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’.
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Acknowledgments
This work benefitted from the support from the “Chaire Modélisation Mathématique et Biodiversité of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X”. Two anonymous reviewers helped us to improve the paper by suggesting small but useful additions and listing a good many small typos in the formulas.
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Collet, P., Méléard, S. & Metz, J.A.J. A rigorous model study of the adaptive dynamics of Mendelian diploids. J. Math. Biol. 67, 569–607 (2013). https://doi.org/10.1007/s00285-012-0562-5
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DOI: https://doi.org/10.1007/s00285-012-0562-5
Keywords
- Individual-based mutation-selection model
- Invasion fitness for diploid populations
- Adaptive dynamics
- Canonical equation
- Polymorphic evolution sequence
- Competitive Lotka–Volterra system