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.     

THE EVOLUTIONARILY STABLE PHENOTYPE DISTRIBUTION IN A RANDOM ENVIRONMENT

In an unpredictably changing environment, phenotypic variability may evolve as a “bet-hedging” strategy. We examine here two models for evolutionarily stable phenotype distributions resulting from stabilizing selection with a randomly fluctuating optimum. Both models include overlapping generations, either survival of adults or a dormant propagule pool. In the first model (mixed-strategies model) we assume that individuals can produce offspring with a distribution of phenotypes, in which case, the evolutionarily stable population always consists of a single genotype. We show that there is a unique evolutionarily stable strategy (ESS) distribution that does not depend on the amount of generational overlap, and that the ESS distribution generically is discrete rather than continuous; that is, there are distinct classes of offspring rather than a continuous distribution of offspring phenotypes. If the probability of extreme fluctuations in the optimum is sufficiently small, then the ESS distribution is monomorphic: a single type fitted to the mean environment. At higher levels of variability, the ESS distribution is polymorphic, and we find stability conditions for dimorphic distributions. For an exponential or similarly broad-tailed distribution of the optimum phenotype, the ESS consists of an infinite number of distinct phenotypes. In the second model we assume that an individual produces offspring with a single, genetically determined phenotype (pure-strategies model). The ESS population then contains multiple genotypes when the environmental variance is sufficiently high. However the phenotype distributions are similar to those in the mixed-strategies model: discrete, with an increasing number of distinct phenotypes as the environmental variance increases.     

Conditional dispersal under kin competition: extension of the Hamilton-May model to brood size-dependent dispersal

I consider a site-based model with contest competition among siblings, and assume that dispersal is conditional on the number of offspring in the natal site. Evolutionarily stable populations contain threshold dispersal strategies, which retain a certain number of offspring in the natal site and disperse the rest (if the actual number of offspring is less than the threshold, then all offspring are retained). Due to the discrete nature of the strategy set (the threshold must be integer), the ESS may not be unique or may not exist. In the latter case, two neighboring threshold strategies coexist in the evolutionarily stable population. Dispersal first decreases and then increases as a function of dispersal mortality, such that all but one offspring should be dispersed both when dispersal mortality is very small or very high. Population-level dispersal fractions are often similar to the unconditional ESS, but differ strongly when fecundity is small and dispersal mortality is high.     

Adaptive diversification of germination strategies.

Evolution of the germination rate (the proportion of newly produced and dormant seeds that germinates every year) of annual plants is investigated, when the environment is temporally stochastic and spatially heterogeneous. The environment consists of two habitats with synchronous stochastic variation in the annual yield and permanent difference in constant seed survival rates. Density dependence operates within the habitats, which are connected via restricted seed dispersal. We find that instead of a single common evolutionarily stable strategy the coexistence of several germination strategies is possible and that in an initially monomorphic population evolutionary branching may occur. During evolutionary branching the population undergoes disruptive selection and splits into two branches of different lineages that converge to the evolutionarily stable coalition of different germination strategies. It is shown that spatial heterogeneity and restricted dispersal are essential for evolutionary branching. Disruptive selection on the germination rate presents yet another possibility for parapatric speciation.     

Global analyses of evolutionary dynamics and exhaustive search for social norms that maintain cooperation by reputation

Reputation formation is a key to understanding indirect reciprocity. In particular, the way to assign reputation to each individual, namely a norm that describes who is good and who is bad, greatly affects the possibility of sustained cooperation in the population. Previously, we have exhaustively studied reputation dynamics that are able to maintain a high level of cooperation at the ESS. However, this analysis examined the stability of monomorphic population and did not investigate polymorphic population where several strategies coexist. Here, we study the evolutionary dynamics of multiple behavioral strategies by replicator dynamics. We exhaustively study all 16 possible norms under which the reputation of a player in the next round is determined by the action of the self and the reputation of the opponent. For each norm, we explore evolutionary dynamics of three strategies: unconditional cooperators, unconditional defectors, and conditional cooperators. We find that only three norms, simple-standing, Kandori, and shunning, can make conditional cooperation evolutionarily stable, hence, realize sustained cooperation. The other 13 norms, including scoring, ultimately lead to the invasion by defectors. Also, we study the model in which private reputation errors exist to a small extent. In this case, we find the stable coexistence of unconditional and conditional cooperators under the three norms.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

Can mixed strategies be stable in asymmetric games?

