Random evolutionarily stable strategies

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.”     

Group selection among alternative evolutionarily stable strategies

Many important models of the evolution of social behavior have more than one evolutionarily stable strategy (ESS). Examples include co-ordination games, contests, mutualism, reciprocity, and sexual selection. Here we show that when there are multiple evolutionarily stable strategies, selection among groups can cause the spread of the strategy that has the lowest extinction rate or highest probability of contributing to the colonization of empty habitats, and that this may occur even when groups are usually very large, migration rates are substantial, and "extinction" entails only the disruption of the group and the dispersal of its members. The main requirements are: (1) individuals drawn from a single surviving group make up a sufficiently large fraction newly formed groups, and (2) the processes increasing the frequency of successful strategies within groups are strong compared to rate of migration among groups. The latter condition suggests that this form of group selection will be particularly important when behavioral variation is culturally acquired.     

Strong stability and density-dependent evolutionarily stable strategies

Stability conditions for equilibria of the evolution of population strategies in a single species are developed by comparing frequency and density dependent fitnesses of pairs of strategies. Stability of such equilibria is shown for general haploid frequency and density dynamics. It is also shown that this stability is stronger than that of multispecies population dynamical systems. A biological interpretation of the conditions is provided in terms of the fitness of invading subpopulations.     

Evolution of conditional dispersal: evolutionarily stable strategies in spatial models

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.     

An explicit approach to evolutionarily stable dispersal strategies: no cost of dispersal

The evolution of dispersal is examined by looking at evolutionarily stable strategies (ESS) for dispersal parameters in discrete time multisite models without any cost of dispersal. ESS are investigated analytically, based on explicit results on sensitivity analysis of matrix models. The basic model considers an arbitrary number of sites and a single age class. An ESS for dispersal parameters is obtained when the spatial reproductive values, calculated at the density-dependent population equilibrium, are equal across sites. From this basic formulation, one derives equivalently that all local populations should be at equilibrium in the absence of migration, and that dispersal between sites should be balanced, i.e., the numbers of individuals arriving to and leaving a site are equal. These results are then generalized to a model with several age classes. Equal age-specific reproductive values do not however imply balanced dispersal in this case. Our results generalize to any number of sites and age classes those available ?M. Doebeli, Dispersal and dynamics, Theoret. Popul. 47 (1995) 82 for two sites and one age class.     

Strong stability and evolutionarily stable strategies with two types of players

Static ESS conditions are developed for the frequency evolution of a two-species haploid system by analyzing the stability of the corresponding dynamics for two pairs of strategies. A dynamic strong stability concept is introduced and shown to be equivalent to the ESS conditions in all cases where a regularity assumption is satisfied.     

Mating group size and evolutionarily stable pattern of sexuality in barnacles

Barnacles, marine crustaceans, have various patterns of sexuality depending on species including simultaneous hermaphroditism, androdioecy (hermaphrodites and dwarf males), and dioecy (females and dwarf males). We develop a model that predicts the pattern of sexuality in barnacles by two key environmental factors: (i) food availability and (ii) the fraction of larvae that settle on the sea floor. Populations in the model consist of small individuals and large ones. We calculate the optimal resource allocation toward male function, female function and growth for small and large barnacles that maximizes each barnacle's lifetime reproductive success using dynamic programming. The pattern of sexuality is defined by the combination of the optimal resource allocations. In our model, the mating group size is a dependent variable and we found that sexuality pattern changes with the food availability through the mating group size: simultaneous hermaphroditism appears in food-rich environments, where the mating group size is large, protandric simultaneous hermaphroditism appears in intermediate food environments, where the mating group size also takes intermediate value, the other sexuality patterns, androdioecy, dioecy, and sex change are observed in food-poor environments, where the mating group size is small. Our model is the first one where small males can control their growth to large individuals, and hence has ability to explain a rich spectrum of sexual patterns found in barnacles.     

Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay

We construct two models of discrete-time replicator dynamics with time delay. In the social-type model, players imitate opponents taking into account average payoffs of games played some units of time ago. In the biological-type model, new players are born from parents who played in the past. We consider two-player games with two strategies and a unique mixed evolutionarily stable strategy. We show that in the first type of dynamics, it is asymptotically stable for small time delays and becomes unstable for big ones when the population oscillates around its stationary state. In the second type of dynamics, however, evolutionarily stable strategy is asymptotically stable for any size of a time delay.     

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.     

