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.     

Bounds on the number of ESSs of a matrix game

It is well known that for any evolutionary game there may be more than one evolutionarily stable strategy (ESS). In general, the more ESSs there are, the more difficult it is to work out how the population will behave (unless there are no ESSs at all). If a matrix game has an ESS which allows all possible pure strategies to be played, referred to as an internal ESS, then no other ESS can exist. In fact, the number of ESSs possible is highly dependent upon how many of the pure strategies each allow to be played, their support size. It is shown that if alpha is the ratio of the mean support size to the number of pure strategies n, then as n tends to infinity the greatest number of ESSs can be represented by a continuous function f(alpha) with useful regularity properties, and bounds are found for both f(alpha) and the value alpha(*), where it attains its maximum. Thus we can obtain a limit on the complexity of any particular system as a function of its mean support size.     

Alternative Predictions for Optimal Dispersal in Response to Local Catastrophic Mortality

What dispersal strategy should be employed by an organism in response to local catastrophic mortality? Here we contrast predictions from an analytical solution derived from an ESS model which optimizes competitive ability (Comins et al., 1980) with those from a stochastic, branching process model (Karlson and Taylor, 1992) which maximizes survivorship of a clonal lineage. The optimal dispersal fraction varies directly with the probability of local extinction in the ESS model, yet varies inversely with this probability over much of the parameter space in the latter model. In order to conform more closely with the assumptions of the ESS model, we have modified the branching process model to have a random, Poisson-distributed number of offspring and compared the predictions of these models. Both models invoke dispersal as an escape from local extinction and predict mixed dispersal strategies over a wide range of conditions. However, increasing local catastrophic mortality favors more dispersal in the ESS model, but it can be so severe in the branching process model that no dispersal strategy is adaptive. In this model, the predicted optimal proportion of dispersed offspring is highest at low to intermediate levels of catastrophic mortality depending on the total number of offspring produced. We suggest that this observed discrepancy is sufficiently large to warrant empirical tests of these qualitatively different predictions.     

Red Queens and ESS: the coevolution of evolutionary rates

Summary The Red Queen principle states that a set of interacting species reaches an evolutionary equilibrium at which all their rates of coevolution exactly balance each other. The lag-load model, which is one way of searching for Red Queens, has, by itself, previously predicted that they do not exist. But this model has assumed that infinite maladaptedness is possible. The lag-load model is improved by assuming that once the lag load of all but one species is determined, so is that of the final species. This assumption eliminates the possibility of infinite maladaptedness. Its result is to allow the lag-load model to yield Red Queen coevolution. It does this whether or not speciation and extinction rates are included. Thus the lag-load model is harmonized with the earlier Red Queen model derived from studies of predation.Because of the intercorrelation of phenotypic traits, the predatory model concluded that the eventual stable rate of coevolution must be zero (except for intermittent bursts after some correlation or compromise is successfully broken). Another model that predicts stable coevolutionary rates of zero is that of evolutionarily stable strategies (ESS).Red Queen assumes that the more extreme a phenotypic trait is, the better it is, and that there are no constraints on the growth of such a phenotypic trait value. Such traits are the key to the Red Queen prediction of progressive coevolution. ESS models make no such assumptions. Eliminating unbounded traits from the model of predator-victim evolution changed its prediction from progressive coevolution to stasis. Before this paper, no model had dealt simultaneously with both unbounded and constrained traits.To handle both sorts of phenotypic traits at the same time in the same model, we abandoned lag load as a measure of evolutionary rate (lag loads do not uniquely determine phenotype). Instead, we used the traditional assumption that rate is proportional to the slope of the adaptive landscape. A model, relying on continuous evolutionary game theory, was developed and simulated under various conditions in two or three species sets, with up to five independent traits coevolving simultaneously. The results were: (1) there was always a set of equilibrium densities eventually achieved by coevolution; if the population interaction represented by this stable coevolutionary state is also stable, then the system should persist whether it evolves further or not; (2) whenever traits were present which were unbounded and best at their most extreme values, then a Red Queen emerged; (3) whenever traits were present which were correlated with each other or constrained below infinity, then an ESS emerged; (4) if both types were present, both results occurred: Red Queen in the unbounded traits and ESS in the constrained ones.Because unbounded traits may not exist, the Red Queen may have no domain. But the domain of ESS is real. ESS should lead to the evolutionary pattern called punctuated equilibrium. The changes in design rules which punctuate stasis should lead to an ever-expanding independence of traits from each other, i.e. to more and more refined differentiation. A single set of design rules which governs a set of species is called a fitness-generating function. Such functions may help to define the concepts of adaptive zone and ecological guild.     

