Journal of Mathematical Biology

, Volume 67, Issue 3, pp 569–607

A rigorous model study of the adaptive dynamics of Mendelian diploids

Authors

  • Pierre Collet
    • CPHT Ecole Polytechnique, CNRS UMR 7644
  • Sylvie Méléard
    • CMAP, Ecole Polytechnique, CNRS
    • Department of Mathematics, Institute of BiologyLeiden University
    • Marine ZoologyNCB Naturalis
    • Ecology and Evolution ProgramInstitute of Applied Systems Analysis
Open AccessArticle

DOI: 10.1007/s00285-012-0562-5

Cite this article as:
Collet, P., Méléard, S. & Metz, J.A.J. J. Math. Biol. (2013) 67: 569. doi:10.1007/s00285-012-0562-5

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’.

Keywords

Individual-based mutation-selection model Invasion fitness for diploid populations Adaptive dynamics Canonical equation Polymorphic evolution sequence Competitive Lotka–Volterra system

Mathematics Subject Classification (2000)

92D25 60J80 37N25 92D15 60J75

Copyright information

© The Author(s) 2012