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 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.     

Single species evolutionary dynamics

Not long after the introduction of evolutionary stable strategy (ESS) concept, it was noticed that dynamic selection did not always lead to the establishment of the ESS. The concept of continuously stable strategy (CSS) was thereafter developed. It was generally accepted that dynamic selection leads to the establishment of an ESS if it is a CSS. Examination of an evolutionary stability concept which is called neighborhood invader strategy (NIS) shows that it may be impossible for an ESS to be established through dynamic selection even if it is a CSS and no polymorphisms occur. We will examine the NIS concept and its implications for two evolutionary game models: root-shoot allocation in plant competition and Lotka–Volterra competition. In the root-shoot model we show that an ESS will be attained through dynamic selection if it is a NIS. Similarly for the Lotka–Volterra model, we show that an ESS will be attained through dynamic selection even if protected dimorphisms occur during the evolutionary process if it is an NIS.     

Alternative Predictions for Optimal Dispersal in Response to Local Catastrophic Mortality

What dispersal strategy should be employed by an organism in response to local catastrophic mortality? Here we contrast predictions from an analytical solution derived from an ESS model which optimizes competitive ability (Comins et al., 1980) with those from a stochastic, branching process model (Karlson and Taylor, 1992) which maximizes survivorship of a clonal lineage. The optimal dispersal fraction varies directly with the probability of local extinction in the ESS model, yet varies inversely with this probability over much of the parameter space in the latter model. In order to conform more closely with the assumptions of the ESS model, we have modified the branching process model to have a random, Poisson-distributed number of offspring and compared the predictions of these models. Both models invoke dispersal as an escape from local extinction and predict mixed dispersal strategies over a wide range of conditions. However, increasing local catastrophic mortality favors more dispersal in the ESS model, but it can be so severe in the branching process model that no dispersal strategy is adaptive. In this model, the predicted optimal proportion of dispersed offspring is highest at low to intermediate levels of catastrophic mortality depending on the total number of offspring produced. We suggest that this observed discrepancy is sufficiently large to warrant empirical tests of these qualitatively different predictions.     

Frequency-dependent selection in sexual family-structured populations

In this paper, a two-phenotype, single-locus, n -allele matrix game diploid model incorporating interactions between full sibs influencing personal fitness is investigated. Necessary and sufficient conditions for an ESS are given. We show that if a strategy is an ESS for this model with the payoff matrix A, then it must be an ESS for the standard game formulation with payoff matrix A+(r/2) A(T) where r is the probability to interact with a sib, but it is also possible that no ESS exists. Moreover, under the assumption of weak selection, the partial change in phenotype frequencies brings the population closer to an ESS when it exists.     

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.     

Robustness of circadian rhythms in the presence of molecular fluctuations: an investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.     

Evolution of virulence when transmission occurs before disease

Most models of virulence evolution assume that transmission and virulence are constant during an infection. In many viral (HIV and influenza), bacterial (TB) and prion (BSE and CWD) systems, disease-induced mortality occurs long after the host becomes infectious. Therefore, we constructed a model with two infected classes that differ in transmission rate and virulence in order to understand how the evolutionarily stable strategy (ESS) depends on the relative difference in transmission and virulence between classes, on the transition rate between classes and on the recovery rate from the second class. We find that ESS virulence decreases when expressed early in the infection or when transmission occurs late in an infection. When virulence occurred relatively equally in each class and there was disease recovery, ESS virulence increased with increased transition rate. In contrast, ESS virulence first increased and then decreased with transition rate when there was little virulence early in the infection and a rapid recovery rate. This model predicts that ESS virulence is highly dependent on the timing of transmission and pathology after infection; thus, pathogen evolution may either increase or decrease virulence after emergence in a new host.     

Effects of territory competition and climate change on timing of arrival to breeding grounds: a game-theory approach

Phenology is an important part of life history that is gaining increased attention because of recent climate change. We use game theory to model phenological adaptation in migratory birds that compete for territories at their breeding grounds. We investigate how the evolutionarily stable strategy (ESS) for the timing of arrival is affected by changes in the onset of spring, the timing of the resource peak, and the season length. We compare the ESS mean arrival date with the environmental optimum, that is, the mean arrival date that maximizes fitness in the absence of competition. When competition is strong, the ESS mean arrival date responds less than the environmental optimum to shifts in the resource peak but more to changes in the onset of spring. Increased season length may not necessarily affect the environmental optimum but can still advance the ESS mean arrival date. Conversely, shifting a narrow resource distribution may change the environmental optimum without affecting the ESS mean arrival date. The ESS mean arrival date and the environmental optimum may even shift in different directions. Hence, treating phenology as an evolutionary game rather than an optimization problem fundamentally changes what we predict to be an adaptive response to environmental changes.     