Long-term stability from fixation probabilities in finite populations: new perspectives for ESS theory

For mixed strategies in finite populations, long-term stability is defined with respect to the probability of fixation of a mutant. Under weak selection, necessary and sufficient conditions are obtained using a diffusion approximation of the Wright-Fisher model or exact solutions for the Moran model. These differ from the usual ESS conditions if the strategies affect fertility instead of viability, leading to a game matrix depending on the population size, or if the mutant mixed strategy uses a new pure strategy. In this case, the mutant deviation must not exceed some threshold value depending on the population size. In a diploid population, long-term stability may not occur unless there is partial dominance. In the case of sex allocation, continuous stability of an even sex ratio is ascertained. If sex allocation is random, an evolutionary decrease of the variance is predicted.     

ESS germination strategies in randomly varying environments. I. Logistic-type models

ESS germination strategies are studied in a model of annual plant population dynamics in a randomly varying environment. The possible strategies are different values of the annual germination fraction G, either constant over time or varying in response to a "cue" correlated with upcoming environmental conditions. The model generalizes D. Cohen's model (1966, J. Theor. Biol. 12, 119-129; 1968, J. Ecol. 56, 219-228) by allowing density-dependent per capita seed yields. ESSs are characterized in terms of the resulting harmonic mean growth rate of population density. The ESS criterion cannot be solved analytically, but qualitative relationships between the value of the ESS and other population parameters are obtained, and environments in which 100% germination is an ESS are identified. Some explicit predictions of the theory are summarized and compared with ideas of M. Westoby (1981, Amer. Nat. 118, 882-885). The results of this study are compared with those of Cohen (op. cit.) in a companion paper.     

Ecotypic variation in the asymmetric Hawk-Dove game: When is Bourgeois an evolutionarily stable strategy?

Summary This paper develops a mathematical model of an iterated, asymmetric Hawk-Dove game with the novel feature that not only are successive pairs of interactants — in the roles of owner and intruder contesting a site — drawn randomly from the population, but also the behaviour adopted at one interaction affects the role of a contestant in the next. Under the assumption that a site is essential for reproduction, the evolutionarily stable strategy (ESS) of the population is found to depend on the probability, w, that the game will continue for at least a further period (which is inversely related to predation risk), and five other parameters; two of them are measures of site scarcity, two are measures of fighting costs, and the last is a measure of resource holding potential (RHP). Among the four strategies — Hawk (H), Dove (D), Bourgeois (B) and anti-Bourgeois (X) — only D is incapable of being an ESS; and regions of parameter space are found in which the ESS can be only H, or only X, or only B; or either H or X; or either X or B; or either H or B; or any of the three. The scarcer the sites or the lower the costs of fighting, or the lower the value of w, the more likely it is that H is an ESS; the more abundant the sites or the higher the costs of fighting, or the higher the value of w, the more likely it is that X or B is an ESS. The different ESSs are interpreted as different ecotypes. The analysis suggests how a non-fighting population could evolve from a fighting population under decreasing risk of predation. If there were no RHP, or if RHP were low, then the ESS in the non-fighting population would be X; only if RHP were sufficiently high would the ESS be B, and the scarcer the sites, the higher the RHP would have to be. These conclusions support the thesis that if long-term territories are essential for reproduction and sites are scarce, then ownership is ruled out not only as an uncorrelated asymmetry for settling disputes in favour of owner, but also as a correlated asymmetry.     

