A rigorous model study of the adaptive dynamics of mendelian diploids

Adaptive dynamics so far has been put on a rigorous footing only for clonal in-heritance. 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 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 perturba-tion 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’. MSC 2000 subject classification: 92D25, 60J80, 37N25, 92D15, 60J75 Key-words: individual-based mutation-selection model, invasion fitness for diploid popula-tions, adaptive dynamics, canonical equation, polymorphic evolution sequence, competitive Lotka-Volterra system