evolutionarily stable strategy (ESS). For a given class of models, it is possible to predict the outcome of this evolutionary process by determining ESSs using an ESS maximum principle. However, heretofore, the proof of this principle has been based on a limited set of conditions. Herein, we generalize the proof by removing certain restrictions and use instead the concept of an ecological stable equilibrium (ESE). Individuals in a biological community will be at an ESE if fixing the strategies used by the individuals results in stable population densities subject to perturbations in those densities. We present both necessary and sufficient conditions for an ESE to exist and then use the ESE concept to provide a very simple proof of the ESS maximum principle (which is a necessary condition for an ESS). A simple example is used to illustrate the difference between a strategy that maximizes fitness and one that satisfies the ESS maximum principle. In general they are different. We also look for ESEs in Lotka—Volterra competition and use the maximum principle to determine when an ESE will be an ESS. Finally, we examine the applicability of these ideas to matrix games.     

On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

The asymmetric war of attrition

The paper, which has an informal discussion at the end, provides a game theoretical analysis of the asymmetric “war of attrition” with incomplete information. This is a contest where animals adopt different roles like “owner” and “intruder” in a territorial conflict, and where the winner is the individual prepared to persist longer. The term incomplete information refers to mistakes in the identification of roles. The idea by Parker & Rubenstein (1981) is mathematically worked out and confirmed that there exists only a single evolutionarily stable strategy (ESS) for the model with a continuum of possible levels of persistence and no discontinuities in the increase of cost during attrition. The ESS prescribes to settle the conflict according to “who has more to gain or less to pay for persistence”. The only evolutionarily stable convention is thus to give the player access to the resource who has the role which is favoured with respect to payoffs. By contrast, it was shown earlier (Hammerstein, 1981) for various asymmetric versions of the “Hawks-Doves” model that an ESS can exist which appears paradoxical with respect to payoffs. The nature of this contrast is further analyzed by introducing elements of discreteness in the asymmetric war of attrition. It turns out that some conditions must be satisfied in order to have the possibility of an alternative ESS which is not of the above simple commonsense type. First, a decision to persist (or escalate) further in a contest must typically commit a contestant to go on fighting for a full “round”, before he can give up without danger. Second, such a “discontinuity” must occur at a level of persistence where the contest is still cheap, and, finally, errors in the identification of roles must be rare.     

The theory of games and the evolution of animal conflicts

The evolution of behaviour patterns used in animal conflicts is discussed, using models based on the theory of games. The paper extends arguments used by Maynard Smith &; Price (1973) showing that ritualized behaviour can evolve by individual selection. The concept of an evolutionarily stable strategy, or ESS, is defined. Two types of ritualized contests are distinguished, “tournaments” and “displays”; the latter, defined as contests without physical contact in which victory goes to the contestant which continues longer, are analyzed in detail. Three main conclusions are drawn. The degree of persistence should be very variable, either between individuals or for the same individual at different times; a negative exponential distribution of persistence times is predicted. Individuals should display with constant intensity, independent of how much longer they will in fact continue. An initial asymmetry in the conditions of a contest can be used to settle it, even if it is irrelevant to the outcome of a more protracted conflict if one were to take place.