Honesty and cheating in cleaning symbioses: evolutionarily stable strategies defined by variable pay-offs

Game-theory models have indicated that the evolution of mixed strategies of cheating and honesty in many mutualisms is unlikely. Moreover, the mutualistic nature of interspecific interactions has often been difficult to demonstrate empirically. We present a game-theory analysis that addresses these issues using cleaning symbioses among fishes as a model system. We show that the assumption of constant pay-offs in existing models prevents the evolution of evolutionarily stable mixed strategies of cheating and honesty. However, when interaction pay-offs are assumed to be density dependent, mixed strategies of cheating and honesty become possible. In nature, cheating by clients often takes the form of retaliation by clients against cheating cleaners, and we show that mixed strategies of cheating and honesty evolve within the cleaner population when clients retaliate. The dynamics of strategies include both negative and positive effects of interactions, as well as density-dependent interactions. Consequently, the effects of perturbations to the model are nonlinear. In particular, we show that under certain conditions the removal of cleaners may have little impact on client populations. This indicates that the underlying mutualistic nature of some interspecific interactions may be difficult to demonstrate using simple manipulation experiments.     

On evolutionarily stable strategies in populations with subpopulations having isolated strategy repertoires

The concept of a population with subpopulations that are isolated with respect to strategies is introduced and applied to a biologically important class of cases, showing that no proper mixed evolutionarily stable strategies can exist in such cases.     

Required parental investment and mating patterns: A quantitative analysis in the context of evolutionarily stable strategies

Abstract

Much social psychological research has been dedicated to understanding mating strategies from the standpoint of genetic‐fitness payout (e.g., Simpson and Gangestad, 2000). The current work is designed to provide a coherent, quantitative model for predicting different classes of mating strategies in both males and females. Specifically, the framework developed in this paper is an elaboration of Dawkins’ (1989) quantitative assessment of different male and female mating strategies. Dawkins suggests that the prevalence of different strategies employed should be predictable in terms of evolutionary stable strategies. In the current work, a quantitative analysis predicting the prevalence of different mating strategies within each sex was conducted. The mathematical functions derived suggest that variability in the costs associated with raising offspring affects the expected prevalence of mating strategies differently for males and females. According to the present model, variability in female strategies should be less affected by changes in parental investment (PI) than variability in male strategies. Important predictions regarding male and female mating strategies across cultures are discussed.     

Required parental investment and mating patterns: a quantitative analysis in the context of evolutionarily stable strategies

Much social psychological research has been dedicated to understanding mating strategies from the standpoint of genetic-fitness payout (e.g., Simpson and Gangestad, 2000). The current work is designed to provide a coherent, quantitative model for predicting different classes of mating strategies in both males and females. Specifically, the framework developed in this paper is an elaboration of Dawkins' (1989) quantitative assessment of different male and female mating strategies. Dawkins suggests that the prevalence of different strategies employed should be predictable in terms of evolutionary stable strategies. In the current work, a quantitative analysis predicting the prevalence of different mating strategies within each sex was conducted. The mathematical functions derived suggest that variability in the costs associated with raising offspring affects the expected prevalence of mating strategies differently for males and females. According to the present model, variability in female strategies should be less affected by changes in parental investment (PI) than variability in male strategies. Important predictions regarding male and female mating strategies across cultures are discussed.     

Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics

An integrodifferential equations model for the distribution of individuals with respect to the age at maturity is considered. Mutation is modeled by an integral operator. Results concerning the behaviour of the steady states and their relation to evolutionarily stable strategies when the mutation rate is small are given. The same results are obtained for a (rather) general class of models that include the one mentioned before.This work was partially supported by DGICYT PB98-0932-C02-02 and DGI BMF2002-04613-C03.     

Descriptions of superparasitism by optimal foraging theory,evolutionarily stable strategies and quantitative genetics

Many parasitoids superparasitize, in which an insect attacks a previously parasitized host, laying an egg in the host even though only one offspring will emerge from the host. In this paper superparasitism is considered from the perspectives of optimal foraging theory, evolutionarily stable strategies, and quantitative genetics. The focal question is: at what point in its life should an individual parasitoid begin attacking previously parasitized hosts? Each of the three theoretical methods can be used to answer the question and by doing so, we see how the three methods are connected. Qualitative, empirical predictions based on the theories are described.     

The status of the conditional evolutionarily stable strategy

The conditional evolutionarily stable strategy (ESS) has proven to be a versatile tool for understanding the production of alternative phenotypes in response to environmental cues. Hence, we would expect the theoretical basis of the conditional strategy to be robust. However, Shuster and Wade have recently criticized the conditional ESS based on Gross's 1996 proposal that most alternative reproductive tactics are conditional and have evolved by 'status-dependent selection.' We critically assess Gross's status-dependent selection model and Shuster and Wade's critique. We find shortcomings and misconceptions in both. We return to the findings of the strategic models behind the conditional ESS and demonstrate how environmental threshold models use a reaction norm approach and quantitative genetic theory to understand the evolution of conditional strategies.