Acceptor control in model ecosystems

The analysis of models of evolutionary games requires explicit consideration of both evolutionary game rules and mutants which infinitesimally break these rules. For example, the Scotch Auction is an evolutionary game which lacks both a rule-obeying evolutionarily stable strategy and an asymptotically stable polymorphism of rule-obeying strategies. However, an infinitesimal rule-breaking, or cheating, mutant can be found which is an evolutionarily stable strategy against rule-obeying strategies. Such cheating strategies can spread through populations initially playing the Scotch Auction, effectively changing the rules of the game. Moreover, the extent of such rule-change will then tend to increase. Thus, the Scotch Auction is a transient evolutionary game, being the initial point of a seemingly orthogenetic game evolutionary process. This sort of transience suggests that the “progressive” nature of evolution may be due in part to those game features of evolutionary processes which make the success of adaptations relative to the level of extant adaptation among competitors, predators, etc.     

Optimal sex ratio as a function of EGG incubation temperature in the crocodilians

Recently, there has been a growing consensus as to the adaptive significance of temperature-dependent sex determination in the Crocodilia. The observationally and experimentally motivated hypotheses are that male fitness depends more strongly on quality of incubation environment than female fitness, and that there is a strong correlation between a female's egg incubation temperature choice and her own egg incubation temperature. A population genetics model based on these hypotheses is derived. A method for finding the optimal sex ratio as a function of temperature, which is an evolutionary stable strategy (ESS), is stated and applied under various assumptions. This extends ESS theory to thefunctional case. Cases where there is no ESS and the population sex ratio oscillates in evolutionary time are discovered. Numerical computation is needed to solve the full problem and the resulting optimal sex ratio is compared to laboratory sex ratio data. The general pattern of TSD in crocodilians (female-male-female with female biased overall sex ratio) agrees well with the theory, but details of the pattern are problematic.     

COEVOLUTION AS AN EVOLUTIONARY GAME

Coevolution is modeled as a continuous game where the fitness-maximizing strategy of an individual is assumed to be a function of the strategy of other individuals who are also under selection to maximize fitness. An evolutionary stable strategy (ESS) is sought such that no rare alternative strategies can invade the community. The approach can be used to model coevolution because the ESS may be composed of a coalition of more than one strategy. This work, by modeling frequency-dependent selection, extends the approach of Roughgarden (1976) which only considered density-dependent selection. In particular, we show that the coevolutionary model of Rummel and Roughgarden (1985) does contain frequency-dependent selection, and thus, their application of Roughgarden's criterion for evolutionary stability to a model for which it is not applicable leads to the erroneous conclusion that the ecological and evolutionary processes are in conflict. The utility of the game theoretic approach is illustrated by two examples. The first considers an ESS composed of a single strategy, the second an ESS composed of a coalition of two strategies. Evolution occurs on a frequency-dependent adaptive landscape. For this reason, the approach is appropriate for modeling competitive speciation (Rosenzweig, 1978). Also, the game theoretic approach is designed to combine the interplay between the background environment (including the biotic components) and the evolutionary potential of the populations or organisms. The actual application of this theory will require knowledge of both.     

Evolutionary Stability for Interactions among Kin under Quantitative Inheritance

The theory of evolutionarily stable strategies (ESS) predicts the long-term evolutionary outcome of frequency-dependent selection by making a number of simplifying assumptions about the genetic basis of inheritance. I use a symmetrized multilocus model of quantitative inheritance without mutation to analyze the results of interactions between pairs of related individuals and compare the equilibria to those found by ESS analysis. It is assumed that the fitness changes due to interactions can be approximated by the exponential of a quadratic surface. The major results are the following. (1) The evolutionarily stable phenotypes found by ESS analysis are always equilibria of the model studied here. (2) When relatives interact, one of the two conditions for stability of equilibria differs between the two models; this can be accounted for by positing that the inclusive fitness function for quantitative characters is slightly different from the inclusive fitness function for characters determined by a single locus. (3) The inclusion of environmental variance will in general change the equilibrium phenotype, but the equilibria of ESS analysis are changed to the same extent by environmental variance. (4) A class of genetically polymorphic equilibria occur, which in the present model are always unstable. These results expand the range of conditions under which one can validly predict the evolution of pairwise interactions using ESS analysis.     

