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.     

Evolution of migration in a metapopulation

In this paper a general deterministic discrete-time metapopulation model with a finite number of habitat patches is analysed within the framework of adaptive dynamics. We study a general model and prove analytically that (i) if the resident populations state is a fixed point, then the resident strategy with no migration is an evolutionarily stable strategy, (ii) a mutant population with no migration can invade any resident population in a fixed point state, (iii) in the uniform migration case the strategy not to migrate is attractive under small mutational steps so that selection favours low migration. Some of these results have been previously observed in simulations, but here they are proved analytically in a general case. If the resident population is in a two-cyclic orbit, then the situation is different. In the uniform migration case the invasion behaviour depends both on the type of the residents attractor and the survival probability during migration. If the survival probability during migration is low, then the system evolves towards low migration. If the survival probability is high enough, then evolutionary branching can happen and the system evolves to a situation with several coexisting types. In the case of out-of-phase attractor, evolutionary branching can happen with significantly lower survival probabilities than in the in-phase attractor case. Most results in the two-cyclic case are obtained by numerical simulations. Also, when migration is not uniform we observe in numerical simulations in the two-cyclic orbit case selection for low migration or evolutionary branching depending on the survival probability during migration.     

Stochastic evolutionary dynamics of direct reciprocity

Evolutionary game theory is the study of frequency-dependent selection. The success of an individual depends on the frequencies of strategies that are used in the population. We propose a new model for studying evolutionary dynamics in games with a continuous strategy space. The population size is finite. All members of the population use the same strategy. A mutant strategy is chosen from some distribution over the strategy space. The fixation probability of the mutant strategy in the resident population is calculated. The new mutant takes over the population with this probability. In this case, the mutant becomes the new resident. Otherwise, the existing resident remains. Then, another mutant is generated. These dynamics lead to a stationary distribution over the entire strategy space. Our new approach generalizes classical adaptive dynamics in three ways: (i) the population size is finite; (ii) mutants can be drawn non-locally and (iii) the dynamics are stochastic. We explore reactive strategies in the repeated Prisoner''s Dilemma. We perform ‘knock-out experiments’ to study how various strategies affect the evolution of cooperation. We find that ‘tit-for-tat’ is a weak catalyst for the emergence of cooperation, while ‘always cooperate’ is a strong catalyst for the emergence of defection. Our analysis leads to a new understanding of the optimal level of forgiveness that is needed for the evolution of cooperation under direct reciprocity.     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

A predator–prey refuge system: Evolutionary stability in ecological systems

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individual’s fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator–prey systems are discussed from this perspective.     

A predator–prey refuge system: Evolutionary stability in ecological systems

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individual’s fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator–prey systems are discussed from this perspective.     

The adaptive dynamics of Lotka-Volterra systems with trade-offs

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.     

The resident strikes back: invader-induced switching of resident attractor

The aim of this paper is two-fold: (a) by way of example, we elucidate the phenomenon of invader-induced switches in a resident attractor; (b) we expose in detail how resonance and phase have a strong impact when semelparous organisms (as, e.g. Pacific salmon) with different life-cycle lengths compete in a self-induced periodically fluctuating environment. We analyse a simple model for the competition between annuals and biennials, focusing on the situation that the annual population in isolation converges to a two-cycle. Well-timed biennial mutants sample the periodically varying environment more efficiently than the annual resident. They can invade successfully even when they are inferior to the resident, in the sense that they have lower viability and/or fertility. Successful invasion can lead to resonance-mediated coexistence if the invader is rather inferior to the resident. Remarkably, for mutants that are less inferior to the resident, successful invasion by a mutant strategy will inevitably be followed by the extinction of the former invader and concurrent re-establishment of the resident. The expulsion of the invader is brought about by an invasion-induced phase shift or attractor switch. We call this phenomenon "the resident strikes back" and say that the resident strategy is invasible, yet invincible. After the resident has struck back, other mutants can successfully invade again. On a longer time-scale, this might lead to an intermittent occurrence of ultimately inferior strategies. The results show that even in a deterministic setting, successful invasion does not necessarily lead to establishment and that mutual invasibility is not always sufficient for coexistence.     

Evolutionary game dynamics in a finite asymmetric two-deme population and emergence of cooperation

We consider evolutionary game dynamics in a finite population subdivided into two demes with both unequal deme sizes and different migration rates. Assuming viability differences in the population according to a linear game within each deme as a result of pairwise interactions, we specify conditions for weak selection favoring a mutant strategy to go to fixation, under the structured-coalescent assumptions, and their connections with evolutionary stability concepts. In the framework of the Iterated Prisoner's Dilemma with strategy ‘tit-for-tat’ as mutant strategy and ‘always defect’ as resident strategy, we deduce a condition under which the emergence of cooperation is favored by selection, when the game matrix is the same in both demes. We show how this condition extends the one-third law for a panmictic population and when an asymmetry in the spatial structure of a two-deme population facilitates the emergence of the cooperative tit-for-tat strategy in comparison with both its symmetric and panmictic population structure counterparts. We find that the condition is less stringent in the asymmetric scenario versus the symmetric scenario if both the fraction of the population in the deme where the mutant was initially introduced, and the expected proportion of migrant offspring in this deme among all migrant offspring after population regulation, are smaller than, or equal to, , provided they are not too small. On the other hand, the condition is less stringent than the one-third law, which holds in the panmictic case, if the latter proportion remains not too close to 1.     

