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.     

Evolutionarily stable sets in symmetric extensive two-person games.

Evolutionarily stable (ES) sets are characterized for evolutionary games in extensive form. It is shown that, for the normal form of games involving informational symmetries or repeated play, the standard approach of determining evolutionarily stable strategies (ESSs) often fails to predict the evolutionary outcome. The dynamic stability of ES sets is proved in both the pure strategy and mixed strategy models. ES sets are shown to also generalize the notion of direct ESSs (an earlier attempt to apply ESS theory to extensive games). The theory is illustrated by three examples of biological games in extensive form.     

Evolutionary stability: States and strategies

Different aspects and modifications of the definition of an evolutionarily stable (ES) strategy that have been considered in the literature can be incorporated in a unifying concept which regards the population context. This concept of evolutionary stability will generally characterize population states in both pure- and mixed-strategist models. In particular, it includes ES strategies, represented as a phenotype unique in an ES population. For an important class of mixed-strategist models, no strict ESS can exist. This will be the case whenever the success of an individual strategy is considered to follow as an average from the successes of its behavioural components. Instead, ESS results may be obtained from what will be called a “degenerate” form of the model, which is simply an ESS model on the level of elementary actions. Then, however, the correct interpretation of an ESS is not an individual phenotype but rather a population mixture of elementary actions. If an ES state exists in a mixed-strategist model it may be determined by an equilibrium condition; if there is an ES strategy, a different approach—mainly maximum considerations—is needed for finding it. An equilibrium condition does not hold for the components of an ES strategy straightforwardly; but it can be derived in terms of an auxiliary ESS model that considers first-order effects of the components. Several examples illustrate the significance of these results. Particularly, two models on “Games between Relatives” are reconsidered in order to display both their formal interrelation and the different meaning of their results in the context of mixed-strategist models.     

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.     

Discrete Clutch Sizes, Local Mate Competition, and the Evolution of Precise Sex Allocation

Optimal sex allocation under a population structure with local mate competition has been studied mainly in deterministic models that are based on the assumption of continuous clutch sizes; Hamilton's (1967) model is the classic example. When clutch sizes are small, however, this assumption is not appropriate. When taking the discrete nature of eggs into account it becomes critically important whether females control only the mean sex ratio (“binomial” females) or the variance as well (“precise” females). As both types of sex ratio control have been found, it is of interest to investigate their evolutionary stability. In particular, it may be questioned whether perfect control of the sex ratio is always favoured by natural selection when mating groups are small. Models based on discrete clutch sizes are developed to determine evolutionarily stable (ES) sex ratios. It is predicted that when all females are of the binomial type they should produce a lower proportion of daughters than predicted by Hamilton's model, especially when clutch size and foundress number are small. When all females are of the precise type, the ES number of sons should generally be either a stable mixed strategy or a pure strategy, but there are special cases (for two foundresses and particular clutch sizes) where the ES number of sons lies in a trajectory of neutrally stable mixed strategies; the predicted mean sex ratios can be either higher or lower than predicted by Hamilton's model. The existence of ES mixed strategies implies that individual females do not necessarily have to produce sex ratios with perfect precision; some level of imperfection can be tolerated (i.e., will not be selected against). When the population consists of both binomial and precise females, the latter always have a selective advantage. This advantage of precision does not disappear when precision approaches fixation in the population. The latter result contradicts the conclusions of Taylor and Sauer (1980) which is due to their way of expressing selective advantage; they define selective advantage as the between-generation increase per allele, which will always become vanishingly small when an allele reaches fixation, irrespective of fitness differences.     

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.     

Sperm competition. II-post-copulatory guarding

A two round sperm competition model is analysed to determine which male strategy is advantageous for fertilization of a given set of eggs; guarding a particular female or searching for another copulation. A guarding male is one who would guard if he mates in the first round (which may not occur) whilst a non-guarding male decides on how much sperm to allocate if given the opportunity to inseminate a female in round one. Guarding behaviour is defined in terms of a probability of preventing a further insemination if challenged by a rival male. Sperm success with a single female obeys the "raffle principle". An evolutionarily stable strategy (ESS) approach is used to ascertain the best non-guarding ejaculation strategy. We show that for each fixed proportion of guarders in the population the strategies are ordered and that only a single guarding strategy need be considered. The model predicts that there will be evolution to either the guarding strategy or a single non-guarding strategy or a polymorphic combination of guarding and some (or all) of the non-guarding strategies. The conditions for coexistence to occur were shown to be rare in comparison to those necessary for a monomorphism. Copyright 1999 Academic Press.     

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.     

Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

Adaptive evolution of anti-predator ability promotes the diversity of prey species: critical function analysis

In this paper, with the method of adaptive dynamics and critical function analysis, we investigate the evolutionary diversification of prey species. We assume that prey species can evolve safer strategies such that it can reduce the predation risk, but this has a cost in terms of its reproduction. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for continuously stable strategy and evolutionary branching in the prey strategy. It is found that if the trade-off curve is globally concave, then the evolutionarily singular strategy is continuously stable. However, if the trade-off curve is concave-convex-concave and the prey's sensitivity to crowding is not strong, then the evolutionarily singular strategy may be an evolutionary branching point, near which the resident and mutant prey can coexist and diverge in their strategies. Second, we find that after branching has occurred in the prey strategy, if the trade-off curve is concave-convex-concave, the prey population will eventually evolve into two different types, which can coexist on the long-term evolutionary timescale. The algebraical analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible for the concave-convex-concave trade-off relationship.     

