Evolutionary Stable State Research Articles

Natural Selection on Fecundity Variance in Subdivided Populations: Kin Selection Meets Bet Hedging

In a series of seminal articles in 1974, 1975, and 1977, J. H. Gillespie challenged the notion that the "fittest" individuals are those that produce on average the highest number of offspring. He showed that in small populations, the variance in fecundity can determine fitness as much as mean fecundity. One likely reason why Gillespie's concept of within-generation bet hedging has been largely ignored is the general consensus that natural populations are of large size. As a consequence, essentially no work has investigated the role of the fecundity variance on the evolutionary stable state of life-history strategies. While typically large, natural populations also tend to be subdivided in local demes connected by migration. Here, we integrate Gillespie's measure of selection for within-generation bet hedging into the inclusive fitness and game theoretic measure of selection for structured populations. The resulting framework demonstrates that selection against high variance in offspring number is a potent force in large, but structured populations. More generally, the results highlight that variance in offspring number will directly affect various life-history strategies, especially those involving kin interaction. The selective pressures on three key traits are directly investigated here, namely within-generation bet hedging, helping behaviors, and the evolutionary stable dispersal rate. The evolutionary dynamics of all three traits are markedly affected by variance in offspring number, although to a different extent and under different demographic conditions.

Is Food Worth Fighting for? ESS’s in Mixed Populations of Kleptoparasites and Foragers

We extend the game theoretic model of kleptoparasitism discussed in Broom et al. (2004), by considering a population of foragers consisting of two groups with different behaviours--those who forage and steal from other feeders, and those who only forage. We a sume that those who do not steal have a better foraging rate than those who are also looking out for opportunities to steal. We also allow either type to resist an attack or not resist. We look for Evolutionary Stable States, of either a mixture of the two behaviours, or where the whole population has just one of these behaviours. We find nine such ESS's, dependent on the environmental parameters, although in fact only five of these are distinguishable. In general, we find that if the overall population density is low, food-stealing becomes less viable, and there is an ESS consisting of only foragers. Conversely, when there are many animals looking for, and finding, food, there is an ESS consisting of just kleptoparasites (which are also foraging). In between, an ESS will contain both pure-foragers and stealers. There is some empirical evidence of such behaviours. We find that when there is a mixture of the two types, they must both have the same resistive behaviour. We can thus have some individuals challenging for food but not resisting challenges, and others not challenging and not resisting. This shows how aggressive behaviour may be context-dependent, as seen in practice.

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

The hawk–dove (HD) game, as defined by Maynard Smith [1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge], allows for a polymorphic fitness equilibrium (PFE) to exist between its two pure strategies; this polymorphism is the attractor of the standard replicator dynamics [Taylor, P.D., Jonker, L., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156; Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge] operating on an infinite population of pure-strategists. Here, we consider stochastic replicator dynamics, operating on a finite population of pure-strategists playing games similar to HD; in particular, we examine the transient behavior of the system, before it enters an absorbing state due to sampling error. Though stochastic replication prevents the population from fixing onto the PFE, selection always favors the under-represented strategy. Thus, we may naively expect that the mean population state (of the pre-absorption transient) will correspond to the PFE. The empirical results of Fogel et al. [1997. On the instability of evolutionary stable states. BioSystems 44, 135–152] show that the mean population state, in fact, deviates from the PFE with statistical significance. We provide theoretical results that explain their observations. We show that such deviation away from the PFE occurs when the selection pressures that surround the fitness-equilibrium point are asymmetric. Further, we analyze a Markov model to prove that a finite population will generate a distribution over population states that equilibrates selection-pressure asymmetry; the mean of this distribution is generally not the fitness-equilibrium state.

The evolution of anisogamy: The adaptive significance of damage, repair and mortality

Classic theory on the evolution of anisogamy focuses on the trade-off between gamete productivity and provisioning and mechanisms associated with post-zygotic survival. In this article, the role of mortality acting on both zygotes and gametes is explored as a factor influencing the evolution of different sized gametes. In particular, variable mortality through differential survival or metabolic damage is shown to affect the persistence of isogamy, the evolution of more than two sexes and the evolution of anisogamy. Evolutionary stable isogamous states are shown to be locally unstable and disruptive selection can induce the evolution of anisogamy. Analysis of both the isogamous and anisogamous ESS points reveals that the persistence of either of these conditions is not always assured. The implications of variable survival on the evolution of anisogamy are discussed.

