Abstract
Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka-Volterra-like dynamics and single locus genetics (as in Kimura's in Proc Natl Acad Sci USA 54: 731-736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the 'long' time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M 1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as 'canonical equation'.
Highlights
Adaptive dynamics (AD) aims at providing an ecology-based framework for scaling up from the micro-evolutionary process of gene substitutions to meso-evolutionary time scales and phenomena
All we can deduce from the canonical equation (5.1) is that for small mutational steps the trait substitution sequence will move to some close neighborhood of an allelic evolutionarily singular strategy
This paper forms part of a series by a varied collection of authors that aim at putting the tools of adaptive dynamics on a rigorous footing (Metz et al 1992, 1996; Dieckmann and Law 1996; Geritz et al 1998; Champagnat et al 2008; Champagnat 2006; Durinx et al 2008; Méléard and Tran 2009; Champagnat and Méléard 2011; Metz 2012; Klebaner et al 2011; Bovier and Champagnat 2012)
Summary
Adaptive dynamics (AD) aims at providing an ecology-based framework for scaling up from the micro-evolutionary process of gene substitutions to meso-evolutionary time scales and phenomena ( called long term evolution in papers on the foundations of ESS theory, that is, meso-evolutionary statics (cf. Eshel 1983, 2012; Eshel et al 1998; Eshel and Feldman 2001). One of the more interesting phenomena that AD has brought to light is the possibility of an emergence of phenotypic diversification at so-called branching points, without the need for a geographical substrate (Metz et al 1996; Geritz et al 1998; Doebeli and Dieckmann 2000) This ecological tendency may in the sexual case induce sympatric speciation (Dieckmann and Doebeli 1999). The theory was first put on a mathematically rigorous footing by Champagnat and Méléard and coworkers (Champagnat et al 2008; Champagnat 2006; Méléard and Tran 2009), and recently from a different perspective by Peter Jagers and coworkers (Klebaner et al 2011) All these papers deal only with clonal models. Our goal is to capture in a simple manner the interplay between these different mechanisms
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