Multi-species evolutionary dynamics

Dynamical attainability of an evolutionarily stable strategy (ESS) through the process of mutations and natural selection has mostly been addressed through the use of the continuously stable strategy (CSS) concept for species evolutionary games in which strategies are drawn from a continuum, and by the adaptive trait dynamics method. We address the issue of dynamical attainability of an ESS in coevolving species through the use of the concept of an ESNIS. It is shown that the definition of an ESNIS coalition for coevolving species is not in general equivalent to other definitions for CSS given in the literature. We show under some additional conditions that, in a dynamic system which involves the strategies of a dimorphic ESNIS coalition and at most two strategies that are not members of ESNIS coalition, the ESNIS coalition will emerge as the winner. In addition an ESNIS will be approached because of the invasion structure of strategies in its neighborhood. This proves that under the above conditions an ESNIS has a better chance of being attained than a strategy coalition which is a CSS. The theory developed is applied to a class of coevolutionary game models with Lotka–Volterra type interactions and we show that for such models, an ESS coalition will be dynamically attainable through mutations and natural selection if the ESS coalition is also an ESNIS coalition.Co-ordinating editor: Metz     

Evolution of the distribution of dispersal distance under distance‐dependent cost of dispersal

Abstract We analyse the evolution of the distribution of dispersal distances in a stable and homogeneous environment in one‐ and two‐dimensional habitats. In this model, dispersal evolves to avoid the competition between relatives although some cost might be associated with this behaviour. The evolutionarily stable dispersal distribution is characterized by an equilibration of the fitness gains among all the different dispersal distances. This cost‐benefit argument has heuristic value and facilitates the comprehension of results obtained numerically. In particular, it explains why some minimal or maximal probability of dispersal may evolve at intermediate distances when the cost of dispersal function is an increasing function of distance. We also show that kin selection may favour long range dispersal even if the survival cost of dispersal is very high, provided the survival probability does not vanish at long distances.     

Unpredictable selection in a structured population leads to local genetic differentiation in evolved reaction norms

Unpredictability during development of the optimum phenotype under future selection leads to a compromise reaction norm with a slope that is shallower than the slope of the optimum reaction norm. Unpredictability of selection can lead to an evolved curved reaction norm when genetic variation for curvature is available even if the optimum reaction norm is linear. This requires asymmetry in the frequency distribution of the habitats of selection; at small population size, stochasticity in the number of individuals per selection habitat is sufficient to generate such asymmetry. Unpredictability of selection in structured populations leads to local genetic differentiation of reaction norms. The mean habitat of a subpopulation is defined as the subpopulation's focal habitat. The evolved mean reaction norm of each subpopulation is anchored at the optimum genotypic value in its focal habitat. Linear reaction norms are parallel if the conditional distribution of adults around the focal habitats is the same for each subpopulation. Adult migration and absence of zygote dispersal represents the ultimate structured population, each habitat playing the role of focal habitat. Absence of zygote dispersal requires that the flow of individuals through the habitats is used instead of the habitats’ frequencies in the prediction of the evolved reaction norm. Adult migration in absence of zygote dispersal leads to an evolved pattern of locally differentiated reaction norms with optimum genotypic value anchored in the focal habitat and, for linear reaction norms, parallel slopes.     

Co-evolution of seed size and seed predation

Using the evolutionarily stable strategy (ESS) approach in a model for the co-evolution of seed size and seed predation, I show that seed size variation within individual plants is favoured if there is a trade-off in the predator's attack rate for different seed sizes. A single seed size is not evolutionarily stable because a predator that is optimally adapted to one particular seed size cannot prevent invasion by plants with a different seed size. The model generates the following predictions. The ESS consists of a continuous range of seed sizes. Small seeds tend to be attacked more frequently than big seeds. Plants with many resources and plants with low (frequency-independent) juvenile mortality have more variable seeds than plants with few resources and a high juvenile mortality. Seed size variation is higher in fluctuating populations regulated by seed predation alone than in stable populations (partially) regulated by seedling competition. Predator searching behaviour does not directly affect the ESS seed size range, but may have an indirect effect by affecting population stability or the significance of seedling competition as a population regulating mechanism. Moreover, seed size distributions are found to be more skewed in favour of small seeds if predation is spatially non-uniform than if predation is more even. Application of the model to systems of several co-evolving plant and predator species is discussed.     