Selten (1980, J. theor. Biol. 84, 93(N)/01) has shown that mixed strategies cannot be evolutionarily stable in asymmetric games. Because every interaction features some asymmetry, this result apparently precludes mixed strategies in an evolutionary setting. In Maynard Smith's Hawk-Dove game (1982, Evolution and the theory of games (UP-Cambridge), for example, Selten's result restricts attention to pure-strategy evolutionarily stable outcomes in which the animals use the ability to condition their actions on asymmetries to coordinate, with one playing Hawk and one playing Dove, and with conflicts in which both animals play Hawk never arising. This result contrasts with the intuition that the mixed equilibrium of the Hawk-Dove game captures important aspects of many animal interactions, including the possibility of conflict. In this paper, we follow Eshel and Sansone (1995, J. theor. Biol. 177, 341-356) in enriching Selten's model to incorporate an important aspect of animal interactions, namely that payoffs and asymmetries may both be imperfectly observed. In the richer model, we find conditions under which effectively mixed strategies are stable in asymmetric games, as well as conditions under which they are not stable. Behavior will be conditioned on asymmetries, leading to pure-strategy equilibria in which conflict is avoided, when there are relatively large, observable asymmetries and small observable variations in payoffs. Under opposite conditions, evolutionarily stable equilibria will appear that are effectively mixed, including the potential for conflict.     

A theoretical model of the evolution of maternal effects under parent-offspring conflict

The evolution of maternal effects on offspring phenotype should depend on the extent of parent-offspring conflict and costs and constraints associated with maternal and offspring strategies. Here, we develop a model of maternal effects on offspring dispersal phenotype under parent-offspring conflict to evaluate such dependence. In the absence of evolutionary constraints and costs, offspring evolve dispersal rates from different patch types that reflect their own, rather than the maternal, optima. This result also holds true when offspring are unable to assess their own environment because the maternal phenotype provides an additional source of information. Consequently, maternal effects on offspring diapause, dispersal, and other traits that do not necessarily represent costly resource investment are more likely to maximize offspring than maternal fitness. However, when trait expression was costly, the evolutionarily stable dispersal rates tended to deviate from those under both maternal and offspring control. We use our results to (re)interpret some recent work on maternal effects and their adaptive value and provide suggestions for future work.     

A dynamic game-theoretic model of parental care

We present a model in which members of a mated pair decide whether to care for their offspring or desert them. There is a breeding season of finite length during which it is possible to produce and raise several batches of offspring. On deserting its offspring, an individual can search for a new mate. The probability of finding a mate depends on the number of individuals of each sex that are searching, which in turn depends upon the previous care and desertion decisions of all population members. We find the evolutionarily stable pattern of care over the breeding season. The feedback between behaviour and mating opportunity can result in a pattern of stable oscillations between different forms of care over the breeding season. Oscillations can also arise because the best thing for an individual to do at a particular time in the season depends on future behaviour of all population members. In the baseline model, a pair splits up after a breeding attempt, even if they both care for the offspring. In a version of the model in which a pair stays together if they both care, the feedback between behaviour and mating opportunity can lead to more than one evolutionarily stable form of care.     

Reproductive skew and indiscriminate infanticide

In communally breeding animals, there is an evolutionary conflict over the partitioning of reproduction within the group. If dominant group members do not have complete control over subordinate reproduction, this conflict may favour the evolution of infanticidal behaviour (by either subordinates or dominants or both). Elimination of offspring, however, is likely to be constrained by the difficulty of discriminating between an individual's own progeny and those of cobreeders. Here, we develop an evolutionarily stable strategy (ESS) model of reproductive partitioning, which demonstrates that killing of young can be favoured, even if such discrimination is not possible. The model predicts that infanticide will typically be associated with elevated levels of offspring production, and is most likely to prove evolutionarily stable when the coefficient of relatedness between cobreeders is low, and offspring are cheap to produce. The effect of infanticide is to release subordinates from the reproductive restraint they would otherwise be forced to exercise, leading to reduced reproductive skew. When infanticide is possible, addition of numerous young to the joint brood will not lower overall productivity, because progeny in excess of the most productive brood size are eliminated. Subordinates are thus free to contribute more young to the brood than would otherwise be the case. In addition, we show that the possibility of infanticide may influence the pattern of reproduction within a group even if no offspring are actually killed at equilibrium. Copyright 1999 The Association for the Study of Animal Behaviour.     