Environmentally induced dispersal under heterogeneous logistic growth

We consider a single-species model which is composed of several habitats connected by linear migration rates and having logistic growth. A spatially varying, temporally constant environment is introduced by the non-homogeneity of its carrying capacity. Under this condition any type of purely diffusive behavior, characterized in our model by symmetric migration rates, produces an unbalanced population distribution, i.e. some locations receive more individuals than can be supported by the environmental carrying capacity, while others receive less. Using an evolutionarily stable strategy (ESS) approach we show that an asymmetric migration mechanism, induced by the heterogeneous carrying capacity of the environment, will be selected. This strategy balances the inflow and outflow of individuals in each habitat (balanced dispersal), as well as 'balancing' the spatial distribution relative to variation in carrying capacity (the Ideal Free Distribution from habitat selection theory). We show that several quantities are maximized or minimized by the evolutionarily stable dispersal strategy.     

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.     

The logic of asymmetric contests

A theoretical analysis is made of the evolution of behavioural strategies in contest situations. It is assumed that behaviour will evolve so as to maximize individual fitness. If so, a population will evolve an ‘evolutionarily stable strategy’, or ESS, which can be defined as a strategy such that, if all members of a population adopt it, no ‘mutant’ strategy can do better. A number of simple models of contest situations are analysed from this point of view. It is concluded that in ‘symmetric’ contests the ESS is likely to be a ‘mixed’ strategy; that is, either the population will be genetically polymorphic or individuals will be behaviourally variable. Most real contests are probably asymmetric, either in pay-off to the contestants, or in size or weapons, or in some ‘uncorrelated’ fashion; i.e. in a fashion which does not substantially bias either the pay-offs or the likely outcome of an escalated contest. An example of an uncorrelated asymmetry is that between the ‘discoverer’ of a resource and a ‘late-comer’. It is shown that the ESS in asymmetric contests will usually be to permit the asymmetric cue to settle the contest without escalation. Escalated contests will, however, occur if information to the contestants about the asymmetry is imperfect.     

Evolution of stepping-stone dispersal rates

We present a general model of the evolution of dispersal in a population with any distribution of dispersal distance. We use this model to analyse evolutionarily stable (ES) dispersal rates for the classical island model of dispersal and for three different stepping-stone models. Using general techniques to compute relatedness coefficients in the different dispersal models which we consider, we find that the distribution of dispersal distance may affect the ES dispersal rate when the cost of dispersal is low. In this case the ES dispersal rate increases with the number of demes that can be reached by one dispersal event. However, for increasing cost the ES dispersal rate converges to a value independent of the distribution of dispersal distance. These results are in contrast to previous analyses of similar models. The effects of the size (number of demes) and shape (ratio between the width and the length) of the population on the evolution of dispersal are also studied. We find that larger and more elongated populations lead generally to higher ES dispersal rates. However, both of these effects can only be observed for extreme parameter values (i.e. for very small and very elongated populations). The direct fitness method and the analytical techniques used here to compute relatedness coefficients provide an efficient way to analyse ES strategies in subdivided populations.     

The evolutionary dynamics of direct phenotypic overdominance: emergence possible, loss probable

An evolutionary dynamical system with explicit diploid genetics is used to investigate the likelihood of observing phenotypically overdominant heterozygotes versus heterozygous phenotypes that are intermediate between the homozygotes. In this model, body size evolves in a population with discrete demographic episodes and with competition limiting reproduction. A genotype-phenotype map for body size is used that can generate the two qualitative types of dominance interactions (overdominance versus intermediate dominance). It is written as a single-locus model with one focal locus and parameters summarizing the effects of alleles at other loci. Two types of evolutionarily stable strategy (ESS; continuously stable strategy, CSS) occur. The ESS is generated either (1) by the population ecology; or (2) by a local maximum of the genotype-phenotype map. Overdominant heterozygotes are expected to arise if the population evolves toward the second type of ESS, where nearly maximum body sizes are found. When other loci with partially dominant inheritance also evolve, the location of the maximum in the genotype-phenotype map repeatedly changes. It is unlikely that an evolving population will track these changes; ESSs of the second type now are at best quasi-stationary states of the evolutionary dynamics. Considering the restrictions on its probability, a pattern of phenotypic overdominance is expected to be rare.     