Polyandry, Life-History Trade-Offs and the Evolution of Imprinting at Mendelian Loci

Genomic imprinting causes parental origin-dependent differential expression of a small number of genes in mammalian and angiosperm plant embryos, resulting in non-Mendelian inheritance of phenotypic traits. The "conflict" theory of the evolution of imprinting proposes that reduced genetic relatedness of paternally, relative to maternally, derived alleles in offspring of polygamous females supports parental sex-specific selection at gene loci that influence maternal investment. While the theory's physiological predictions are well supported by observation, the requirement of polyandry in the evolution of imprinting from an ancestral Mendelian state has not been comprehensively analyzed. Here, we use diallelic models to examine the influence of various degrees of polyandry on the evolution of both Mendelian and imprinted autosomal gene loci that influence trade-offs between maternal fecundity and offspring viability. We show that, given a plausible assumption on the physiological relationship between maternal fecundity and offspring viability, low levels of polyandry are sufficient to reinforce exclusively the fixation of "greedy" paternally imprinted alleles that increase offspring viability at the expense of maternal fecundity and "thrifty" maternally imprinted alleles of opposite effect. We also show that, for all levels of polyandry, Mendelian alleles at genetic loci that influence the trade-off between maternal fecundity and offspring viability reach an evolutionary stable state, whereas pairs of reciprocally imprinted alleles do not.

Coincidence of ESAD and ESS in dominant–recessive hereditary systems

The paper deals with the following question: when do the phenotypic evolutionarily stable state (ESS) and the evolutionarily stable allele distribution (ESAD) coincide? It is supposed that for a sexual population, in dominant–recessive inheritance system, n allele at one autosomal locus determine n possible pure individual phenotypes and each pure phenotype is obtained as the phenotype of a homozygote. Under these conditions, earlier results of the authors imply that, if a phenotype distribution is an ESS then the allele distribution generating it is an ESAD. In this paper, apart from a certain degenerate pay-off matrices, the inverse statement is also proved: if a distribution is an ESAD then the corresponding phenotypic distribution is an ESS.

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 3 $\times$ 3 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 3 $\times$ 3 payoff matrix.

A stochastic model of selection on selfing rates in structured populations

SummaryPrevious theoretical studies of the evolution of the selfing rate have shown that mixed mating systems are not evolutionary stable states. Such models have, however, not included the effects of population structure and thus biparental inbreeding together with the evolution of selfing rates and inbreeding depression. In order to examine selection on selfing rates in structured populations, a stochastic model simulating a finite population with partial selfing and restricted pollen and seed dispersal has been developed. Selection on the mating system was followed by introducing modifiers affecting the selfing rate. The major result was that, with density dependent recruitment, a process which maintains the population structure necessary for biparental inbreeding to occur, a mixed mating system could be maintained. This result was associated with an increase of the mutation load with high selfing rates, and the selected selfing rate depended on the degree of population structure rather than on the initial selfing rate. With low dominance of deleterious alleles, complete allogamy can be selected for. Further studies showed that the more general condition of spatial heterogeneity of recruitment can lead to similar results, the most important condition being the maintenance of genetic structure within populations. A brief survey of the empirical literature shows that a positive relationship between the magnitude of inbreeding depression and the inbreeding coefficient within populations has been observed, in support of the present model.

