Journal of Mathematical Biology

, Volume 67, Issue 3, pp 569–607

A rigorous model study of the adaptive dynamics of Mendelian diploids

  • Pierre Collet
  • Sylvie Méléard
  • Johan A. J. Metz
Open Access
Article

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 M1 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 

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Pierre Collet
    • 1
  • Sylvie Méléard
    • 2
  • Johan A. J. Metz
    • 3
    • 4
    • 5
  1. 1.CPHT Ecole Polytechnique, CNRS UMR 7644Palaiseau CedexFrance
  2. 2.CMAP, Ecole Polytechnique, CNRSPalaiseau CedexFrance
  3. 3.Department of Mathematics, Institute of BiologyLeiden UniversityLeidenThe Netherlands
  4. 4.Marine ZoologyNCB NaturalisLeidenThe Netherlands
  5. 5.Ecology and Evolution ProgramInstitute of Applied Systems AnalysisLaxenburgAustria

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