Evolutionary dynamics of fearfulness and boldness: a stochastic simulation model

A stochastic simulation model is investigated for the evolution of anti-predator behavior in birds. The main goal is to reveal the effects of population size, predation threats, and energy lost per escape on the evolutionary dynamics of fearfulness and boldness. Two pure strategies, fearfulness and boldness, are assumed to have different responses for the predator attacks and nonlethal disturbance. On the other hand, the co-existence mechanism of fearfulness and boldness is also considered. For the effects of total population size, predation threats, and energy lost per escape, our main results show that: (i) the fearful (bold) individuals will be favored in a small (large) population, i.e. in a small (large) population, the fearfulness (boldness) can be considered to be an ESS; (ii) in a population with moderate size, fearfulness would be favored under moderate predator attacks; and (iii) although the total population size is the most important factor for the evolutionary dynamics of both fearful and bold individuals, the small energy lost per escape enables the fearful individuals to have the ability to win the advantage even in a relatively large population. Finally, we show also that the co-existence of fearful and bold individuals is possible when the competitive interactions between individuals are introduced.     

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.     

The evolutionary dynamics of direct phenotypic overdominance: emergence possible, loss probable

An evolutionary dynamical system with explicit diploid genetics is used to investigate the likelihood of observing phenotypically overdominant heterozygotes versus heterozygous phenotypes that are intermediate between the homozygotes. In this model, body size evolves in a population with discrete demographic episodes and with competition limiting reproduction. A genotype-phenotype map for body size is used that can generate the two qualitative types of dominance interactions (overdominance versus intermediate dominance). It is written as a single-locus model with one focal locus and parameters summarizing the effects of alleles at other loci. Two types of evolutionarily stable strategy (ESS; continuously stable strategy, CSS) occur. The ESS is generated either (1) by the population ecology; or (2) by a local maximum of the genotype-phenotype map. Overdominant heterozygotes are expected to arise if the population evolves toward the second type of ESS, where nearly maximum body sizes are found. When other loci with partially dominant inheritance also evolve, the location of the maximum in the genotype-phenotype map repeatedly changes. It is unlikely that an evolving population will track these changes; ESSs of the second type now are at best quasi-stationary states of the evolutionary dynamics. Considering the restrictions on its probability, a pattern of phenotypic overdominance is expected to be rare.     

An Evolutionarily Stable Strategy (ESS) model of postcopulatory guarding in insects

A simple Evolutionarily Stable Strategy (ESS) model for promiscuous insect species is analyzed to obtain the optimal strategy for the duration of male guarding behavior after copulation with a female. Such guarding behavior prevents other males from copulating with that female. Predictions of the model are (i) that the ESS is either a non-guarding strategy, a perfect guarding strategy until oviposition, or a polymorphic equilibrium between the two types, and (ii) that the perfect guarding strategy has more advantages than the non-guarding strategy when (a) the ratio of males to females is large, (b) the searching efficiency is high, (c) the population density is high, and (d) the preoviposition period is short. Male guarding behavior in several species seems to agree with the predictions of the model.     

Size‐dependent resource allocation and sex allocation in herbaceous perennial plants

Individuals within a population often differ considerably in size or resource status as a result of environmental variation. In these circumstances natural selection would favour organisms not with a single, genetically determined allocation, but with a genetically determined allocation rule specifying allocation in relation to size or environment. Based on a graphical analysis of a simple evolutionarily stable strategy (ESS) model for herbaceous perennial plants, we aim to determine how cosexual plants within a population should simultaneously adjust their reproductive allocation and sex allocation to their size. We find that if female fitness gain is a linear function of resource investment, then a fixed amount of resources should be allocated to male function, and to post‐breeding survival as well, for individuals above a certain size threshold. The ESS resource allocation to male function, female function, and post‐breeding survival positively correlate if both male and female fitness gains are a saturating function of resource investment. Plants smaller than the size threshold are expected to be either nonreproductive or functionally male only.     

Arms races in evolution—An ESS to the opponent-independent costs game

The opponent-independent cost game is an animal contest in which the bigger and more heavily-armed opponent wins a disputed resource without significant fighting costs. A strategy is a choice of investment level in armament. Increasing armament is assumed to have fitness costs that are unrelated to contests; i.e. the cost of an individual's investment in arms is independent of the strategy played by an opponent.Previous work with this model showed that no ESS exists if a strategy prescribes an arms level exactly. This is equivalent to the notion that there is no environmental variation in the arms level attained by a given strategy. If environmental variation is introduced, a pure ESS can generally exist. A strategy is assumed to prescribe an exact investment cost, but this is translated into a probability distribution of arms levels attained, rather than an exact arms level. Increasing investment increases the mean of the arms level distribution. The ESS investment level depends both on how environmental factors distribute arms levels, and on the shape of the cost function (i.e. on the way that costs increase with investment); in some instances there is no ESS. Two types of model are investigated; in one fitness is additive (benefit-cost), in the other it is multiplicative (benefit × survivorship). The multiplicative model is likely to apply to the case where contests are between males for access to females. Here the ESS investment level (an ESS degree of risk that a male sustains as a result of armament) increases as fewer individuals guard the available resources. Thus sexual size dimorphism (male/female size) and relative male armament should increase as harem size increases. The ESS investment level will also be highest if most individuals are small and poorly armed, as would often be the case where size increases throughout life.The model can be applied to coevolutionary arms races between two classes of opponent, such as prey and predator, or parent and offspring. Here the ESS is likely to be a pair of ESS arms levels, one for each class of opponent.     