Phenotypic variation, sexual reproduction and evolutionary population dynamics

AbstractI studied the effects of introducing phenotypic variation into a well‐known single species model for a population with discrete, non‐overlapping generations. The phenotypes differed in their dynamic behaviour. The analysis was made under the assumption that the population was in an evolutionary stable state. Differences in the timing of the competitive impacts of the phenotypes on each other had a strong simplifying effect on the dynamics. This result could also be applied to competition between species. The effect of sexual reproduction on the dynamics of the population was analysed by assuming the simplest genetic model of one locus with two alleles. Sexual reproduction made the system much more stable in the (mathematical) sense that the number of attractors was reduced and their basins of attraction enlarged. In a dominant system sex tended to increase the frequency of the recessive allele, and in an overdominant system it induced gene frequencies of 1/2. Whether the attractors in the dominant system tended to be simpler or more complex than the attractors in the asexual system depended on the phenotype of the recessive homozygote. The overdominant sexual system tended to have simpler dynamics than the corresponding asexual population. A 2‐locus model was used to study whether sexuals can invade an asexual population and vice versa. One locus coded for sexual and asexual reproduction, while the other coded for the dynamics. Enhanced stability through sexual reproduction seemed to be the reason why there was a clear asymmetry favouring sex in this evolutionary context.

Parental care as a game

AbstractA new game model of parental care is presented where fitnesses of males and females depend on frequencies of care and no‐care strategies in the population, as well as strategies of individuals and those of their mates. Evolutionarily stable states (ESS) of biparental‐care (BC), female‐care (FC), male‐care (MC), and no‐care (NC) are represented as regions in a 3‐dimensional parameter space, comprising the ratio of the number of offspring raised by FC or MC to that by BC, the ratio of that by NC to that by FC or MC, and the sex ratio. ESSs are not necessarily uniquely determined: ESS regions of FC and MC overlap in parts, as well as those of BC and NC. In a region where none of the four states is ever an ESS, a polymorphism (a mixed state of care and no‐care strategies in one or both sexes) is stable. Although an asymmetry between prezygotic investment times of males and females results in the difference in sizes of the ESS regions of FC and MC, it is not responsible for determining whether FC or MC evolves from other modes of parental care as a result of slowly changing parameters. The key factor is the sex ratio of adults in the population. By a slightly modified model, it is also shown that paternity uncertainty at fertilization of eggs is highly important for evolution of FC.

Sex Ratio in the Social Hymenoptera: A Population-Genetics Study of Long-Term Evolution

We analyze a two-locus population-genetics model for the queen-workers conflict over the sex ratio in social Hymenoptera. We assume that the total number of reproductives and the proportion of females among them are both functions of a queen's trait α and of a workers' trait β, which are controlled by one locus each. We determine the conditions for a pair {α,β} to be an evolutionarily stable state (ESS), that is, to be such that no combination of mutations at the two loci is able to invade a population fixed for the pair of genes determining {α,β}. Our results, in some respects, validate the alternative game-theoretic approach, based on maximization of a given pair of queen's and workers' inclusive fitnesses, while showing the limitations of this approach in some other respects. If all mutations are assumed to have a small effect and if linkage between the two loci is not too tight, we show that, indeed, for any ESS {α,β}, α and β maximize the queen's and the workers' payment functions, H and G, respectively, but the ESS predicted by H and G does not always coincide with the ESS predicted by the queen's and workers' inclusive fitness functions. Very strong linkage, however, may destabilize some of the maxima of H and G but preserve as ESSs those among them that maximize a joint, queen-workers payment function, T. We show that ESSs can be of two kinds: (1) isolated points, in which the proportion of females is intermediate between one-half and three-fourths and the colony productivity is less than maximal, because of conflict between queen and workers; (2) a continuum of points, with a corresponding spread of female proportions, usually within [1/2, 3/4] but also, rarely, beyond this interval. At these points there is full agreement between the two parties, and the colony productivity is accordingly maximal.

On Conditions for Evolutionary Stability for a Continuously Varying Character

The two conditions for stability of an evolutionary equilibrium, the m-stability and the δ-stability conditions, are discussed. The m-stability condition is a condition for the convergence of the population toward the equilibrium, and the δ-stability condition coresponds to a local version of the classic evolutionarily stable strategy (ESS) condition. Together the two conditions provide the condition for a continuous stable strategy. The convergence stability condition corresponds to the requirement for convergence due to initial increase of rare alleles in a monomorphic population, and the local ESS stability condition corresponds to the stability of a monomorphic population at the evolutionary equilibrium against the increase of rare alleles. In this way, an evolutionary equilibrium that is convergence stable, but not local ESS stable, will tend to become polymorphic. The local ESS stability condition therefore contributes more to a description of the dynamics of variation at an evolutionary equilibrium than to the description of the stability of the evolutionary equilibrium. However, the characterization of a polymorphic evolutionary equilibrium cannot be reached by studying the initial increase of rare variant alleles, just as this method cannot describe all aspects of the convergence stability either. Combining these analyses provides a powerful tool in the initial exploration of evolutionary equilibriums of complicated systems, and convergence stable and local ESS unstable equilibriums point toward very interesting polymorphic evolutionary stable states. The analysis is illustrated on a model for intraspecific exploitative competition that may show monomorphic and polymorphic evolutionarily stable equilibriums.

