Evolutionary game dynamics with impulsive effects

The jumps in population size due to the occurrence of an unfavorable physical environment (e.g. the effects of periodic climate disaster on the population size), or due to the intrinsic physiological and reproductive mechanisms of the population (e.g. the seasonal reproduction of most animal populations), can be called impulsive perturbations. A two-phenotype evolutionary game dynamics with impulsive effects is investigated. The main goal is to show how the evolutionary game dynamics is affected by the impulsive perturbations. The results show that the impulsive perturbations not only result in periodic behavior, but also it is possible that an ESS strategy based on the traditional concept of evolutionary stability can be replaced successfully by a non-ESS strategy.     

Superparasitism as a differential game

Superparasitism refers to a female parasitoid laying an egg in a host already parasitized by a conspecific. In solitary species, only one offspring per host is expected to complete development, hence the game. Hosts are often clumped in patches and several females exploiting such an aggregate of resource make its state change over time, hence the dynamical character of the game. Two coupled questions arise: (i) Is it worth accepting a parasitized host? (ii) When to leave the host patch? Through these decisions (i) the competition for healthy hosts and (ii) the trade-off between leaving in quest of a better patch and staying to make the patch less profitable for other parasitoids (this is a way to lower superparasitism likely to occur after having left the patch) are addressed. The aim of this work is to characterize a strategy that would be evolutionarily relevant in such a situation, as it directly concerns females' reproductive success. Investigating a (synchronous) nonzero-sum two-player differential game allows us to characterize candidate dynamic evolutionarily stable policies in terms of both oviposition and patch-leaving decisions. For that matter, the game is (in the most part of the parameter space) completely solved if the probability that superparasitism succeeds is assumed to be close to one-half, a fair value under direct competition. The strategic equilibrium consists, for each females, in (i) superparasitizing consistently upon arrival on the patch, and (ii) leaving when the loss of fitness due to superparasitism likely to occur after its departure is reduced to zero. The competing females are thus expected to leave the patch as they arrived: synchronously. Superparasitism does not necessarily lead to a war of attrition.     

A large population parental care game: Polymorphisms and feedback between patterns of care and the operational sex ratio

This article presents a game theoretic model of parental care which models the feedback between patterns of care and the operational sex ratio. It is assumed here that males can be in one of two states: searching for a mate or breeding (including caring for their offspring). Females can be in one of three states: receptive (searching), non-receptive or breeding. However, these sets of states can be adapted to the physiology of a particular species. The length of time that an individual remains in the breeding state depends on the level of care an individual gives. When in the searching state, individuals find partners at a rate dependent on the proportion of members of the opposite sex searching. These rates are defined to satisfy the Fisher condition that the total number of offspring of males equals the total number of offspring of females. The operational sex ratio is not defined exogenously, but can be derived from the adult sex ratio and the pattern of parental care. Pure strategy profiles and so-called single sex stable polymorphisms, in which behaviour is varied within one sex, are derived analytically. The difference between mixed evolutionarily stable strategies and stable polymorphisms within this framework is highlighted. The effects of various physiological and demographic parameters on patterns of care are considered.     

Higher disease prevalence can induce greater sociality: a game theoretic coevolutionary model

There is growing evidence that communicable diseases constitute a strong selective force on the evolution of social systems. It has been suggested that infectious diseases may determine upper limits of host sociality by, for example, inducing territoriality or early juvenile dispersal. Here we use game theory to model the evolution of host sociality in the context of communicable diseases. Our model is then augmented with the evolution of virulence to determine coevolutionarily stable strategies of host sociality and pathogen virulence. In contrast to a controversial hypothesis by Ewald (1994), our analysis indicates that pathogens may become more virulent when contact rates are low, and their prevalence can ultimately induce greater sociality.     

The patch distributed producer-scrounger game

Grouping in animals is ubiquitous and thought to provide group members antipredatory advantages and foraging efficiency. However, parasitic foraging strategy often emerges in a group. The optimal parasitic policy has given rise to the producer-scrounger (PS) game model, in which producers search for food patches, and scroungers parasitize the discovered patches. The N-persons PS game model constructed by Vickery et al. (1991. Producers, scroungers, and group foraging. American Naturalist 137, 847-863) predicts the evolutionarily stable strategy (ESS) of frequency of producers that depends on the advantage of producers and the number of foragers in a group. However, the model assumes that the number of discovered patches in one time unit never exceeds one. In reality, multiple patches could be found in one time unit. In the present study, we relax this assumption and assumed that the number of discovered patches depends on the producers’ variable encounter rate with patches (λ). We show that strongly depends on λ within a feasible range, although it still depends on the advantage of producer and the number of foragers in a group. The basic idea of PS game is the same as the information sharing (parasitism), because scroungers are also thought to parasitize informations of locations of food patches. Horn (1968) indicated the role of information-parasitism in animal aggregation (Horn, H.S., 1968. The adaptive significance of colonial nesting in the Brewer's blackbird (euphagus cyanocephalus). Ecology 49, 682-646). Our modified PS game model shows the same prediction as the Horn's graphical animal aggregation model; the proportion of scroungers will increase or animals should adopt colonial foraging when resource is spatiotemporally clumped, but scroungers will decrease or animals should adopt territorial foraging if the resource is evenly distributed.     