Evolution of extraordinary female-biased sex ratios: the optimal schedule of sex ratio in local mate competition

Female-biased sex ratio in local mate competition has been well studied both theoretically and experimentally. However, some experimental data show more female-biased sex ratios than the theoretical predictions by Hamilton [1967. Science 156, 477-488] and its descendants. Here we consider the following two effects: (1) lethal male-male combat and (2) time-dependent control (or schedule) of sex ratio. The former is denoted by a male mortality being an increasing function of the number of males. The optimal schedule is analytically obtained as an evolutionarily stable strategy (ESS) by using Pontrjagin's maximum principle. As a result, an ESS is a schedule where only males are produced first, then the proportion of females are gradually increased, and finally only females are produced. Total sex ratio (sex ratio averaged over the whole reproduction period) is more female-biased than the Hamilton's result if and only if the two effects work together. The bias is stronger when lethal male combat is severer or a reproduction period is longer. When male-male combat is very severe, the sex ratio can be extraordinary female-biased (less than 5%). The model assumptions and the results generally agree with experimental data on Melittobia wasps in which extraordinary female-biased sex ratio is observed. Our study might provide a new basis for the evolution of female-biased sex ratios in local mate competition.     

The evolution of parasite virulence and transmission rate in a spatially structured population

If the transmission occurs through local contact of the individuals in a spatially structured population, the evolutionarily stable (ESS) traits of parasite might be quite different from what the classical theory with complete mixing predicts. In this paper, we theoretically study the ESS virulence and transmission rate of a parasite in a lattice-structured host population, in which the host can send progeny only to its neighboring vacant site, and the transmission occurs only in between the infected and the susceptible in the nearest-neighbor sites. Infected host is assumed to be infertile. The analysis based on the pair approximation and the Monte Carlo simulation reveal that the ESS transmission rate and virulence in a lattice-structured population are greatly reduced from those in completely mixing population. Unlike completely mixing populations, the spread of parasite can drive the host to extinction, because the local density of the susceptible next to the infected can remain high even when the global density of host becomes very low. This demographic viscosity and group selection between self-organized spatial clusters of host individuals then leads to an intermediate ESS transmission rate even if there is no tradeoff between transmission rate and virulence. The ESS transmission rate is below the region of parasite-driven extinction by a finite amount for moderately large reproductive rate of host; whereas, the evolution of transmission rate leads to the fade out of parasite for small reproductive rate, and the extinction of host for very large reproductive rate.     

Parental care as a game

A new game model of parental care is presented where fitnesses of males and females depend on frequencies of care and no-care strategies in the population, as well as strategies of individuals and those of their mates. Evolutionarily stable states (ESS) of biparental-care ( BC ), female-care ( FC ), male-care ( MC ), and no-care ( NC ) are represented as regions in a 3-dimensional parameter space, comprising the ratio of the number of offspring raised by FC or MC to that by BC , the ratio of that by NC to that by FC or MC , and the sex ratio. ESSs are not necessarily uniquely determined: ESS regions of FC and MC overlap in parts, as well as those of BC and NC . In a region where none of the four states is ever an ESS, a polymorphism (a mixed state of care and no-care strategies in one or both sexes) is stable. Although an asymmetry between prezygotic investment times of males and females results in the difference in sizes of the ESS regions of FC and MC , it is not responsible for determining whether FC or MC evolves from other modes of parental care as a result of slowly changing parameters. The key factor is the sex ratio of adults in the population. By a slightly modified model, it is also shown that paternity uncertainty at fertilization of eggs is highly important for evolution of FC .     

