Multi-species evolutionary dynamics

Dynamical attainability of an evolutionarily stable strategy (ESS) through the process of mutations and natural selection has mostly been addressed through the use of the continuously stable strategy (CSS) concept for species evolutionary games in which strategies are drawn from a continuum, and by the adaptive trait dynamics method. We address the issue of dynamical attainability of an ESS in coevolving species through the use of the concept of an ESNIS. It is shown that the definition of an ESNIS coalition for coevolving species is not in general equivalent to other definitions for CSS given in the literature. We show under some additional conditions that, in a dynamic system which involves the strategies of a dimorphic ESNIS coalition and at most two strategies that are not members of ESNIS coalition, the ESNIS coalition will emerge as the winner. In addition an ESNIS will be approached because of the invasion structure of strategies in its neighborhood. This proves that under the above conditions an ESNIS has a better chance of being attained than a strategy coalition which is a CSS. The theory developed is applied to a class of coevolutionary game models with Lotka–Volterra type interactions and we show that for such models, an ESS coalition will be dynamically attainable through mutations and natural selection if the ESS coalition is also an ESNIS coalition.Co-ordinating editor: Metz     

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.     

Genetic versus phenotypic models of selection: can genetics be neglected in a long-term perspective?

Game theoretical concepts in evolutionary biology have been criticized by populations geneticists, because they neglect such crucial aspects as the mating system or the mode of inheritance. In fact, the dynamics of natural selection does not necessarily lead to a fitness maximum or an ESS if genetic constraints are taken into account. Yet, it may be premature to conclude that game theoretical concepts do not have a dynamical justification. The new paradigm of long-term evolution postulates that genetic constraints, which may be dominant in a short-term perspective, will in the long run disappera in the face of the ongoing influx of mutations. Two basic results (see Hammerstein; this issue) seem to reconcile the dynamical approach of long-term population genetics with the static approach of evolutionary game theory: (1) only populations at local fitness optima (Nash strategies) can be long-term stable; and (2) in monomorphic populations, evolutionary stability is necessary and sufficient to ensure long-term dynamic stability. The present paper has a double purpose. On the one hand, it is demonstrated by fairly general arguments that the scope of the results mentioned above extends to non-linear frequency dependent selection, to multiple loci, and to quite general mating systems. On the other hand, some limitations of the theory of long-term evolution will also be stressed: (1) there is little hope for a game theoretical characterization of stability in polymorphic populations; (2) many interesting systems do not admit long-term stable equilibria; and (3) even if a long-term stable equilibrium exists, it is not at all clear whether and how it is attainable by a series of gene substition events.     

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.     

Population structure,spite, and the iterated Prisoner's Dilemma

Evolutionary stability (sensu Maynard Smith: Evolution and the Theory of Games, Cambridge: Cambridge University Press, 1982) of TIT FOR TAT (TFT) under the social ecology of the iterated Prisoner's Dilemma is a function of the number of pure TFT groups (dyads) in the population, relative to the social position of a focal invading defector. Defecting against TFT always raises the defector's relative intragroup fitness; when Axelrod's (Am. Polit. Sci. Rev. 75:306–318, 1981; The Evolution of Cooperation. New York: Basic Books, 1984) Evolutionary stable strategy (ESS) conditions are met, defection also lowers the absolute fitness of the defector. Here the retaliatory (punishing) character of TFT converts defection into spite, permitting pure TFT groups to sufficiently outproduce the defector for the latter's evolutionary suppression. Increasing the relative impact of spiteful defection on a population lowers the range of evolutionary stability for TFT. When individuals participate in multiple dyads, those participating in the greatest number of dyads are most likely to provide a vehicle for the successful invasion of defection. Within social networks, ESS conditions for TFT are thus individual specific. This logic is generalized to the context of an interated n-person Prisoner's Dilemma, providing a cooperative solution conceptually identical with TFT in the two-person game.     

On the relation between the instability of ESS in discrete dynamics and segregation distortion

There is a simple correspondence between discrete dynamical systems associated with evolutionary game dynamics and general locus multiallele selection models with non-Mendelian segregation. When interpreted properly the payoff matrix has two components, a fitness matrix component and a segregation matrix component. The presence of segregation distortion which corresponds to a non-symmetric payoff matrix, is a source of instability. With non-symmetric payoff an ESS does not usually correspond to a stable equilibrium. It is always externally stable but does not necessarily have an internally stable equilibrium.     

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.     

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.     

