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Random Modeling of Adaptive Dynamics and Evolutionary Branching

Chapter
Part of the Mathematics and Biosciences in Interaction book series (MBI)

Abstract

We are interested in modeling the Darwinian dynamics of a polymorphic asexual population, as driven by the interplay of phenotypic variation and natural selection through ecological interactions. Our modeling is based on a stochastic individual-based model that details the dynamics of heritable traits characterizing each individual. We consider the specific scales of the biological framework of adaptive dynamics: rare mutations and large population. We prove that under a good combination of these two scales, the population process is approximated in an evolution long time scale by a Markov pure jump process describing successive equilibria of the population. Then we consider this polymorphic evolution process in the limit of small mutations. From a fine study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching.

Keywords

Mutation-selection individual-based model fitness of invasion adaptive dynamics polymorphic evolution sequence evolutionary branching 

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.CMAP, Ecole Polytechnique, CNRSPalaiseau CedexFrance

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