Evolutionary stability of first-order-information indirect reciprocity in sizable groups

Indirect reciprocity is considered as a key mechanism for explaining the evolution of cooperation in situations where the same individuals interact only a few times. Under indirect reciprocity, an individual who helps others gets returns indirectly from others who know her good reputation. Recently, many studies have discussed the effect of reputation criteria based only on the former actions of the others (first-order information) and of those based also on the former reputation of opponents of the others (second-order information) on the evolution of indirect reciprocity. In this study, we investigate the evolutionary stability of the indirectly reciprocal strategy (discriminating strategy: DIS), which cooperates only with opponents who have good reputations, in -person games where more than two individuals take part in a single group (interaction). We show that in n-person games, DIS is an evolutionarily stable strategy (ESS) even under the image-scoring reputation criterion, which is based only on first-order information and where cooperations (defections) are judged to be good (bad). This result is in contrast to that of 2-person games where DIS is not an ESS under reputation criteria based only on first-order information.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

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.     

Stable equilibrium strategies and penalty functions in a game of attrition

Maynard-Smith (1974) has presented a game of attrition model for animal conflict. He assumed that the penalty function, giving the cost in terms of fitness, of displaying for a given time period, is a linear function of the time of display. Under this assumption he shows that an evolutionary stable strategy (ESS) always exists: one such is a mixed strategy for which display times have a negative exponential distribution. Given the diversity of reproductive strategies and patterns of agonistic behavior in nature, it is reasonable to consider games with different types of cost functions. In this paper it is shown that if more general cost functions are allowed (not necessarily linear), then ESS's still exist and give a great variety of distributions of display time. Supporting data are presented to suggest that these distributions may be found in nature. It is suggested that the interrelations between an animal's fitness budget and the game's penalty function will determine the nature of an ESS for different kinds of games.     

Evolutionary stability concepts for N-species frequency-dependent interactions.

The classical static concept of an evolutionarily stable strategy (ESS) for a single species gives rise to two new notions when there are more than two species (called an N-species ESS and RL-stability). The paper relates these to the dynamic stability of monomorphic and polymorphic evolutionary systems. It is shown that RL-stability implies the global asymptotic stability of either system with or without mutations. However, the N-species ESS only implies stability of the monomorphic system.     

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.     

Differential games in evolutionary theory: Height growth strategies of trees

Differential game theory is applied to the analysis of evolutionarily stable strategies (ESS) in this article. A general form for the evolutionary differential game is introduced in the case of intra-specific competition, and a connection between the ESS and the mathematical Nash solution concept is indicated. A dynamic ESS is found for the height growth strategies of trees. A hierarchical model is introduced to account for different time constants in simultaneous selection processes. Differential evolutionary games are compared with static evolutionary games utilizing the hierarchical approach.     

Evolution in heterogeneous environments: Effects of migration on habitat specialization

Summary Richard Levins introduced fitness sets as a tool for investigating evolution within heterogeneous environments. Evolutionary game theory permits a synthesis and generalization of this approach by considering the evolutionary response of organisms to any scale of habitat heterogeneity. As scales of heterogeneity increase from fine to coarse, the evolutionary stable strategy (ESS) switches from a single generalist species to several species that become increasingly specialized on distinct habitats. Depending upon the organisms' ecology, the switch from one to two species may occur at high migration rates (relatively fine-grained environment), or may only occur at very low migration rates (coarse-grained environment). At the ESS, the evolutionary context of a species is the entire landscape, while its ecological context may be a single habitat.Evolution towards the ESS can be represented with adaptive landscapes. In the absence of frequency-dependence, shifting from a single strategy ESS to a two strategy ESS poses the problem of evolving across valleys in the adaptive surface to occupy new peaks (hence, Sewell Wright's shifting balance theory). Frequency-dependent processes facilitate evolution across valleys. If a system with a two strategy ESS is constrained to possess a single strategy, the population may actually evolve a strategy that minimizes fitness. Because the population now rests at the bottom of a valley, evolution by natural selection can drive populations to occupy both peaks.     

Single species evolutionary dynamics

Not long after the introduction of evolutionary stable strategy (ESS) concept, it was noticed that dynamic selection did not always lead to the establishment of the ESS. The concept of continuously stable strategy (CSS) was thereafter developed. It was generally accepted that dynamic selection leads to the establishment of an ESS if it is a CSS. Examination of an evolutionary stability concept which is called neighborhood invader strategy (NIS) shows that it may be impossible for an ESS to be established through dynamic selection even if it is a CSS and no polymorphisms occur. We will examine the NIS concept and its implications for two evolutionary game models: root-shoot allocation in plant competition and Lotka–Volterra competition. In the root-shoot model we show that an ESS will be attained through dynamic selection if it is a NIS. Similarly for the Lotka–Volterra model, we show that an ESS will be attained through dynamic selection even if protected dimorphisms occur during the evolutionary process if it is an NIS.     

