Analysis of disruptive selection in subdivided populations

BMC Evolutionary Biology BioMed Central Open Access Research article Analysis of disruptive selection in subdivided populations Émile Ajar* Address: Laboratoire Génétique et Environnement, Institut des Sciences de l'Évolution, CC065, USTL, Place E. Bataillon, 34095 Montpellier Cedex 05, France Email: Émile Ajar* - * Corresponding author Published: 06 November 2003 BMC Evolutionary Biology 2003, 3:22 Received: 30 June 2003 Accepted: 06 November 2003 This article is available from: http://www.biomedcentral.com/1471-2148/3/22 © 2003 Ajar; licensee BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL. Abstract Background: Analytical methods have been proposed to determine whether there are evolutionarily stable strategies (ESS) for a trait of ecological significance, or whether there is disruptive selection in a population approaching a candidate ESS. These criteria do not take into account all consequences of small patch size in populations with limited dispersal. Results: We derive local stability conditions which account for the consequences of small and constant patch size. All results are derived from considering Rm, the overall production of successful emigrants from a patch initially colonized by a single mutant immigrant. Further, the results are interpreted in term of concepts of inclusive fitness theory. The condition for convergence to an evolutionarily stable strategy is proportional to some previous expressions for inclusive fitness. The condition for evolutionary stability stricto sensu takes into account effects of selection on relatedness, which cannot be neglected. It is function of the relatedness between pairs of genes in a neutral model and also of a three-genes relationship. Based on these results, I analyze basic models of dispersal and of competition for resources. In the latter scenario there are cases of global instability despite local stability. The results are developed for haploid island models with constant patch size, but the techniques demonstrated here would apply to more general scenarios with an island mode of dispersal. Conclusions: The results allow to identity and to analyze the relative importance of the different selective pressures involved. They bridge the gap between the modelling frameworks that have led to the Rm concept and to inclusive fitness. Background Various criteria have been proposed to compute the stable states of the evolutionary dynamics of traits of ecological significance. Previous works ("adaptive dynamics", e.g., [1-6]) have highlighted the need to distinguish different kinds of stability. A strategy is convergence stable if the population evolves towards it by allelic substitutions. A convergence stable strategy is evolutionarily stable (noninvasible) if rare deviants are selected against. Otherwise, there is disruptive selection, and "branching" of the distribution of phenotypes in the population may occur [3,6]. When fitness can be evaluated exactly, the different kinds of stability can be evaluated. However, in many cases approximations are useful, either because exact results are not available or because they are too complex to allow better understanding of evolution. This occurs when populations are structured in patches occupied by a small number of individuals. In such a case, a widely used Page 1 of 12 (page number not for citation purposes) BMC Evolutionary Biology 2003, 3 measure of fitness effects is inclusive fitness. Inclusive fitness measures fitness effects as the effect of a deviant strategy on the fitness of an individual which expresses this strategy, plus the effect on the fitness of an individual when the strategy is expressed by other individuals in the patch, the latter effect being weighted by a measure of genetic similarity of individuals within a patch [7]. Although the inclusive fitness approach often allows to identify selective pressures, it is desirable to integrate it in a more general framework where the different kinds of dynamics are distinguished [8]. Can inclusive fitness be used to compute convergence stability, evolutionary stability, or both? Some works made no distinction between the concepts of convergence and of evolutionary stability [9], while others have found that inclusive fitness is suitable for evaluating convergence stability but not for evolutionary stability [10,11]. There have been some attempts to derive evolutionary stability conditions using inclusive fitness concepts (see [10] and references therein) but further insight into the above issues has been limited by a dearth of well-established results which could be compared to some alternative approach. This paper will provide such results, using the Rm concept introduced in ref. [12]. Rm is the overall production of successful emigrants from a patch, descended from a single mutant immigrant. Ref. [12] presents an exact numerical method to compute Rm in complex metapopulation models (also used in ref. [13]), but analytical conditions for convergence and evolutionary stability can also be deduced from Rm. In this paper I show how this can be done. In particular, a new result is the analytical condition for local invasibility versus non-invasibility (i.e. evolutionary stability) of a convergence stable strategy for the island model of dispersal. Kin selection effects are taken into account in this computation, as the kin interactions that occur in the patch all the way from colonization to local allele extinction. Thus, we should also be able to recover known inclusive fitness expressions from Rm, but how we can do that is not a priori obvious. We will see that inclusive fitness can be derived as a measure of local convergence stability from Rm. But the evolutionary stability condition can also be understood in terms of the concepts of inclusive fitness theory. Rather than considering the complex metapopulation model of ref. [12], I will consider discrete-time models which assume a constant number N of haploid adults per patch. This should help to see the logic of the method. Within this setting, I will analyze some basic models widely considered in previous works, dealing with the evolution of dispersal and with disruptive selection under competition for resources. http://www.biomedcentral.com/1471-2148/3/22 Results We consider here models where N adults reproduce within each of a large ("infinite") number of patches. A large number of juveniles are produced by each adult. A fraction of them disperse, in which case they disperse randomly over all patches, following an "island" or "global" mode of dispersal. The juveniles then compete for access to reproduction so that exactly N of them survive this competition in each patch. No other exact assumption about reproduction, competition and dispersal is done at this stage (this is done (...truncated)