The evolutionary stability of automimicry

Internal defences such as toxins cannot be detected from a distance by a predator, and are likely to be costly to produce and maintain. Populations of well-defended prey may therefore be vulnerable to invasion from rare 'cheater' mutants that do not produce the toxin themselves but obtain some protection from their resemblance to their better defended conspecifics (automimicry). Although it is well established that well-defended and weakly defended morphs may coexist stably in protected dimorphisms, recent theoretical work suggests that such dimorphisms would not be resistant to invasion by novel mutants with defence levels intermediate to those present. Given that most defences (including toxins) are likely to be continuous traits, this implies that automimicry may tend to be a transitory phenomenon, and thus less likely to explain variation in defence levels in nature. In contrast to this, we show that automimicry can also be evolutionarily stable for continuous traits, and that it may evolve under a wide range of conditions. A recently developed geometric method allows us to determine directly from a trade-off curve whether an evolutionarily stable defence dimorphism is at all possible, and to make some qualitative inferences about the ecological conditions that may favour it.     

Game theoretical modelling of survival strategies of Candida albicans inside macrophages

The polymorphic fungus Candida albicans can live as an aggressive pathogen that causes a wide variety of diseases in humans. Host resistance against these infections is mediated predominantly by phagocytes, namely neutrophils and macrophages. This report provides two game theoretical models of ingested C. albicans cells in macrophages. Two strategies are available for each pathogenic yeast cell: avoiding lysis transiently (called silencing) or forming hyphae and escaping (called piercing because the macrophage is pierced from inside). In dependence on parameter values, two different outcomes can be derived from the model: when the difference of the costs of the two strategies is low, all fungal cells inside a macrophage will play the piercing strategy, while in the high-cost case, a mixed population of piercing and silencing cells is the only stable solution. Further, the role of the SAP gene family encoding secreted proteinases and the Sap proteins is investigated with the help of known studies and is put in relation to the costs of the strategies, the most important parameter of this model. Our results are in agreement with wet-lab results presented by other groups and the model parameters can be estimated from experimental data.     

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.     

The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

Robustness of optimal mixed strategies

 Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in ‘ideal’ and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small. Received: 27 October 1996 / Revised version: 5 March 1997     

Evolution of alternative mating tactics: conditional versus mixed strategies

Intrasexual polymorphisms have evolved in a wide range of organisms.Most of them have been interpreted as the product of conditionalstrategies in which the tactic an individual adopts is determinedby some aspect of state (e.g., age, size, condition). However,there are a few examples that appear to represent an evolutionarilystable mixture of heritable pure strategies that are maintainedby frequency-dependent selection. In the present study, we producea model of a mating system with two morphs: a territorial morphand a sneak morph. By varying the costs and limits associatedwith conditional strategies, mating skew, and the proportionof matings obtained by sneaking males, we examine the conditionsthat favor the evolution of conditional versus pure strategies.Contrary to current thinking, our results show that as longas either costs or limits are greater than zero, conditionalstrategists are never able to entirely replace pure strategists,and equilibrium populations may frequently consist of a mixtureof conditional and pure strategists. Our results suggest thatconditional strategists will be most frequent at intermediatelevels of mating skew. Polymorphisms in which conditional strategistsare rare or absent are most likely to evolve when mating skewis extremely high, the costs and limits of plasticity are veryhigh, or the benefits of being conditional are very low. Thelimited data available suggest that high mating skew is probablythe most important factor.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

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.     

On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

Evolutionary stable strategies and game dynamics for n-person games

This note contains a generalization of the definition of an evolutionary stable strategy and of the corresponding game dynamics from 2-person to n-person games. This broader framework also allows modelling of several interacting populations or of populations containing different types of individuals, for example males and females.     

On the Evolution of Behavioral Heterogeneity in Individuals and Populations

A wide range of ecological and evolutionary models predict variety in phenotype or behavior when a population is at equilibrium. This heterogeneity can be realized in different ways. For example, it can be realized through a complex population of individuals exhibiting different simple behaviors, or through a simple population of individuals exhibiting complex, varying behaviors. In some theoretical frameworks these different realizations are treated as equivalent, but natural selection distinguishes between these two alternatives in subtle ways. By investigating an increasingly complex series of models, from a simple fluctuating selection model up to a finite population hawk/dove game, we explore the selective pressures which discriminate between pure strategists, mixed at the population level, and individual mixed strategists. Our analysis reveals some important limitations to the ESS framework often employed to investigate the evolution of complex behavior.     

