Random modeling of adaptive dynamics and evolutionary branching. (English) Zbl 1379.92039
Chalub, Fabio A. C. C. (ed.) et al., The mathematics of Darwin’s legacy. Basel: Birkhäuser (ISBN 978-3-0348-0121-8/hbk; 978-3-0348-0122-5/ebook). Mathematics and Biosciences in Interaction, 175-192 (2011).
Summary: 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.
For the entire collection see [Zbl 1220.92045].
For the entire collection see [Zbl 1220.92045].
MSC:
| 92D15 | Problems related to evolution |
| 92D25 | Population dynamics (general) |
| 60J85 | Applications of branching processes |
Keywords:
mutation-selection individual-based model; fitness of invasion; adaptive dynamics; polymorphic evolution sequence; evolutionary branchingReferences:
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