Fighting for food: a dynamic version of the Hawk-Dove game

Summary The Hawk-Dove game (Maynard Smith, 1982) has been used to analyse conflicts over resources such as food. At the evolutionarily stable strategy (ESS), either a proportionp* of animals always play Hawk, or each animal has a probabilityp* of playing Hawk. We modify the standard Hawk-Dove game to include a state variable,x, that represents the animal's level of energy reserves. A strategy is now a rule for choosing an action as a function ofx and time of day. We consider a non-reproductive period and adopt the criterion of minimizing mortality over this period. We find the ESS, which has the form play Hawk if reserves are belowc* (t) at timet, otherwise play Dove. This ESS is very different from the ESS in the standard Hawk-Dove game. It is a pure ESS that depends on the animal's state and on time. Furthermore, it is characterized by the strong condition that any single mutant that does not adopt the ESS suffers a reduction in fitness. The standard Hawk-Dove game assumes pay-offs that are related to fitness; our approach starts from a definition of fitness and derives the pay-offs in the process of finding the ESS. When the environment becomes worse (e.g. food becomes less reliable or energy expenditure increases) the ESS changes in such a way as to increase the proportion of animals that will play Hawk.     

Stability in an evolutionary game

A behavior or strategy which is evolutionarily stable must be both optimal and stable. The strategy must be optimal in that it maximizes the expected fitness of all the individuals using it. In addition, the strategy must be resistant to invasion by a mutant. The difference between the Nash solution of game theory and the ESS used in ecology is that the Nash solution only satisfies an optimality criterion and not an evolutionary stability criterion. We extend the ESS definition of Maynard Smith and Price so that it can be applied directly to two-strategy evolutionary games. The concept of a balanced game is introduced, and necessary conditions are derived which are similar to the Nash necessary conditions. The balanced game necessary conditions may be used for direct calculation of ESS candidates. These results are used to examine the optimal flowering time of an annual plant experiencing competition from neighboring plants. The plant competition model is general, and the results may be applied to a wide range of interference competition problems.     

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.     

Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models

This paper studies the correspondence between Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models. Whenever the fitness function is sufficiently continuous, any large-population ESS corresponds to a symmetric Nash equilibrium in the game that describes the simultaneous interaction of the individuals in the population, and any strict, symmetric Nash equilibrium in that game corresponds to a large-population ESS. This correspondence continues to hold, approximately, in finite populations; and it holds exactly for strict pure-strategy equilibria in sufficiently large finite populations. By contrast, a sequence of (mixed-strategy) finite-population ESSs can converge, as the population grows, to a limit that is not a large-population ESS, and a large-population ESS need not be the limit of any sequence of finite-population ESSs.     

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.     

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.     

HABITAT SELECTION AS AN EVOLUTIONARY GAME

Under habitat selection, mobile foragers may not only possess behavioral flexibility that allows them to utilize habitats selectively or opportunistically, but they may also possess heritable traits that influence their performance within each habitat. A game theoretic model is developed that investigates this evolutionary dimension of habitat selection. The model follows that of Rosenzweig (1987b) and considers a patchy environment containing two distinct habitat types. Behaviorally, foragers may be selective or opportunistic; morphologically, foragers possess traits that represent a trade-off between performance in the two habitat types. Depending on the environment's structure, one of three types of communities emerges as the ESS: (1) a single generalist species that behaves opportunistically, (2) two species that are extreme specialists on habitat 1 and 2, respectively; behaviorally, these species are selective on their respective habitat types, and (3) one generalist species that behaves opportunistically and one specialist species that behaves selectively on its preferred habitat. Community (1) emerges when habitat selection is costly, community (2) emerges when habitat selection is cost-free, and community (3) emerges when the relative abundances or productivities of the two habitat types are lopsided.     

Stochastic Fluctuations Through Intrinsic Noise in Evolutionary Game Dynamics

A one-step (birth–death) process is used to investigate stochastic noise in an elementary two-phenotype evolutionary game model based on a payoff matrix. In this model, we assume that the population size is finite but not fixed and that all individuals have, in addition to the frequency-dependent fitness given by the evolutionary game, the same background fitness that decreases linearly in the total population size. Although this assumption guarantees population extinction is a globally attracting absorbing barrier of the Markov process, sample trajectories do not illustrate this result even for relatively small carrying capacities. Instead, the observed persistent transient behavior can be analyzed using the steady-state statistics (i.e., mean and variance) of a stochastic model for intrinsic noise that assumes the population does not go extinct. It is shown that there is good agreement between the theory of these statistics and the simulation results. Furthermore, the ESS of the evolutionary game can be used to predict the mean steady state.     

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.     

Ecological stability,evolutionary stability and the ESS maximum principle

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

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.     

A simple model of host-parasite evolutionary relationships. Parasitism: compromise or conflict?

The evolutionary biology of host-parasite relationships are considered here using a simple game-theory model in which hosts play against parasite and vice versa. In this model, the players can choose between two strategies (aggressive or not aggressive) and the utility of the game is envisaged in terms of fitness and selective costs. The game solutions suggest that the two types of confrontation are encountered in symbiotic relationships and thus constitute two Evolutionary Stable Strategies (ESS). These observations lead us to discuss: (i) the status of different kinds of symbiotic relationships (i.e. parasitoidism; parasitism, commensalism and mutualism) related to selective costs and (ii) the position of coevolution in this game theory context.