The ideal free distribution as an evolutionarily stable state in density‐dependent population games

In classical games that have been applied to ecology, individual fitness is either density independent or population density is fixed. This article focuses on the habitat selection game where fitness depends on the population density that evolves over time. This model assumes that changes in animal distribution operate on a fast time scale when compared to demographic processes. Of particular interest is whether it is true, as one might expect, that resident phenotypes who use density‐dependent optimal foraging strategies are evolutionarily stable with respect to invasions by mutant strategies. In fact, we show that evolutionary stability does not require that residents use the evolutionarily stable strategy (ESS) at every population density; rather it is the combined resident–mutant system that must be at an evolutionary stable state. That is, the separation of time scales assumption between behavioral and ecological processes does not imply that these processes are independent. When only consumer population dynamics in several habitats are considered (i. e. when resources do not undergo population dynamics), we show that the existence of optimal foragers forces the resident‐mutant system to approach carrying capacity in each habitat even though the mutants do not die out. Thus, the ideal free distribution (IFD) for the single‐species habitat selection game becomes an evolutionarily stable state that describes a mixture of resident and mutant phenotypes rather than a strategy adopted by all individuals in the system. Also discussed is how these results are affected when animal distribution and demographic processes act on the same time scale.     

THE EVOLUTION OF STRATEGY VARIATION: WILL AN ESS EVOLVE?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.     

Fixation probabilities for the Moran process with three or more strategies: general and coupling results

We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

     

Stability in N-species coevolutionary systems

Stability criteria have recently been developed for coevolutionary Lotka-Volterra systems where individual fitness functions are assumed to be linear in the population state. We extend these criteria as part of a general theory of coevolution (that combines effects of ecology and evolution) based on arbitrary (i.e. nonlinear) fitness functions and a finite number of individual phenotypes. The central role of the stationary density surface where species' densities are at equilibrium is emphasized. In particular, for monomorphic resident systems, it is shown coevolutionary stability is equivalent to ecological stability combined with evolutionary stability on the stationary density surface. Also discussed is how our theory relates to recent treatments of phenotypic coevolution via adaptive dynamics when there is a continuum of individual phenotypes.     

EVOLUTION OF DISPERSAL RATES IN METAPOPULATION MODELS: BRANCHING AND CYCLIC DYNAMICS IN PHENOTYPE SPACE

We study the evolution of dispersal rates in a two patch metapopulation model. The local dynamics in each patch are given by difference equations, which, together with the rate of dispersal between the patches, determine the ecological dynamics of the metapopulation. We assume that phenotypes are given by their dispersal rate. The evolutionary dynamics in phenotype space are determined by invasion exponents, which describe whether a mutant can invade a given resident population. If the resident metapopulation is at a stable equilibrium, then selection on dispersal rates is neutral if the population sizes in the two patches are the same, while selection drives dispersal rates to zero if the local abundances are different. With non-equilibrium metapopulation dynamics, non-zero dispersal rates can be maintained by selection. In this case, and if the patches are ecologically identical, dispersal rates always evolve to values which induce synchronized metapopulation dynamics. If the patches are ecologically different, evolutionary branching into two coexisting dispersal phenotypes can be observed. Such branching can happen repeatedly, leading to polymorphisms with more than two phenotypes. If there is a cost to dispersal, evolutionary cycling in phenotype space can occur due to the dependence of selection pressures on the ecological attractor of the resident population, or because phenotypic branching alternates with the extinction of one of the branches. Our results extend those of Holt and McPeek (1996), and suggest that phenotypic branching is an important evolutionary process. This process may be relevant for sympatric speciation.     

Stability in -species coevolutionary systems

Stability criteria have recently been developed for coevolutionary Lotka–Volterra systems where individual fitness functions are assumed to be linear in the population state. We extend these criteria as part of a general theory of coevolution (that combines effects of ecology and evolution) based on arbitrary (i.e. nonlinear) fitness functions and a finite number of individual phenotypes. The central role of the stationary density surface where species’ densities are at equilibrium is emphasized. In particular, for monomorphic resident systems, it is shown coevolutionary stability is equivalent to ecological stability combined with evolutionary stability on the stationary density surface. Also discussed is how our theory relates to recent treatments of phenotypic coevolution via adaptive dynamics when there is a continuum of individual phenotypes.     

How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

Forms of density regulation and (quasi-) stationary distributions of population sizes in birds

The theta-logistic model of density regulation is an especially flexible class of density regulation models where different forms of non-linear density regulation can be expressed by only one parameter, θ. Estimating the parameters of the theta-logistic model is, however, challenging. This is mainly due to the need for information concerning population growth at low densities as well as data on fluctuations around the carrying capacity K in order to estimate the strength of density regulation. Here we estimate parameters of the theta-logistic model for 28 populations of three species of birds that have grown from very small population sizes followed by a period of fluctuations around K. We then use these parameters to estimate the quasi-stationary distribution of population size. There were often large uncertainties in these parameters specifying the form of density regulation that were generally independent of the duration of the study period. In contrast, precision in the estimates of environmental variance increased with the length of the time series. In most of the populations, a large proportion of the probability density of the (quasi-) stationary distribution of population sizes was located at intermediate population sizes relative to K. Thus, we suggest that the (quasi-) stationary distribution of population sizes represents a useful summary statistic that in many cases provides a more robust characterisation of basic population dynamics (e.g. range of variation in population fluctuations or proportion of time spent close to K) than can be obtained from analyses of single model parameters.