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Polymorphic evolution sequence and evolutionary branching
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  • Published: 14 April 2010

Polymorphic evolution sequence and evolutionary branching

  • Nicolas Champagnat1 &
  • Sylvie Méléard2 

Probability Theory and Related Fields volume 151, pages 45–94 (2011)Cite this article

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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.

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Author information

Authors and Affiliations

  1. EPI TOSCA, INRIA Sophia Antipolis, Méditerranée, 2004 route des Lucioles, BP. 93, 06902, Sophia Antipolis Cedex, France

    Nicolas Champagnat

  2. CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128, Palaiseau Cedex, France

    Sylvie Méléard

Authors
  1. Nicolas Champagnat
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  2. Sylvie Méléard
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Correspondence to Nicolas Champagnat.

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Champagnat, N., Méléard, S. Polymorphic evolution sequence and evolutionary branching. Probab. Theory Relat. Fields 151, 45–94 (2011). https://doi.org/10.1007/s00440-010-0292-9

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  • Received: 26 November 2009

  • Revised: 22 March 2010

  • Published: 14 April 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0292-9

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Keywords

  • Mutation-selection individual-based model
  • Fitness of invasion
  • Adaptive dynamics
  • Polymorphic evolution sequence
  • Competitive Lotka–Volterra system
  • Evolutionary branching

Mathematics Subject Classification (2000)

  • 92D25
  • 60J80
  • 37N25
  • 92D15
  • 60J75
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