Life-history variation in contrasting habitats: flowering decisions in a clonal perennial herb (Veratrum album)

Quantifying intraspecific demographic variation provides a powerful tool for exploring the diversity and evolution of life histories. We investigate how habitat-specific demographic variation and the production of multiple offspring types affect the population dynamics and evolution of delayed reproduction in a clonal perennial herb with monocarpic ramets (white hellebore). In this species, flowering ramets produce both seeds and asexual offspring. Data on ramet demography are used to parameterize integral projection models, which allow the effects of habitat-specific demographic variation and reproductive mode on population dynamics to be quantified. We then use the evolutionarily stable strategy (ESS) approach to predict the flowering strategy-the relationship between flowering probability and size. This approach is extended to allow offspring types to have different demographies and density-dependent responses. Our results demonstrate that the evolutionarily stable flowering strategies differ substantially among habitats and are in excellent agreement with the observed strategies. Reproductive mode, however, has little effect on the ESSs. Using analytical approximations, we show that flowering decisions are predominantly determined by the asymptotic size of individuals rather than variation in survival or size-fecundity relationships. We conclude that habitat is an important aspect of the selective environment and a significant factor in predicting the ESSs.     

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.     

Evolution of cannibalistic traits: scenarios derived from adaptive dynamics

The evolution of cannibalistic traits in consumer populations is studied in this paper with the approach of adaptive dynamics theory. The model is kept at its minimum complexity by eliminating some environmental characteristics, like heterogeneity and seasonalities, and by hiding the size-structure of the population. Evolutionary dynamics are identified through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the canonical equation of adaptive dynamics. The result is a rich catalog of evolutionary scenarios involving evolutionary stable strategies and branching points both in the monomorphic and dimorphic dynamics. The possibility of evolutionary extinction of highly cannibalistic populations is also ascertained. This allows one to explain why cannibalism can be a transient stage of evolution.     

Body condition dependent dispersal in a heterogeneous environment

We find the evolutionarily stable dispersal behaviour of a population that inhabits a heterogeneous environment where patches differ in safety (the probability that a juvenile individual survives until reproduction) and productivity (the total competitive weight of offspring produced by the local individual), assuming that these characteristics do not change over time. The body condition of clonally produced offspring varies within and between families. Offspring compete for patches in a weighted lottery, and dispersal is driven by kin competition. Survival during dispersal may depend on body condition, and competitive ability increases with increasing body condition. The evolutionarily stable strategy predicts that families abandon patches which are too unsafe or do not produce enough successful dispersers. From families that invest in retaining their natal patches, individuals stay in the patch that are less suitable for dispersal whereas the better dispersers disperse. However, this clear within-family pattern is often not reflected in the population-wide body condition distribution of dispersers or non-dispersers. This may be an explanation why empirical data do not show any general relationship between body condition and dispersal. When all individuals are equally good dispersers, then there exist equivalence classes defined by the competitive weight that remains in a patch. An equivalence class consists of infinitely many dispersal strategies that are selectively neutral. This provides an explanation why very diverse patterns found in body condition dependent dispersal data can all be equally evolutionarily stable.     

Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating

We consider mating strategies for females who search for males sequentially during a season of limited length. We show that the best strategy rejects a given male type if encountered before a time‐threshold but accepts him after. For frequency‐independent benefits, we obtain the optimal time‐thresholds explicitly for both discrete and continuous distributions of males, and allow for mistakes being made in assessing the correct male type. When the benefits are indirect (genes for the offspring) and the population is under frequency‐dependent ecological selection, the benefits depend on the mating strategy of other females as well. This case is particularly relevant to speciation models that seek to explore the stability of reproductive isolation by assortative mating under frequency‐dependent ecological selection. We show that the indirect benefits are to be quantified by the reproductive values of couples, and describe how the evolutionarily stable time‐thresholds can be found. We conclude with an example based on the Levene model, in which we analyze the evolutionarily stable assortative mating strategies and the strength of reproductive isolation provided by them.     

Egg size evolution and energetic constraints on population dynamics.

We use population models that are based on dynamic energy budget models for individuals in order to study the evolution of offspring size and its relationship to the evolution of population dynamics. We show the existence of alternative evolutionarily stable strategies for offspring investment strategy resulting from a trade off between offspring number and time-to-maturity. The model predicts egg energy in Daphnia magna well, and suggests that the observed egg energy in D. magna is the result of selection for minimal egg investment constrained by minimum viable egg energy, combined with selection for a juvenile energy reserve. The selection for minimal egg size pushes populations toward chaotic dynamics. However, the minimum viable egg size combined with low efficiency of conversion of energy to new biomass is sufficient to keep population dynamics out of chaos.     

