The importance of intraspecific frequency-dependent selection in modelling competitive coevolution

Summary The coevolution of competitors has been analyzed by two different types of fitness-maximization techniques; ESS methods (Lawlor and Maynard Smith, 1976), and CSS methods (Roughgarden, 1979). This paper argues that CSS methods generally do not predict the outcome of competitive coevolution. Even when there is relatively little variability within species, fitness maximization leads to an ESS rather than a CSS. A simple model is analyzed to show that ESS and CSS predictions about character displacement can differ qualitatively. Previous results of CSS analyses are discussed.     

Evolutionary stability for large populations

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

Strong evolutionary equilibrium and the war of attrition

In developing the concept of an evolutionarily stable strategy, Maynard Smith proposed formal conditions for stability. These conditions have since been shown to be neither necessary nor sufficient for evolutionary stability in finite populations. This paper provides a strong stability condition which is sensitive to the population size. It is then demonstrated that in the war of attrition with uncertain rewards there is a unique “strong evolutionary equilibrium” strategy. As the population becomes large this is shown to approach the solution strategy proposed by Bishop, Cannings and Maynard Smith.The analysis is then extended to wars of attrition between different populations. It is concluded that for such contests there is a whole family of potential strong evolutionary equilibria.     

ALTRUISM IN MENDELIAN POPULATIONS DERIVED FROM SIBLING GROUPS: THE HAYSTACK MODEL REVISITED

A group-selection model is presented in which each group is initiated by a single fertilized female and persists for several generations before dispersal. Maynard Smith (1964) concluded that altruism could not plausibly evolve under these circumstances. I show that his conclusion is an artifact of a simplifying assumption that amounts to a worst-case scenario for group selection. When the standard donor-recipient equations for altruistic behavior are used in Maynard Smith's model, Mendelian populations derived from sibling groups are often more favorable for the evolution of altruism than are the sibling groups themselves. In general, long-term and large-scale aspects of population structure may at times be important in the evolution of altruistic and other group-advantageous behaviors.     

The Maynard Smith model of sympatric speciation

The paper entitled "Sympatric speciation," which was published by John Maynard Smith in 1966, initiated the development of mathematical models aiming to identify the conditions for sympatric speciation. A part of that paper was devoted to a specific two-locus, two-allele model of sympatric speciation in a population occupying a two-niche system. Maynard Smith provided some initial numerical results on this model. Later, Dickinson and Antonovics (1973) and Caisse and Antonovics (1978) performed more extensive numerical studies on the model. Here, I report analytical results on the haploid version of the Maynard Smith model. I show how the conditions for sympatric and parapatric speciation and the levels of resulting genetic divergence and reproductive isolation are affected by the strength of disruptive selection and nonrandom mating, recombination rate, and the rates of male and female dispersal between the niches.     

Does frequency-dependent selection optimize fitness?

In evolutionary biology, the axiom that natural selection tends ideally to maximize inclusive fitness of the individual or some other suitable quantity is often advanced (Cody, 1974; Maynard Smith, 1978; Krebs & McCleery, 1984; Houston et al., 1988). Moreover, the evolutionists generally distinguish two situations (Dawkins, 1980; Maynard Smith, 1982): one in which fitness is independent of the frequency of the phenotypes present in the population (frequency-independent selection), and one in which it does depend on this frequency (frequency-dependent selection). This led some authors such as Parker (1984), and more recently Parker & Maynard Smith (1990), to consider "a 2-speed optimization": frequency-independent selection should lead to a "simple optimum" at the end of the selective process, since all the individuals should have the same strategy and the mean fitness of the population should be maximized; frequency-dependent selection, formulated in terms of the theory of games, should lead to a "competitive optimum" even though the "evolutionary stable strategy" (or "ESS"; Maynard Smith & Price, 1973) characterizing the equilibrium "is not the strategy that maximizes fitness in a population sense" (Parker & Maynard Smith, 1990: 30). Our aim in this short communication is to criticize the concept of "competitive optimum" by Parker & Maynard Smith, as well as the general ability of natural selection to "maximize fitness", even in "phenotypic models" (Lloyd, 1977). These models, devoid of genetic constraints since each strategist is assumed to reproduce its own kind, are especially suitable for examining the ideal effect of natural selection.     