Evolution of alternative reproductive strategies: frequency-dependent sexual selection in male bluegill sunfish

This study provides empirical evidence in a wild population for frequency-dependent sexual selection between alternative male reproductive strategies. The bluegill sunfish ( Lepomis macrochirus ) has two male reproductive strategies, cuckolder or parental, used by different males to compete in fertilizing the same eggs. As the density of cuckolders in colonies of parental males increases, the average mating success of cuckolders initially peaks but then declines. The cuckolder density at which their success peaks is determined by ecological characteristics of each colony. A theoretical analysis assuming random and omniscient cuckolder distributions among ecologically different colonies shows that cuckolders will fertilize decreasing proportions of eggs, relative to parental males, as cuckolders increase in frequency in the population. This supports evolutionary models that assume negative frequency-dependent selection between the competing strategies. Cuckolder and parental strategies may therefore have evolved as an Evolutionarily Stable State (ESST).

Age structure, environmental fluctuations, and hermaphroditic sex allocation

This paper studies sex allocation in an age-structured population of hermaphrodites living in a temporally fluctuating environment. The general condition for the evolutionary stable state (ESS) of allocation is derived for density-independent dynamics. This condition is used to determine the effect on the deterministic ESS of a dependence of survival rates on allocation. It is also used to identify the special conditions under which a stochastic ESS is given by a product rule and show how demographic structure and the correlation structure of vital rates determines the stochastic ESS.

The evolution of habitat preference II. Evolutionary genetic stability under soft selection

A previous analysis of a single-locus, two-allele model of the evolution of habitat selection revealed that preference evolves toward equalization of parental investment in offspring placed in two different habitats in which population size is regulated independently (soft selection). That analysis is extended here to the case of a single locus with multiple alleles. The major result of this analysis is that regardless of how many alleles are present at equilibrium, new alleles will invade if and only if they tend to make parental investment in the two habitats more equal. In other words, the population evolves toward an Evolutionary Stable State (ESS) characterized by equalization of parental investment. The model may explain the maintenance of genetic variation for host (habitat) preference exhibited by many insect species.

Ant Reproductive Strategies and Sex Allocation Theory

Trivers and Hare (1976) predicted that the degree of genetic relatedness between hymenopteran workers and their brood is a fundamental factor in influencing the amount of resource allocated to the sexes. An analysis of a wide body of literature on ants (Family: Formicidae) strongly supports this genetic relatedness hypothesis (GRH). Their predictions on the differences in investment between monogynous, polygynous, slavemaking, and parasitic colonies are supported qualitatively and quantitatively both across and within species. Another explanation of female bias, local mate competition (LMC) (e.g., Alexander and Sherman, 1977), is found to be inconsistent with the data. The ratio of investment in species with non-conspecific brood care, most colonies producing either extremely male or female biased broods, as well as the suggestion that male ants may be highly limited in the number of females they can fertilize, are incompatible with LMC predictions. A limitation in the number of matings on the part of an individual male may mean that mating strategies in ant sexes could be quite different from those in most other species. The bimodal sexual investment pattern appears to be a mixed "evolutionary stable strategy and state" and seems proximately determined by the amount of food resources. Abundance favors females, scarcity favors males. Further utilization of game theory approaches may be quite useful in explaining both individual colony and population-level sexual investment values.

Genetic and strategic models for the evolution of mating systems

Male and female fitnesses in the Shaw-Mohler equation are partitioned into components which putatively determine mating systems. The resultant genetic models provide criteria for evolutionary stable population states and yield strategic models based on maximization principles and fitness sets.