Population differences in the timing of diapause: a test of hypotheses

Summary The reproductive phenology of the freshwater copepod Diaptomus sanguineus differs markedly between populations residing in two Rhode Island ponds. In a permanent pond the population switches abruptly from making subitaneous (immediately hatching) eggs to diapausing eggs at the end of March each year. In contrast, a temporary pond population switches egg types in May, returns to production of subitaneous eggs in June, and concludes the reproductive season by making diapausing eggs in July. An ESS model suggests that the pattern of diapause expected of a copepod population is a function of annual variation in the onset of harsh conditions (catastrophe date). When variation is relatively low, the superior strategy is for diapause to begin a constant period before the mean catastrophe date. When variation is high, females should make first subitaneous eggs and then diapausing eggs irrespective of the expected catastrophe date. With discrete generations, such a population would alternate between egg types. In the permanent pond, variation of catastrophe date the spring onset of planktivory by sunfish is low, whereas in the temporary pond variation of the catastrophe (pond drying) is high. The model predicts well the phenology of the two copepod populations.In the research reported here, we tested the hypothesis that copepods from the permanent pond, which switch to diapause at the same time every year, are cued by the environment to begin diapause (i.e. by photoperiod, temperature, or both), whereas those from the temporary pond make both egg types regardless of environmental conditions. In opposition to our hypothesis, experimental results indicate that diapause in both populations is cued by the environment. The distinct reproductive phenologies documented in the two populations apparently result from the copepods responding to different environmental cues, rather than one being responsive to the environment while the other is not.     

Evolutionarily stable stealing: game theory applied to kleptoparasitism

We present an individual-based model of a group of foraginganimals. Individuals can obtain food either by discovering itthemselves or by stealing it from others (kleptoparasitism).Given that challenging another individual for a discovered fooditem costs time (which could otherwise be spent searching foran undiscovered item), attempting to steal from another maynot always be efficient We show that there is generally a uniquestrategy that maximizes uptake rate—always or never challengingothers. For any combination of parameter values, we can identifywhich strategy is appropraite. As a corollary to this, we predictthat small changes in ecolgical conditions can, under some circumstances,cause a dramatic change in the aggressive behavior of individuals.Further, we investigate situations where searching for undiscoveredfood and searching for potential opportunities for stealingare mutually exclusive activities (i.e., success at one canonly be improved at the expense of the other). Using game theory,we are able to find the evolutionarily stable strategy for investmentin these two activities in terms of the ecological parametersof the model.     

Rare but severe concerted punishment that favors cooperation

As one of the mechanisms that are supposed to explain the evolution of cooperation among unrelated individuals, costly punishment, in which altruistic individuals privately bear the cost to punish defection, suffers from such drawbacks as decreasing individuals’ welfare, inducing second-order free riding, the difficulty of catching defection, and the possibility of triggering retaliation. To improve this promising mechanism, here we propose an extended Public Goods game with rare but severe concerted punishment, in which once a defector is caught punishment is triggered and the cost of punishment is equally shared among the remainder of the group. Analytical results show that, when the probability for concerted punishment is above a threshold, cooperating is, while defecting is not, an evolutionarily stable strategy in finite populations, and that this way of punishment can considerably decrease the total cost of inhibiting defection, especially in large populations.     

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms'' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin''s arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.     

The ideal free distribution: a review and synthesis of the game-theoretic perspective

The Ideal Free Distribution (IFD), introduced by Fretwell and Lucas in [Fretwell, D.S., Lucas, H.L., 1970. On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica 19, 16-32] to predict how a single species will distribute itself among several patches, is often cited as an example of an evolutionarily stable strategy (ESS). By defining the strategies and payoffs for habitat selection, this article puts the IFD concept in a more general game-theoretic setting of the “habitat selection game”. Within this game-theoretic framework, the article focuses on recent progress in the following directions: (1) studying evolutionarily stable dispersal rates and corresponding dispersal dynamics; (2) extending the concept when population numbers are not fixed but undergo population dynamics; (3) generalizing the IFD to multiple species.For a single species, the article briefly reviews existing results. It also develops a new perspective for Parker’s matching principle, showing that this can be viewed as the IFD of the habitat selection game that models consumer behavior in several resource patches and analyzing complications involved when the model includes resource dynamics as well. For two species, the article first demonstrates that the connection between IFD and ESS is now more delicate by pointing out pitfalls that arise when applying several existing game-theoretic approaches to these habitat selection games. However, by providing a new detailed analysis of dispersal dynamics for predator-prey or competitive interactions in two habitats, it also pinpoints one approach that shows much promise in this general setting, the so-called “two-species ESS”. The consequences of this concept are shown to be related to recent studies of population dynamics combined with individual dispersal and are explored for more species or more patches.     