Evolutionary limits to the frequency of aggression between related or unrelated conspecifics in diploid species with simple mendelian inheritance

Game theory has been used by some authors to analyse evolutionary limits to the expression of aggression in theoretical haploid parthenogenetic species. Others have examined frequency dependent selection, of which aggression may be a case, by applying population genetic models to diploid species. A model is presented which attempts to combine these two approaches. Game theory is used to determine evolutionarily stable strategies and corresponding stable polymorphisms for a two-strategy game played by members of a diploid sexual species, when choice of strategy is determined by two alleles at a single locus. Results are given for dominant, co-dominant and recessive determination of choice of the more aggressive of two strategies, for two levels of relationship: unrelated players and sibs. It is found that for a range of models of single locus inheritance the evolutionarily stable strategy (ESS) determined for haploid species remains the stable population strategy for diploid sexual species, when players are unrelated. In sibling contestants aggression is reduced. The mixed strategy haploid ESS underestimates, but the pure strategy haploid ESS provides a good indication of the degree to which relatedness lessens aggression in diploid species. For both haploid and diploid species there may be a considerable advantage to confining conflicts to kin.     

The ideal free distribution as an evolutionarily stable state in density‐dependent population games

In classical games that have been applied to ecology, individual fitness is either density independent or population density is fixed. This article focuses on the habitat selection game where fitness depends on the population density that evolves over time. This model assumes that changes in animal distribution operate on a fast time scale when compared to demographic processes. Of particular interest is whether it is true, as one might expect, that resident phenotypes who use density‐dependent optimal foraging strategies are evolutionarily stable with respect to invasions by mutant strategies. In fact, we show that evolutionary stability does not require that residents use the evolutionarily stable strategy (ESS) at every population density; rather it is the combined resident–mutant system that must be at an evolutionary stable state. That is, the separation of time scales assumption between behavioral and ecological processes does not imply that these processes are independent. When only consumer population dynamics in several habitats are considered (i. e. when resources do not undergo population dynamics), we show that the existence of optimal foragers forces the resident‐mutant system to approach carrying capacity in each habitat even though the mutants do not die out. Thus, the ideal free distribution (IFD) for the single‐species habitat selection game becomes an evolutionarily stable state that describes a mixture of resident and mutant phenotypes rather than a strategy adopted by all individuals in the system. Also discussed is how these results are affected when animal distribution and demographic processes act on the same time scale.     

Evolutionarily stable leaf area production in plant populations

Using an analytical model, it was shown that for a given amount of nitrogen in the canopy of a stand (N(T)), there exists an evolutionarily stable leaf area index (ES-LAI), and therefore an evolutionarily stable average leaf nitrogen content (n(ES)(av);n(ES)(av) =N(T)/ES-LAI), at which no individual plant in the stand can increase its photosynthesis by changing its leaf area. It was also shown that this ES-LAI is always greater than the optimal LAI that maximizes photosynthesis per unit N(T) of the stand. This illustrates that the canopy structure that maximizes photosynthesis of a population is not the same as the canopy structure that maximizes photosynthesis of individuals within a population. It was further derived that the ES-LAI at given N(T) increases with the ratio between the light-saturated photosynthesis and the N content per unit leaf area (leaf-PPNUE) and that it decreases with the canopy extinction coefficient for light (K(L)), the light availability and the apparent quantum yield (phi). These hypotheses were tested by comparing calculated ES-LAI and n(ES)(av) values to actual LAIs and leaf N contents measured for stands of a large variety of herbaceous plants. There was a close correspondence between the calculated and measured values. As predicted by the model, plants with high leaf-PPNUEs produced more leaf area per unit nitrogen than those with low leaf-PPNUEs while plants with horizontal leaves, forming stands with higher K(L) values, produced less leaf area than those with more vertically inclined leaves. These results suggest that maximization of individual plant photosynthesis per unit of nitrogen plays an important role in determining leaf area production of plants and the resulting canopy structure of stands of vegetation. They further suggest this optimization to be a mechanism by which leaf traits such as leaf-PPNUE and leaf inclination angle are causally related to structural characteristics of the population, i.e. the leaf area index of the stand.     

Cheating as a mixed strategy in a simple model of aggressive communication

The possibility that frequency-dependent cheating can persist in an evolutionarily stable communication system has frequently been proposed. Although there is empirical evidence for this idea, however, it has not been investigated in terms of game theory. In the present paper I show for a simple symmetric game that cheating can be part of a mixed evolutionarily stable strategy (ESS). Furthermore, despite the widespread assumption that cheaters must be rare, I show that most of the population can be cheaters, while the signalling system remains evolutionarily stable. Consequences for signalling theory and experiments to detect such mixed ESS are discussed. Copyright 2000 The Association for the Study of Animal Behaviour.     

Mistakes allow evolutionary stability in the repeated prisoner's dilemma game

The repeated prisoner's dilemma game has been widely used in analyses of the evolution of reciprocal altruism. Recently it was shown that no pure strategy could be evolutionarily stable in the repeated prisoner's dilemma. Here I show that if there is always some probability that individuals will make a mistake, then a pure strategy can be evolutionarily stable provided that it is "strong perfect equilibria" against itself. To be a strong perfect equilibrium against itself, a strategy must be the best response to itself after every possible sequence of behavior. I show that both unconditional defection and a modified version of tit-for-tat have this property.