A theory for the evolutionary game

We present an evolutionary game theory. This theory differs in several respects from current theories related to Maynard Smith's pioneering work on evolutionary stable strategies (ESS). Most current work deals with two person matrix games. For these games the strategy set is finite. We consider evolutionary games which are defined over a continuous strategy set and which permit any number of players. Matrix games are included as a bilinear continuous game. However, under our definition, such games will not posses an ESS on the interior of the strategy set. We extend previous work on continuous games by developing an ESS definition which permits the ESS to be composed of a coalition of several strategies. This definition requires that the coalition must not only be stable with respect to perturbations in strategy frequencies which comprise the coalition, but the coalition must also satisfy the requirement that no mutant strategies can invade. Ecological processes are included in the model by explicitly considering population size and density dependent selection.     

Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time

The uninvadability concept, that was originally introduced through static comparisons of individual fitness in resident-mutant systems for a single species, is developed for multi-species models with frequency-dependent fitness by extending its equivalent single-species dynamic characterization. This multi-species definition is then reinterpreted in terms of individual fitness functions based on intra and interspecific interactions. The resultant concept is discussed in relation to that of an N-species ESS (evolutionarily stable strategy) and to dynamic stability of monomorphic and polymorphic evolutionary systems.     

Evolutionary dynamics of the Trivers–Willard effect: A nonparametric approach

The Trivers–Willard hypothesis (TWH) states that parents in good condition tend to bias their offspring sex ratio toward the sex with a higher variation in reproductive value, whereas parents in bad condition favor the opposite sex. Although the TWH has been generalized to predict various Trivers–Willard effects (TWE) depending on the life cycle of a species, existing work does not sufficiently acknowledge that sex‐specific reproductive values depend on the relative abundances of males and females in the population. If parents adjust their offspring sex ratio according to the TWE, offspring reproductive values will also change. This should affect the long‐term evolutionary dynamics and might lead to considerable deviations from the original predictions.In this paper, I model the full evolutionary dynamics of the TWE, using a published two‐sex integral projection model for the Columbian ground squirrel (Urocitellus columbianus). Offspring sex ratio is treated as a nonparametric continuous function of maternal condition. Evolutionary change is treated as the successive invasion of mutant strategies. The simulation is performed with varying starting conditions until an evolutionarily stable strategy (ESS) is reached.The results show that the magnitude of the evolving TWE can be far greater than previously predicted. Furthermore, evolutionary dynamics show considerable nonlinearities before settling at an ESS. The nonlinear effects depend on the starting conditions and indicate that evolutionary change is fastest when starting at an extremely biased sex ratio and that evolutionary change is weaker for parents of high condition. The results show neither a tendency to maximize average population fitness nor to minimize the deviation between offspring sex ratio and offspring reproductive value ratio.The study highlights the importance of dynamic feedback in models of natural selection and provides a new methodological framework for analyzing the evolution of continuous strategies in structured populations.     

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.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Evolutionarily stable strategies in food selection models with fitness sets

Most current models for optimal food selection apply to ecological and behavioural optimization. In this paper optimal food selection theory is extended to apply to evolutionary optimization. A general evolutionary model for optimal food selection must incorporate the concept of fitness sets--or that variables, changing as a result of natural selection in evolutionary time, cannot, in general, vary independently of each other. A "Charnov type" optimal food selection model with a fitness set is investigated, and evolutionarily stable strategy (ESS) solutions of the evolutionary variables (i.e., the efficiencies of using available food types) are found. From this analysis it follows that the relative frequency of various food types in the environment may, under specified conditions, influence the evolutionarily optimal diet. Secondly, the analysis demonstrates that a food type not in the optimal diet may, in evolutionary time, be added to this by becoming more abundant. Thirdly, it follows from the analysis that the ecological result of MacArthur and Pianka, that food types are worth eating even if there is competition for them, is not generally applicable when referring to an evolutionary time scale. Finally, it is pointed out that for the diet to be an ESS, it is necessary that the consumer's density is stable and that the consumer's population dynamics are subjected to some density-dependent factor.