Evolutionary stability of optimal foraging: Partial preferences in the diet and patch models

In this article the patch and diet choice models of the optimal foraging theory are reanalyzed with respect to evolutionary stability of the optimal foraging strategies. In their original setting these fundamental models consider a single consumer only and the resulting fitness functions are both frequency and density independent. Such fitness functions do not allow us to apply the classical game theoretical methods to study an evolutionary stability of optimal foraging strategies for competing animals. In this article frequency and density dependent fitness functions of optimal foraging are derived by separation of time scales in an underlying population dynamical model and corresponding evolutionarily stable strategies are calculated. Contrary to the classical foraging models the results of the present article predict that partial preferences occur in optimal foraging strategies as a consequence of the ecological feedback of consumer preferences on consumer fitness. In the case of the patch occupation model these partial preferences correspond to the ideal free distribution concept while in the case of the diet choice model they correspond to the partial inclusion of the less profitable prey type in predators diet.     

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.     

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 framework for modelling and analysing conspecific brood parasitism

Recently several papers that model parasitic egg-laying by birds in the nests of others of their own species have been published. Whilst these papers are concerned with answering different questions, they approach the problem in a similar way and have a lot of common features. In this paper a framework is developed which unifies these models, in the sense that they all become special cases of a more general model. This is useful for two main reasons; firstly in order to aid clarity, in that the assumptions and conclusions of each of the models are easier to compare. Secondly it provides a base for further similar models to start from. The basic assumptions for this framework are outlined and a method for finding the ESSs of such models is introduced. Some mathematical results for the general, and more specific, models are considered and their implications discussed. In addition we explore the biological consequences of the results that we have obtained and suggest possible questions which could be investigated using models within or very closely related to our framework.M. Broom is also a member of the Centre for the Study of Evolution at the University of Sussex.     

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

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

Negotiation may lead selfish individuals to cooperate: the example of the collective vigilance game

Game-theoretical models have been highly influential in behavioural ecology. However, these models generally assume that animals choose their action before observing the behaviour of their opponents while, in many natural situations, individuals in fact continuously react to the actions of others. A negotiation process then takes place and this may fundamentally influence the individual attitudes and the tendency to cooperate. Here, I use the classical model system of vigilance behaviour to demonstrate the consequences of such behavioural negotiation among selfish individuals, by predicting patterns of vigilance in a pair of animals foraging under threat of predation. I show that the game played by the animals and the resulting vigilance strategies take radically different forms, according to the way predation risk is shared in the pair. In particular, if predators choose their target at random, the prey respond by displaying moderate vigilance and taking turns scanning. By contrast, if the individual that takes flight later in an attack endures a higher risk of being targeted, vigilance increases and there is always at least one sentinel in the pair. Finally, when lagging behind its companion in fleeing from an attacker becomes extremely risky, vigilance decreases again and the animals scan simultaneously.     

On sperm competition games: raffles and roles revisited

 In principle there are two approaches to modelling a trade-off between the positive and negative outcomes of a behavior: after suitably defining a value for the behavior in the absence of any trade-off, one can either multiply that value by an appropriate discount or subtract an appropriate cost. In a prospective analysis of sperm competition, Parker (Proc. Roy. Soc. Lond. B (1990) 242, 120–126) adopted the multiplicative approach to model the trade-off between the value of a mating and the cost of its acquisition. He obtained two paradoxical results. First, if two males ‘know’ whether they are first or second to mate, but these roles are assigned randomly, then sperm numbers should be the same for both males whether the ‘raffle’ for fertilization is fair or unfair. Second, if mating order is constant, then a favored male should expend less on sperm. His results are puzzling not only in terms of intuition about nature, but also in terms of his model’s consistency. In other words, they present both an external and an internal paradox. Parker assumed the fairness of the raffle to a disfavored male to be independent of how much sperm a favored male deposits. This article both generalizes Parker’s analysis by allowing fairness to decrease with sperm expenditure by the favored male and compares Parker’s results to those obtained by the additive approach. In many respects, results are similar. Nevertheless, if the costs of mating are assumed to increase with sperm expenditure but not to depend on the role in which sperm is expended, as Parker assumed, then the additive approach is more fundamentally correct. In particular, Parker’s constant-role paradox is an artifact of his approach. His random-role paradox is internally rationalized in terms of standard microeconomic theory. When fairness decreases, however slightly, with sperm expenditure by the favored male, both models demonstrate that the evolutionarily stable strategy is for more sperm to be deposited during a favored mating than during a disfavored mating. The lower the costs, the greater the divergence. Thus a possible resolution of the external paradox is that fairness is not constant in nature. Received: 7 December 1998     

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

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