Adaptive evolution of cooperation through Darwinian dynamics in Public Goods games

The linear or threshold Public Goods game (PGG) is extensively accepted as a paradigmatic model to approach the evolution of cooperation in social dilemmas. Here we explore the significant effect of nonlinearity of the structures of public goods on the evolution of cooperation within the well-mixed population by adopting Darwinian dynamics, which simultaneously consider the evolution of populations and strategies on a continuous adaptive landscape, and extend the concept of evolutionarily stable strategy (ESS) as a coalition of strategies that is both convergent-stable and resistant to invasion. Results show (i) that in the linear PGG contributing nothing is an ESS, which contradicts experimental data, (ii) that in the threshold PGG contributing the threshold value is a fragile ESS, which cannot resist the invasion of contributing nothing, and (iii) that there exists a robust ESS of contributing more than half in the sigmoid PGG if the return rate is relatively high. This work reveals the significant effect of the nonlinearity of the structures of public goods on the evolution of cooperation, and suggests that, compared with the linear or threshold PGG, the sigmoid PGG might be a more proper model for the evolution of cooperation within the well-mixed population.     

Conditional dispersal under kin competition: extension of the Hamilton-May model to brood size-dependent dispersal

I consider a site-based model with contest competition among siblings, and assume that dispersal is conditional on the number of offspring in the natal site. Evolutionarily stable populations contain threshold dispersal strategies, which retain a certain number of offspring in the natal site and disperse the rest (if the actual number of offspring is less than the threshold, then all offspring are retained). Due to the discrete nature of the strategy set (the threshold must be integer), the ESS may not be unique or may not exist. In the latter case, two neighboring threshold strategies coexist in the evolutionarily stable population. Dispersal first decreases and then increases as a function of dispersal mortality, such that all but one offspring should be dispersed both when dispersal mortality is very small or very high. Population-level dispersal fractions are often similar to the unconditional ESS, but differ strongly when fecundity is small and dispersal mortality is high.     

Co-evolution of seed size and seed predation

Using the evolutionarily stable strategy (ESS) approach in a model for the co-evolution of seed size and seed predation, I show that seed size variation within individual plants is favoured if there is a trade-off in the predator's attack rate for different seed sizes. A single seed size is not evolutionarily stable because a predator that is optimally adapted to one particular seed size cannot prevent invasion by plants with a different seed size. The model generates the following predictions. The ESS consists of a continuous range of seed sizes. Small seeds tend to be attacked more frequently than big seeds. Plants with many resources and plants with low (frequency-independent) juvenile mortality have more variable seeds than plants with few resources and a high juvenile mortality. Seed size variation is higher in fluctuating populations regulated by seed predation alone than in stable populations (partially) regulated by seedling competition. Predator searching behaviour does not directly affect the ESS seed size range, but may have an indirect effect by affecting population stability or the significance of seedling competition as a population regulating mechanism. Moreover, seed size distributions are found to be more skewed in favour of small seeds if predation is spatially non-uniform than if predation is more even. Application of the model to systems of several co-evolving plant and predator species is discussed.     

A mixed strategy model for the emergence and intensification of social learning in a periodically changing natural environment

Based on a population genetic model of mixed strategies determined by alleles of small effect, we derive conditions for the evolution of social learning in an infinite-state environment that changes periodically over time. Each mixed strategy is defined by the probabilities that an organism will commit itself to individual learning, social learning, or innate behavior. We identify the convergent stable strategies (CSS) by a numerical adaptive dynamics method and then check the evolutionary stability (ESS) of these strategies. A strategy that is simultaneously a CSS and an ESS is called an attractive ESS (AESS). For certain parameter sets, a bifurcation diagram shows that the pure individual learning strategy is the unique AESS for short periods of environmental change, a mixed learning strategy is the unique AESS for intermediate periods, and a mixed learning strategy (with a relatively large social learning component) and the pure innate strategy are both AESS's for long periods. This result entails that, once social learning emerges during a transient era of intermediate environmental periodicity, a subsequent elongation of the period may result in the intensification of social learning, rather than a return to innate behavior.     