The evolutionary ecology of attachment organization

Life history theory’s principle of allocation suggests that because immature organisms cannot expend reproductive effort, the major trade-off facing juveniles will be the one between survival, on one hand, and growth and development, on the other. As a consequence, infants and children might be expected to possess psychobiological mechanisms for optimizing this trade-off. The main argument of this paper is that the attachment process serves this function and that individual differences in attachment organization (secure, insecure, and possibly others) may represent facultative adaptations to conditions of risk and uncertainty that were probably recurrent in the environment of human evolutionary adaptedness. An early version of this paper was presented in the symposium “Childhood in Life-history Perspective: Developing Views” organized by Gilda Morelli and Paula Ivey for the Annual Meeting of the Society for Cross-Cultural Research in Santa Fe, New Mexico, February 16–20, 1994. James S. Chisholm recently joined the Department of Anatomy and Human Biology at the University of Western Australia. Previously he taught in the Department of Anthropology at the University of New Mexico and in the Division of Human Development at the University of California, Davis. He is a biosocial anthropologist whose research interests lie in the fields of human behavioral biology, evolutionary ecology, and life history theory, where he focuses on infant social-emotional development and the development of reproductive strategies in adolescence and young adulthood. In addition to numerous articles he is the author ofNavajo Infancy: An Ethological Study of Child Development (Aldine de Gruyter, 1983).     

Evolutionary dynamics of biological auctions

Many scenarios in the living world, where individual organisms compete for winning positions (or resources), have properties of auctions. Here we study the evolution of bids in biological auctions. For each auction, n individuals are drawn at random from a population of size N. Each individual makes a bid which entails a cost. The winner obtains a benefit of a certain value. Costs and benefits are translated into reproductive success (fitness). Therefore, successful bidding strategies spread in the population. We compare two types of auctions. In “biological all-pay auctions”, the costs are the bid for every participating individual. In “biological second price all-pay auctions”, the cost for everyone other than the winner is the bid, but the cost for the winner is the second highest bid. Second price all-pay auctions are generalizations of the “war of attrition” introduced by Maynard Smith. We study evolutionary dynamics in both types of auctions. We calculate pairwise invasion plots and evolutionarily stable distributions over the continuous strategy space. We find that the average bid in second price all-pay auctions is higher than in all-pay auctions, but the average cost for the winner is similar in both auctions. In both cases, the average bid is a declining function of the number of participants, n. The more individuals participate in an auction the smaller is the chance of winning, and thus expensive bids must be avoided.     

The behavioural ecology of personality: consistent individual differences from an adaptive perspective

Individual humans, and members of diverse other species, show consistent differences in aggressiveness, shyness, sociability and activity. Such intraspecific differences in behaviour have been widely assumed to be non‐adaptive variation surrounding (possibly) adaptive population‐average behaviour. Nevertheless, in keeping with recent calls to apply Darwinian reasoning to ever‐finer scales of biological variation, we sketch the fundamentals of an adaptive theory of consistent individual differences in behaviour. Our thesis is based on the notion that such ‘personality differences’ can be selected for if fitness payoffs are dependent on both the frequencies with which competing strategies are played and an individual's behavioural history. To this end, we review existing models that illustrate this and propose a game theoretic approach to analyzing personality differences that is both dynamic and state‐dependent. Our motivation is to provide insights into the evolution and maintenance of an apparently common animal trait: personality, which has far reaching ecological and evolutionary implications.     

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a "one-third law" of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 13. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the "number"13 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.     

Stochastic evolutionary dynamics of bimatrix games

Evolutionary game dynamics of two-player asymmetric games in finite populations is studied. We consider two roles in the game, roles α and β. α-players and β-players interact and gain payoffs. The game is described by a pair of matrices, which is called bimatrix. One's payoff in the game is interpreted as its fecundity, thus strategies are subject to natural selection. In addition, strategies can randomly mutate to others. We formulate a stochastic evolutionary game dynamics of bimatrix games as a frequency-dependent Moran process with mutation. We analytically derive the stationary distribution of strategies under weak selection. Our result provides a criterion for equilibrium selection in general bimatrix games.     

The evolutionary ecology of Plasmodium

Plasmodium, the aetiological agent of malaria, imposes a substantial public health burden on human society and one that is likely to deteriorate. Hitherto, the recent Darwinian medicine movement has promoted the important role evolutionary biology can play in issues of public health. Recasting the malaria parasite two‐host life cycle within an evolutionary framework has generated considerable insight into how the parasite has adapted to life within both vertebrate and insect hosts. Coupled with the rapid advances in the molecular basis to host–parasite interactions, exploration of the evolutionary ecology of Plasmodium will enable identification of key steps in the life cycle and highlight fruitful avenues of research for developing malaria control strategies. In addition, elucidating the extent to which Plasmodium can respond to short‐ and long‐term changes in selection pressures, i.e. its adaptive capacity, is even more crucial in predicting how the burden of malaria will alter with our rapidly evolving ecology.