Delayed female reproduction in equilibrium and chaotic populations

Behavioural and life history polymorphisms are often observed in animal populations. We analyse the timing of maturation and reproduction in risky and resource-limited environments. Field and laboratory evidence suggests that female voles and mice, for example, can adjust their breeding according to the level of risk to their own survival and to survival probabilities and recruitment of young produced under different environmental conditions. Under risky or harsh conditions breeding can be postponed until later in the current breeding season or even to the next breeding season. We develop a population dynamics and life history model for polymorphism in reproduction (co-existence of breeding and non-breeding behaviours) of females in an age-structured population, with two temporally distinct mating events within the breeding season. We assume that, after overwintering, the females can breed in spring and again in summer or they can delay breeding in spring and breed in summer only. Young females born in spring can either mature and breed in summer or stay immature and postpone breeding over the winter to the next breeding season. We show that an evolutionarily stable breeding strategy is either an age-structured combination of pure breeding behaviours (old females breed and young delay maturity) or a mixed breeding behaviour within age-classes (a fraction of females breed and the rest of the age class postpones breeding). Co-occurrence of mixed reproductive behaviour in spring and summer within a single breeding season is observed in fluctuating populations only. The reproductive patterns depend on intraspecific, possibly interspecific, and ecological factors. The density dependence (e.g. social suppression) and predation risk are shown to be possible evolutionary mechanisms in adjusting the relative proportions of the different but co-existing reproductive behaviours.     

Polymorphism in heterogeneous environments,evolution of habitat selection and sympatric speciation: Soft and hard selection models

Summary The adaptation to a variable environment has been studied within soft and hard selection frameworks. It is shown that an epistatically determined habitat preference, following a Markovian process, always leads to the maintenance of an adaptive polymorphism, in a soft selection context. Although local mating does not alter the conditions for polymorphism maintenance, it is shown that, in that case, habitat selection also leads to the evolution of isolated reproductive units within each available habitat. Habitat selection, however, cannot evolve in the total absence of adaptive polymorphism. This represents a theoretical problem for all models assuming habitat selection to be an initially fixed trait, and means that within a soft selection framework, all the available habitats will be exploited, even the less favourable ones.On the other hand, polymorphism cannot be maintained when selection is hard, even when all individuals select their habitat. Here, the evolution of habitat selection does not need any prerequisite polymorphism, and always leads to the exploitation of only one habitat by the most specialized genotype. It appears then that hard selection can account for the existence of empty habitat and for an easier evolution of habitat specialization.     

Evolution of airborne infectious diseases according to changes in characteristics of the host population

The authors predicted evolutionary changes in airborne infectious diseases according to changes in the characteristics of the host population. The predictions were based upon a mathematical model of infectious diseases and the validity of the predictions was verified against the history of man and pathogens. The feature of this model is that it involves a density of pathogens in the environment as an additional variable which can be regarded as more suitable to airborne infectious diseases. In spite of this modification, this study reached a similar conclusion to the threshold density theory: that is, susceptible host density in the absence of the pathogen must be larger than that in the presence of the pathogen, for the pathogen to be persistent. Moreover the authors concluded that one type of pathogen cannot be replaced by another type of pathogen as long as the susceptible host density of the former type is the mininum one. The predictions were considered to be valid for a wide range of infectous diseases. Making use of these principles, the authors predicted that the variety of infectious diseases should increase as host density increases and that pathogens should evolve to be less virulent as the host life-span increases. The finalidea discussed is whether or nor the history of man and pathogen can be verified by the predictions.     

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms'' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin''s arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.     

Evolutionarily stable strategies of migration in heterogeneous environments

By means of a simulation model we are showing that the rates of migration can be related to avoidance of competition between relatives, especially in clonal organisms. This could result in a strong selective pressure for migration, even at a high cost. In addition, if the habitat is fragmented, migration can strongly affect local dynamics and result in a dramatic decrease of the densities in some places. In parthenogenetically reproducing organisms like aphids, the level of relatedness in local populations is expected to be very high and therefore they can serve as a good model group for testing these hypotheses. This revised version was published online in July 2006 with corrections to the Cover Date.     

On several conjectures from evolution of dispersal

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17–36 [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]] concerning the dynamics of a diffusion–advection–competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. It was shown in [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.     

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.     

ESS emergence pattern of male butterflies in stochastic environments

Summary The evolutionarily stable (or ESS) emergence schedule for males of univoltine butterflies is analysed in an environment in which the female emergence schedule fluctuates stochastically between years. The ESS emergence curve, computed using the mutant invadability criterion, is shown to be the one that maximizes mean logarithmic lifetime mating success in the population in which it dominates. If males have accurate information about the female emergence schedule within each year, their emergence curve would evolve to the one predicted by a deterministic game model. The male emergence curve would then shift between years, closely following year to year changes in the female emergence pattern. If, instead, males have uncertainty about the female emergence schedule, the ESS male emergence curve becomes broader than the one predicted by the deterministic game model and will not track the between-year fluctuation of female emergence well. In a special case, we show how the between-year variation of mean emergence date, the variance of emergence date, the sexual difference in mean emergence dates (protandry) and the between-year correlation of mean emergence dates of both sexes should change with the degree of accuracy of information available to males.