Evolution of handling time can destroy the coexistence of cycling predators

Several consumers (predators) with Holling type II functional response may robustly coexist even if they utilize the same resource (prey), provided that the population exhibits nonequilibrium dynamics and the handling time of predators is sufficiently different. We investigate the evolution of handling time and, in particular, its effect on coexistence. Longer handling time is costly in terms of lost foraging time, but allows more nutrients to be extracted from a captured prey individual. Assuming a hyperbolically saturating relationship between handling time and the number of new predators produced per prey consumed, we obtain three results: (i) There is a globally evolutionarily stable handling time; (ii) At most two predator strategies can coexist in this model; (iii) When two predators coexist, a mutant with intermediate handling time can always invade. This implies that there is no evolutionarily stable coexistence, and the evolution of handling time eventually leads to a single evolutionarily stable predator. These results are proven analytically and are valid for arbitrary (not only small) mutations; they however depend on the relationship between handling time and offspring production and on the assumption that predators differ only in their prey handling strategy.     

Multiple patterns of parental care

Many animals show multiple patterns of parental care, where more than one of the four basic patterns (biparental care, uniparental care by males or females, or no care) is present within a single population during a single breeding season. We consider three reasons for the existence of multiple patterns of parental care: (1) mixed-strategy behaviours; (2) time-dependent behaviour with parents changing their care decision during the breeding season; and (3) quality differences between individuals leading to different care decisions being made depending on the qualities of both parents. The basic framework we use to investigate these is a two-stage game-theoretical model, and we highlight the importance of including feedback between the parental care decisions made by population members and the probability that a deserting individual will find a new mate. Including this feedback may introduce a nonlinear dependence of the fitness payoffs on the frequencies with which the pure strategies ('care' and 'desert') are played by each of the sexes. This can have important consequences for the existence of evolutionarily stable strategies (ESSs). For example, mixed-strategy ESSs may exist (an outcome forbidden if the feedback is not included) and, in one model, the feedback also prevents uniparental care by either sex from being evolutionarily stable. We also point out that decisions made by animals without dependent offspring can have important consequences for observed parental care behaviour. Copyright 1999 The Association for the Study of Animal Behaviour.     

Evolutionary branching of virulence in a single-infection model

This study explores the evolutionary dynamics of pathogen virulence in a single-infection model with density-dependent mortality. Although virulence is not an adaptation of the pathogen per se, it is generally believed to be an inevitable by-product of a pathogen's need to propagate and transmit to new hosts: an increase in virulence will parallel an increase in transmission efficacy. The exact characteristics of the trade-off curve defined by this relationship are important with respect to possible evolutionary scenarios. We conduct a critical function analysis, a method that exposes the evolutionary outcome resulting from trade-offs of arbitrary shape, and find that this simple model can display a wide variety of evolutionary dynamics; comprising multiple stable attractors, evolutionary repellors, and most notably, evolutionary branching points. We identify the conditions under which the different evolutionary outcomes are realised. Our analysis furthermore considers the evolution of coexisting strains, and identifies the trade-off characteristics that will support an evolutionarily stable dimorphic state. We find that an evolutionarily stable dimorphism may exist also in the absence of a branching point in the monomorphic state. The analysis reveals that an evolutionarily stable dimorphism will always be attracting and that no further branching is possible under this model. We discuss our results in relation to the dimension of the environmental feedback inherent in the model, and to results from previous studies and models of evolution of virulence.     

Host–parasite interactions and the evolution of nonrandom mating

Some species mate nonrandomly with respect to alleles underlying immunity. One hypothesis proposes that this is advantageous because nonrandom mating can lead to offspring with superior parasite resistance. We investigate this hypothesis, generalizing previous models in four ways: First, rather than only examining invasibility of modifiers of nonrandom mating, we identify evolutionarily stable strategies. Second, we study coevolution of both haploid and diploid hosts and parasites. Third, we allow for maternal parasite transmission. Fourth, we allow for many alleles at the interaction locus. We find that evolutionarily stable rates of assortative or disassortative mating are usually near zero or one. However, for one case, in which assumptions most closely match the major histocompatibility complex (MHC) system, intermediate rates of disassortative mating can evolve. Across all cases, with haploid hosts, evolution proceeds toward complete disassortative mating, whereas with diploid hosts either assortative or disassortative mating can evolve. Evolution of nonrandom mating is much less affected by the ploidy of parasites. For the MHC case, maternal transmission of parasites, because it creates an advantage to producing offspring that differ from their parents, leads to higher evolutionarily stable rates of disassortative mating. Lastly, with more alleles at the interaction locus, disassortative mating evolves to higher levels.