A rigorous model study of the adaptive dynamics of Mendelian diploids

Collet P, Meleard S, & Metz JAJ (2013). A rigorous model study of the adaptive dynamics of Mendelian diploids. Journal of Mathematical Biology 67 (3): 569-607. DOI:10.1007/s00285-012-0562-5.

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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".

Item Type: Article
Uncontrolled Keywords: Individual-based mutation-selection model; Invasion fitness for diploid populations; Adaptive dynamics; Canonical equation; Polymorphic evolution sequence; Competitive Lotka-Volterra system
Research Programs: Evolution and Ecology (EEP)
Bibliographic Reference: Journal of Mathematical Biology; 67(3):569-607 (September 2013) (Published online 21 July 2012)
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:48
Last Modified: 24 Feb 2016 14:49
URI: http://pure.iiasa.ac.at/10367

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