A rigorous model study of the adaptive dynamics of Mendelian diploids

作者：Collet Pierre; Meleard Sylvie; Metz Johan A J^*

来源：Journal of Mathematical Biology, 2013, 67(3): 569-607.

DOI：10.1007/s00285-012-0562-5

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

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%26apos;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 %26apos;long%26apos; 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 %26apos;canonical equation%26apos;.

• 出版日期2013-9