ESS equations sometimes do not specify an ESS

The equations used to find an evolutionarily stable strategy in the basic game theory model (Maynard Smith, 1974, 1982; Maynard Smith & Price, 1973), and in sexual conflict models (Maynard Smith, 1977; Parker, 1979) do not, in fact, specify an ESS when the expected number of contests entered is not the same for each strategy. This means that the conclusions of many game theory models may be incorrect. This is particularly likely to be true when the mean durations of contests for different strategies are not the same, or when the probability that an individual enters a contest is not the same for all strategies. New ESS equations are developed which incorporate the expected number of contests entered.     

Finite populations choose an optimal language

This paper studies the evolution of a proto-language in a finite population under the frequency-dependent Moran process. A proto-language can be seen as a collection of concept-to-sign mappings. An efficient proto-language is a bijective mapping from objects of communication to used signs and vice versa. Based on the comparison of fixation probabilities, a method for deriving conditions of evolutionary stability in a finite population [Nowak et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 246-650], it is shown that efficient proto-languages are the only strategies that are protected by selection, which means that no mutant strategy can have a fixation probability that is greater than the inverse population size. In passing, the paper provides interesting results about the comparison of fixation probabilities as well as Maynard Smith's notion of evolutionary stability for finite populations [Maynard Smith, 1988. Can a mixed strategy be stable in a finite population? J. Theor. Biol. 130, 247-251] that are generally true for games with a symmetric payoff function.     

Recombination or mutational hot spots in human mtDNA?

Awadalla, Eyre-Walker, and Maynard Smith (1999) recently argued that there might be recombination in human mitochondrial DNA (mtDNA). Their claim was based on their observation of decaying linkage disequilibrium (LD) as a function of physical distance. Their study was much criticized, and follow-up studies have failed to find any evidence for recombination. We argue that the criticisms levied, even if correct, could not possibly explain the findings of Awadalla, Eyre-Walker, and Maynard Smith (1999). Nonetheless, the test proposed by Awadalla, Eyre-Walker, and Maynard Smith (1999 ) is not robust because recombination is not the only explanation for decay of LD. We show that such a pattern can be caused by mutational hot spots as well. However, a closer look at the data suggests that the pattern observed was not caused by mutational hot spots but rather by chance. Thus, there appears to be no evidence for recombination in the mtDNA polymorphism data. In conclusion, we discuss the possibility of detecting recombination in mtDNA and the implications of its existence.     

How phenotypic matching based on neutral mating cues enables speciation in locally adapted populations

Maynard Smith's (American Naturalist, 1966, 100, 637) suggestion that in some cases a prerequisite for speciation is the existence of local ecological adaptations has not received much attention to date. Here, we test the hypothesis using a model like that of Maynard Smith but differing in the way animals disperse between niches. In previous studies, males disperse randomly between niches but females stay put in their natal niche. As a first step toward generalizing the model, we here analyze the case that equal proportions of the two sexes disperse between niches before breeding. Supporting Maynard Smith's (1966) hypothesis, we find that once local adaptations are established, a neutral mating cue at an independent locus can rapidly enable speciation in populations with a suitable mechanism for phenotype matching. We find that stable ecological polymorphisms are relatively insensitive to the strength of selection, but depend crucially on the extent of dispersal between niches, with a threshold of ~5% if population sizes in two niches are equal. At higher levels of dispersal, ecological differentiation is lost. These results contrast with those of earlier studies and shed light on why parapatric speciation is limited by the extent of gene flow. Our testable model provides a candidate explanation for the rapid speciation rates, diversity of appearance and occurrence of “species flocks” observed among some African cichlids and neotropical birds and may also have implications for the occurrence of punctuational change on phylogenies.     

The strategy concept and John Maynard Smith’s influence on theoretical biology

Here we argue that the concept of strategies, as it was introduced into biology by John Maynard Smith, is a prime illustration of the four dimensions of theoretical biology in the post-genomic era. These four dimensions are: data analysis and management, mathematical and computational model building and simulation, concept formation and analysis, and theory integration. We argue that all four dimensions of theoretical biology are crucial to future interactions between theoretical and empirical biologists as well as with philosophers of biology.     

