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From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics

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  • Chalub, Fabio A.C.C.
  • Souza, Max O.

Abstract

We study the large population limit of the Moran process, under the assumption of weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.

Suggested Citation

  • Chalub, Fabio A.C.C. & Souza, Max O., 2009. "From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics," Theoretical Population Biology, Elsevier, vol. 76(4), pages 268-277.
    • Handle: RePEc:eee:thpobi:v:76:y:2009:i:4:p:268-277
    DOI: 10.1016/j.tpb.2009.08.006
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    References listed on IDEAS

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    1. U. Dieckmann & R. Law, 1996. "The Dynamical Theory of Coevolution: A Derivation from Stochastic Ecological Processes," Working Papers wp96001, International Institute for Applied Systems Analysis.
    2. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
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    Cited by:

    1. Flávio L Pinheiro & Jorge M Pacheco & Francisco C Santos, 2012. "From Local to Global Dilemmas in Social Networks," PLOS ONE, Public Library of Science, vol. 7(2), pages 1-6, February.

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