Evolutionarily stable strategies in differential and difference equation models

Summary A definition for an evolutionarily stable strategy (ESS) is given which is applicable to a general differential equation population model and two difference equation analogs. With the introduction of a fitnessgenerating function, it is possible to develop necessary conditions for the determination of an ESS for each of these systems. In most situations, an ESS for one system will also be an ESS for the other. Necessary conditions for an ESS are obtained. Under certain restrictions, they are shown to be valid, even under an unstable equilibrium in population density. the results are illustrated with an example which has the same ESS solution whether a continuous or discrete model is used. The behavior of the ESS for the discrete model is then examined under unstable equilibrium conditions in population density.     

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.     

COEVOLUTION AS AN EVOLUTIONARY GAME

Coevolution is modeled as a continuous game where the fitness-maximizing strategy of an individual is assumed to be a function of the strategy of other individuals who are also under selection to maximize fitness. An evolutionary stable strategy (ESS) is sought such that no rare alternative strategies can invade the community. The approach can be used to model coevolution because the ESS may be composed of a coalition of more than one strategy. This work, by modeling frequency-dependent selection, extends the approach of Roughgarden (1976) which only considered density-dependent selection. In particular, we show that the coevolutionary model of Rummel and Roughgarden (1985) does contain frequency-dependent selection, and thus, their application of Roughgarden's criterion for evolutionary stability to a model for which it is not applicable leads to the erroneous conclusion that the ecological and evolutionary processes are in conflict. The utility of the game theoretic approach is illustrated by two examples. The first considers an ESS composed of a single strategy, the second an ESS composed of a coalition of two strategies. Evolution occurs on a frequency-dependent adaptive landscape. For this reason, the approach is appropriate for modeling competitive speciation (Rosenzweig, 1978). Also, the game theoretic approach is designed to combine the interplay between the background environment (including the biotic components) and the evolutionary potential of the populations or organisms. The actual application of this theory will require knowledge of both.     

On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

Evolution of condition-dependent dispersal: A genetic-algorithm search for the ESS reaction norm

Many insects produce two types (winged and wingless) of offspring that greatly differ in dispersal ability. The ratio of the two types often depends on the quality of the local habitat and the crowding experienced by the mother. Here we studied the condition-dependent dispersal that is evolutionarily stable. The model is also applicable to annual plants that produce two types of seeds differing in dispersal rates. The model assumptions are: the population is composed of a number of sites each occupied by a single adult. The total number of offspring produced by a mother depends on the environmental quality of the site that varies over the years and between sites. The ESS fraction of dispersing type as a function of the quality of the habitat (or ESS reaction norm) states that dispersers should not be produced if habitat qualitym is smaller than a critical valuek. Ifm is larger thank, the number of dispersers should increase withm and that of nondispersers should be kept constant. Second, we developed an alternative way of searching for the ESS: the reaction norm is represented as a three-layered neural network, and the parameters (weights and biases) are chosen by genetic algorithm (GA). This method can be extended easily to the cases of multiple environmental factors. There was an optimal (relatively wide) range of mutation rates for weights and biases, outside of which the convergence of the network to the valid ESS was likely to fail. Recombination, or crossing-over, was not effective in improving the success rate. The learned network often shows several characteristic ways of deviation from the ESS. We also examined the case in which the quality of different sites was correlated. In this case the ESS fraction of dispersers increases both with the quality of the site and with the average quality of the whole population in that year.     

Time-dependent animal conflicts: 1. The symmetric case

Animal conflicts are often characterized by time-dependent strategy sets. This paper considers the following type of animal conflicts: a member of a group is at risk and needs the assistance of another member to be saved. As long as assistance is not provided, the individual which is at risk has a positive, time-dependent rate of dying. Each of the other group members is a potential helper. Assisting this individual accrues a cost, but losing him decreases the inclusive fitness of each group member. A potential helper's interval between the moment an individual finds itself at risk and the moment it assists is a random variable, hence its strategy is to choose the probability distribution for this random variable. Assuming that each of the potential helpers knows the others' strategies, we show that the ability to observe their realizations influences the evolutionarily stable strategies (ESS) of the game. According to our results, where the realizations can be observed ESS always exist: immediate assistance, no assistance and delayed assistance. Where the realizations cannot be observed ESS do not always exist, immediate assistance and no assistance are possible ESS, while delayed assistance cannot be an ESS. We apply our model to the n brothers' problem and to the parental investment conflict.     