Individual-based simulations of the War of Attrition

The War of Attrition model of John Maynard Smith predicts a single, mixed evolutionarily stable strategy (ESS) for animal contests which are settled by conventional displays with no assessment of the opponent's fighting ability. We test the predictions of the model by simulating the evolution of strategies in a finite population of animals under various assumptions on how possible strategies are coded and mutated. While our simulations for the most part confirm the predictions of the model, we also discovered some significant deviations from the theoretically predicted ESS. Specifically, we found that if inheritance of strategies is somewhat imprecise, then a population can evolve that achieves on average a higher payoff than a population at the theoretically predicted ESS. Moreover, if the ESS is realized as a polymorphism of fixed persistence times, then for small populations, sufficiently stringent statistical tests will reject the hypothesis that these times are distributed as theoretically predicted.     

ESS-theory for finite populations

On the basis of some principles from the philosophy of science, the inadequacy of the ESS-theory as introduced by Maynard Smith and Price as a biological theory is discussed, and an improved ESS-theory for finite populations is presented which can adopt the ideas of the original formalism, although modified. Resulting are explicit conditions on the population sizes that ensure certain strategies to be evolutionarily stable.     

Strategic analysis in evolutionary genetics and the theory of games

This paper is written in memory of John Maynard Smith. In a brief survey it discusses essential aspects of how game theory in biology relates to its counterpart in economics, the major transition in game theory initiated by Maynard Smith, the discrepancies between genetic and phenotypic models in evolutionary biology, and a balanced way of reconciling these models. In addition, the paper discusses modern problems in understanding games at the genetic level using the examples of conflict between endosymbionts and their hosts, and the molecular interactions between parasites and the mammalian immune system.     

Sex ratio under the haystack model

This paper investigates the evolution of the sex ratio under an extension of the haystack model of Maynard Smith (1964). At the beginning of each season a stack is colonized by a number of fertilized females, and their offspring breed there for several generations until new haystacks are available for colonization. We intend this as a model for populations which undergo periodical population explosions and crashes. With mating before dispersal, the number of generations in the stack has little effect on the equilibrium sex ratio, but it has a marked effect with mating after dispersal. This model is then used to investigate the evolutionary stability of the mechanism of sex determination found in the wood lemming which leads to a population sex ratio of three females to one male.     

Signature Function for Predicting Resonant and Attenuant Population 2-cycles

Populations are either enhanced via resonant cycles or suppressed via attenuant cycles by periodic environments. We develop a signature function for predicting the response of discretely reproducing populations to 2-periodic fluctuations of both a characteristic of the environment (carrying capacity), and a characteristic of the population (inherent growth rate). Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the carrying capacity and the demographic characteristic. Periodic environments are deleterious for populations when the signature function is negative. However, positive signature functions signal favorable environments. We compute the signature functions of six classical discrete-time single species population models, and use the functions to determine regions in parameter space that are either favorable or detrimental to the populations. The two-parameter classical models include the Ricker, Beverton-Holt, Logistic, and Maynard Smith models.     

Evolutionary equilibrium strategies.

This paper re-examines the concept of evolutionary stability proposed by Maynard Smith.For any finite population it is shown that a strategy which is stable in the sense of Maynard Smith may have a lower fitness than a mutant strategy regardless of the proportion of contestants using the latter.Two alternative concepts of evolutionary stability are then proposed. A strategy is described as being strongly stable if no mutant is able to invade because of its higher fitness and weakly stable if it has a higher fitness whenever the contestants using any particular mutant strategy become sufficiently numerous.For the “war of attrition” between contestants of a given species Maynard Smith and others have argued that the evolutionarily stable strategy is for a contestant to bid (for food or territory) by attempting to wait out its opponent according to an exponential mixed strategy. This paper establishes that such a strategy is only weakly stable and that for any number n there exists a mutant strategy with a higher fitness until the number of mutants in the population exceeds n.The final section reconsiders the stability issue when a natural informational asymmetry is introduced. Each contestant is assumed to be uncertain as to the value its opponent places on the object over which they are competing. In contrast to the symmetric case it is shown that there is a strategy which is strongly stable with respect to any feasible mutant as long as the population is sufficiently large.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.