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Polymorphic evolution sequence and evolutionary branching.

Authors :
Champagnat, Nicolas
Méléard, Sylvie
Source :
Probability Theory & Related Fields; Oct2011, Vol. 151 Issue 1/2, p45-94, 50p
Publication Year :
2011

Abstract

We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence (TSS), is called polymorphic evolution sequence (PES). Then we introduce a scaling of the size of mutations and we study the PES in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. This phenomenon corresponds to the situation where the population, initially essentially single modal, is driven by the selective forces to divide into two separate subpopulations. To this end we finely analyze the asymptotic behavior of three-dimensional competitive Lotka–Volterra systems. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
01788051
Volume :
151
Issue :
1/2
Database :
Complementary Index
Journal :
Probability Theory & Related Fields
Publication Type :
Academic Journal
Accession number :
156759418
Full Text :
https://doi.org/10.1007/s00440-010-0292-9