A mixed strategy model for the emergence and intensification of social learning in a periodically changing natural environment

Based on a population genetic model of mixed strategies determined by alleles of small effect, we derive conditions for the evolution of social learning in an infinite-state environment that changes periodically over time. Each mixed strategy is defined by the probabilities that an organism will commit itself to individual learning, social learning, or innate behavior. We identify the convergent stable strategies (CSS) by a numerical adaptive dynamics method and then check the evolutionary stability (ESS) of these strategies. A strategy that is simultaneously a CSS and an ESS is called an attractive ESS (AESS). For certain parameter sets, a bifurcation diagram shows that the pure individual learning strategy is the unique AESS for short periods of environmental change, a mixed learning strategy is the unique AESS for intermediate periods, and a mixed learning strategy (with a relatively large social learning component) and the pure innate strategy are both AESS's for long periods. This result entails that, once social learning emerges during a transient era of intermediate environmental periodicity, a subsequent elongation of the period may result in the intensification of social learning, rather than a return to innate behavior.     

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.     

Maternal control of offspring sex and male morphology in the Otitesella fig wasps

Models concerning the evolution of alternative mating tactics commonly assume that individuals determine their own strategies. Here we develop a computer-based ESS model that allows mothers, ovipositing in discrete patches, to choose both the sex and the male mating tactics (natal-patch mating or dispersing) of their offspring based only on how many other mothers have used the specific patch before them. Data for three species of nonpollinating fig wasps from the Otitesella genus agree quantitatively with the model's assumptions and predictions. This suggests that females respond to population densities at the level of individual figs. The alternative male tactics in the species we studied are probably a result of a conditional strategy exercised by the mother that laid them. In addition, as females were only allowed to lay one egg per patch, our results suggest a new mechanism that can skew population sex ratios towards a female bias.     

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.     

Inherent high correlation of individual motility enhances population dispersal in a heterotrophic, planktonic protist

Quantitative linkages between individual organism movements and the resulting population distributions are fundamental to understanding a wide range of ecological processes, including rates of reproduction, consumption, and mortality, as well as the spread of diseases and invasions. Typically, quantitative data are collected on either movement behaviors or population distributions, rarely both. This study combines empirical observations and model simulations to gain a mechanistic understanding and predictive ability of the linkages between both individual movement behaviors and population distributions of a single-celled planktonic herbivore. In the laboratory, microscopic 3D movements and macroscopic population distributions were simultaneously quantified in a 1L tank, using automated video- and image-analysis routines. The vertical velocity component of cell movements was extracted from the empirical data and used to motivate a series of correlated random walk models that predicted population distributions. Validation of the model predictions with empirical data was essential to distinguish amongst a number of theoretically plausible model formulations. All model predictions captured the essence of the population redistribution (mean upward drift) but only models assuming long correlation times (minute), captured the variance in population distribution. Models assuming correlation times of 8 minutes predicted the least deviation from the empirical observations. Autocorrelation analysis of the empirical data failed to identify a de-correlation time in the up to 30-second-long swimming trajectories. These minute-scale estimates are considerably greater than previous estimates of second-scale correlation times. Considerable cell-to-cell variation and behavioral heterogeneity were critical to these results. Strongly correlated random walkers were predicted to have significantly greater dispersal distances and more rapid encounters with remote targets (e.g. resource patches, predators) than weakly correlated random walkers. The tendency to disperse rapidly in the absence of aggregative stimuli has important ramifications for the ecology and biogeography of planktonic organisms that perform this kind of random walk.     

Evolutionary stability for large populations

We present a revision of Maynard Smith's evolutionary stability criteria for populations which are very large (though technically finite) and of unknown size. We call this the large population ESS, as distinct from Maynard Smith's infinite population ESS and Schaffer's finite population ESS. Building on Schaffer's finite population model, we define the large population ESS as a strategy which cannot be invaded by any finite number of mutants, as long as the population size is sufficiently large. The large population ESS is not equivalent to the infinite population ESS: we give examples of games in which a large population ESS exists but an infinite population ESS does not, and vice versa. Our main contribution is a simple set of two criteria for a large population ESS, which are similar (but not identical) to those originally proposed by Maynard Smith for infinite populations.