The evolution of intermittent breeding

A central issue in life history theory is how organisms trade off current and future reproduction. A variety of organisms exhibit intermittent breeding, meaning sexually mature adults will skip breeding opportunities between reproduction attempts. It’s thought that intermittent breeding occurs when reproduction incurs an extra cost in terms of survival, energy, or recovery time. We have developed a matrix population model for intermittent breeding, and use adaptive dynamics to determine under what conditions individuals should breed at every opportunity, and under what conditions they should skip some breeding opportunities (and if so, how many). We also examine the effect of environmental stochasticity on breeding behavior. We find that the evolutionarily stable strategy (ESS) for breeding behavior depends on an individual’s expected growth and mortality, and that the conditions for skipped breeding depend on the type of reproductive cost incurred (survival, energy, recovery time). In constant environments there is always a pure ESS, however environmental stochasticity and deterministic population fluctuations can both select for a mixed ESS. Finally, we compare our model results to patterns of intermittent breeding in species from a range of taxonomic groups.     

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.     

Ecological stability,evolutionary stability and the ESS maximum principle

Summary Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals in the community, the essential element of a game exists. This game is an evolutionary game where the individual organisms (players) inherit their strategies from continuous play of the game through time. Here, the strategies are assumed to be constants associated with certain adaptive parameters (such as sunlight conversion efficiency for plants or body length in animals) in a set of differential equations which describe the population dynamics of the community. By means of natural selection, these parameters will evolve to a set of strategy values that natural selection, by itself, can no longer modify, i.e. an evolutionarily stable strategy (ESS). For a given class of models, it is possible to predict the outcome of this evolutionary process by determining ESSs using an ESS maximum principle. However, heretofore, the proof of this principle has been based on a limited set of conditions. Herein, we generalize the proof by removing certain restrictions and use instead the concept of an ecological stable equilibrium (ESE). Individuals in a biological community will be at an ESE if fixing the strategies used by the individuals results in stable population densities subject to perturbations in those densities. We present both necessary and sufficient conditions for an ESE to exist and then use the ESE concept to provide a very simple proof of the ESS maximum principle (which is a necessary condition for an ESS). A simple example is used to illustrate the difference between a strategy that maximizes fitness and one that satisfies the ESS maximum principle. In general they are different. We also look for ESEs in Lotka—Volterra competition and use the maximum principle to determine when an ESE will be an ESS. Finally, we examine the applicability of these ideas to matrix games.     

OPTIMAL REPRODUCTIVE EFFORT IN STOCHASTIC,DENSITY-DEPENDENT ENVIRONMENTS

The amount of effort organisms should put into reproducing at any given time has been a matter of debate for many years. Early models suggested a simple rule of thumb: iteroparity should be favored when juvenile survival is relatively variable and semelparity when adult survival is relatively variable. When more mathematically complex models were developed, these simple conclusions were found to be special cases. Variability can select toward iteroparity or semelparity depending on a number of factors irrespective of relative adult/juvenile survival (e.g, the density-independent models of Orzack and Tuljapurkar). Using new techniques, we estimate the ESS reproductive effort for stage-structured models in density-dependent and stochastic conditions. We find that variability causes significant changes in reproductive effort, these changes are often small (± 10% of determinstic ESS effort, but up to 50% change in some instances), and the amount that effort increases or decreases depends on many factors (e.g., the deterministic population dynamics, the vital rates affected by density, the amount of variation, the correlations between the vital rates, the distribution from which the variation is drawn, and the deterministic ESS effort). In a variable environment, semelparity is the ESS in only 3.5% of cases; iteroparity is the rule.     

Evolution of unconditional dispersal in periodic environments

Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against 'unconditional' dispersal in spatially heterogeneous environments. 'Unconditional' means individuals disperse at a rate independent of their location. We prove that if within-patch fitness varies spatially and between two values temporally, then there is selection for unconditional dispersal: any evolutionarily stable strategy (ESS) or evolutionarily stable coalition (ESC) includes a dispersive phenotype. Moreover, at this ESS or ESC, there is at least one sink patch (i.e. geometric mean of fitness less than one) and no sources patches (i.e. geometric mean of fitness greater than one). These results coupled with simulations suggest that spatial-temporal heterogeneity is due to abiotic forcing result in either an ESS with a dispersive phenotype or an ESC with sedentary and dispersive phenotypes. In contrast, the spatial-temporal heterogeneity due to biotic interactions can select for higher dispersal rates that ultimately spatially synchronize population dynamics.     