The Exon Splicing Silencer in Human Immunodeficiency Virus Type 1 Tat Exon 3 Is Bipartite and Acts Early in Spliceosome Assembly

Inefficient splicing of human immunodeficiency virus type 1 (HIV-1) RNA is necessary to preserve unspliced and singly spliced viral RNAs for transport to the cytoplasm by the Rev-dependent pathway. Signals within the HIV-1 genome that control the rate of splicing include weak 3′ splice sites, exon splicing enhancers (ESE), and exon splicing silencers (ESS). We have previously shown that an ESS present within tat exon 2 (ESS2) and a suboptimal 3′ splice site together act to inhibit splicing at the 3′ splice site flanking tat exon 2. This occurs at an early step in spliceosome assembly. Splicing at the 3′ splice site flanking tat exon 3 is regulated by a bipartite element composed of an ESE and an ESS (ESS3). Here we show that ESS3 is composed of two smaller elements (AGAUCC and UUAG) that can inhibit splicing independently. We also show that ESS3 is more active in the context of a heterologous suboptimal splice site than of an optimal 3′ splice site. ESS3 inhibits splicing by blocking the formation of a functional spliceosome at an early step, since A complexes are not detected in the presence of ESS3. Competitor RNAs containing either ESS2 or ESS3 relieve inhibition of splicing of substrates containing ESS3 or ESS2. This suggests that a common cellular factor(s) may be required for the inhibition of tat mRNA splicing mediated by ESS2 and ESS3.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

Red Queens and ESS: the coevolution of evolutionary rates

Summary The Red Queen principle states that a set of interacting species reaches an evolutionary equilibrium at which all their rates of coevolution exactly balance each other. The lag-load model, which is one way of searching for Red Queens, has, by itself, previously predicted that they do not exist. But this model has assumed that infinite maladaptedness is possible. The lag-load model is improved by assuming that once the lag load of all but one species is determined, so is that of the final species. This assumption eliminates the possibility of infinite maladaptedness. Its result is to allow the lag-load model to yield Red Queen coevolution. It does this whether or not speciation and extinction rates are included. Thus the lag-load model is harmonized with the earlier Red Queen model derived from studies of predation.Because of the intercorrelation of phenotypic traits, the predatory model concluded that the eventual stable rate of coevolution must be zero (except for intermittent bursts after some correlation or compromise is successfully broken). Another model that predicts stable coevolutionary rates of zero is that of evolutionarily stable strategies (ESS).Red Queen assumes that the more extreme a phenotypic trait is, the better it is, and that there are no constraints on the growth of such a phenotypic trait value. Such traits are the key to the Red Queen prediction of progressive coevolution. ESS models make no such assumptions. Eliminating unbounded traits from the model of predator-victim evolution changed its prediction from progressive coevolution to stasis. Before this paper, no model had dealt simultaneously with both unbounded and constrained traits.To handle both sorts of phenotypic traits at the same time in the same model, we abandoned lag load as a measure of evolutionary rate (lag loads do not uniquely determine phenotype). Instead, we used the traditional assumption that rate is proportional to the slope of the adaptive landscape. A model, relying on continuous evolutionary game theory, was developed and simulated under various conditions in two or three species sets, with up to five independent traits coevolving simultaneously. The results were: (1) there was always a set of equilibrium densities eventually achieved by coevolution; if the population interaction represented by this stable coevolutionary state is also stable, then the system should persist whether it evolves further or not; (2) whenever traits were present which were unbounded and best at their most extreme values, then a Red Queen emerged; (3) whenever traits were present which were correlated with each other or constrained below infinity, then an ESS emerged; (4) if both types were present, both results occurred: Red Queen in the unbounded traits and ESS in the constrained ones.Because unbounded traits may not exist, the Red Queen may have no domain. But the domain of ESS is real. ESS should lead to the evolutionary pattern called punctuated equilibrium. The changes in design rules which punctuate stasis should lead to an ever-expanding independence of traits from each other, i.e. to more and more refined differentiation. A single set of design rules which governs a set of species is called a fitness-generating function. Such functions may help to define the concepts of adaptive zone and ecological guild.     

Models on butterfly protandry: Virgin females are at risk to die

Current models on protandry in butterflies assume that females are mated instantaneously upon eclosion. However, for most butterfly species this assumption is not realistic. In this paper a model is formulated in which the mating rate depends on both male and female density. Given the female presence curve, protandry is an evolutionarily stable strategy (ESS) for males. The evolutionarily stable amount of protandry decreases with increasing death rate and decreasing encounter rate. Given the male presence curve, protandry also is an ESS for females. However, male and female ESS are not identical; moreover, in the present model a simultaneous ESS does not exist. Protandry critically depends on the assumption that females mate only once, whereas males are capable of multiple mating. If females too are capable of multiple mating, absence of protandry is the ESS for males as well as females. The model predicts that protandry depends on population density: protandry should be more pronounced in populations with high density than in populations with low density. Protandry also depends on sex ratio. It becomes more pronounced when the proportion of males among emerging adults increases.     

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.