The ESS under spatial variation with applications to sex allocation

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.     

The Comparative Biology of Genetic Variation for Conditional Sex Ratio Behavior in a Parasitic Wasp, Nasonia Vitripennis

Using genetic markers, we tracked the sex ratio behavior of individual females of the parasitic wasp, Nasonia vitripennis, in foundress groups of size 1, 2, 4, 8 and 16. Comparison of 12 isofemale strains extracted from a natural population reveals significant between-strain heterogeneity of sex ratios produced in all sizes of foundress group. Under simple assumptions about population structure, this heterogeneity results in heterogeneity of fitnesses. The strains differ in their conditional sex ratio behavior (the sex ratio response of a female to foundress groups of different sizes). Females of some strains produce more males as foundress group size increases (up to size eight). Females of another strain produce more males when not alone but do not respond differentially to group size otherwise. Females of two other strains show no conditional sex ratio behavior. Females of only two strains behave differently in foundress groups of size 8 and 16. Correlation and regression analyses indicate that the strains differ significantly in their fit to the predictions of an evolutionarily stable strategy (ESS) model of conditional sex ratio behavior. Such heterogeneity contradicts the notion that females of this species possess conditional sex ratio behavior that is optimal in the ESS sense. The results imply that this ESS model is useful but not sufficient for understanding the causal basis of the evolution of this behavior in this species. This is the first report on the sex ratio behavior of individual females in multiple foundress groups in any species of parasitic wasp. Data of this type (and not foundress group or ``patch'''' sex ratios) are essential for testing evolutionary models that predict the sex ratio behaviors of individuals. We suggest that a test for an ESS model include the answers to two important questions: 1) is the model quantitatively accurate? and 2) is there reasonable evidence to indicate that natural selection has caused individuals to manifest the ESS behavior?     

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.     

On fitness in structured metapopulations

In this paper we derive a general expression measuring fitness in general structured metapopulation models. We apply the theory to a model structured by local population size and in which local dynamics is explicitly modelled. In particular, we calculate the evolutionarily stable dispersal strategy for individuals that can assess the local population density in the case where only dispersal is subject to evolutionary control but all other model ingredients are assumed fixed. We show that there exists a threshold size such that at ESS everyone should stay as long as the population size is below the threshold and everyone should disperse immediately as the population size reaches the threshold. Received: 13 August 1999 / Revised version: 2 May 2001 / Published online: 12 October 2001     

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.     

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.     

The logic of asymmetric contests

A theoretical analysis is made of the evolution of behavioural strategies in contest situations. It is assumed that behaviour will evolve so as to maximize individual fitness. If so, a population will evolve an ‘evolutionarily stable strategy’, or ESS, which can be defined as a strategy such that, if all members of a population adopt it, no ‘mutant’ strategy can do better. A number of simple models of contest situations are analysed from this point of view. It is concluded that in ‘symmetric’ contests the ESS is likely to be a ‘mixed’ strategy; that is, either the population will be genetically polymorphic or individuals will be behaviourally variable. Most real contests are probably asymmetric, either in pay-off to the contestants, or in size or weapons, or in some ‘uncorrelated’ fashion; i.e. in a fashion which does not substantially bias either the pay-offs or the likely outcome of an escalated contest. An example of an uncorrelated asymmetry is that between the ‘discoverer’ of a resource and a ‘late-comer’. It is shown that the ESS in asymmetric contests will usually be to permit the asymmetric cue to settle the contest without escalation. Escalated contests will, however, occur if information to the contestants about the asymmetry is imperfect.     

Adaptive evolution of cooperation through Darwinian dynamics in Public Goods games

The linear or threshold Public Goods game (PGG) is extensively accepted as a paradigmatic model to approach the evolution of cooperation in social dilemmas. Here we explore the significant effect of nonlinearity of the structures of public goods on the evolution of cooperation within the well-mixed population by adopting Darwinian dynamics, which simultaneously consider the evolution of populations and strategies on a continuous adaptive landscape, and extend the concept of evolutionarily stable strategy (ESS) as a coalition of strategies that is both convergent-stable and resistant to invasion. Results show (i) that in the linear PGG contributing nothing is an ESS, which contradicts experimental data, (ii) that in the threshold PGG contributing the threshold value is a fragile ESS, which cannot resist the invasion of contributing nothing, and (iii) that there exists a robust ESS of contributing more than half in the sigmoid PGG if the return rate is relatively high. This work reveals the significant effect of the nonlinearity of the structures of public goods on the evolution of cooperation, and suggests that, compared with the linear or threshold PGG, the sigmoid PGG might be a more proper model for the evolution of cooperation within the well-mixed population.