2004/07/08 by Michel Benaïm, Michel Benaim, Sebastian J. Schreiber +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Social Sciences · #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Mathematical and Theoretical Epidemiology and Ecology Models #math.PR #msc:60J10 #msc:92D25
paper · pdf · doi:10.1214/105051604000000422
published as Annals of Probability 2004, Vol. 14, No. 3, 1455-1478
arxiv created 2004/07/08 · openalex publication_date 2004/07/14 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Generalized Pólya urn models can describe the dynamics of finite populations of interacting genotypes. Three basic questions these models can address are: Under what conditions does a population exhibit growth? On the event of growth, at what rate does the population increase? What is the long-term behavior of the distribution of genotypes? To address these questions, we associate a mean limit ordinary differential equation (ODE) with the urn model. Previously, it has been shown that on the event of population growth, the limiting distribution of genotypes is a connected internally chain recurrent set for the mean limit ODE. To determine when growth and convergence occurs with positive probability, we prove two results. First, if the mean limit ODE has an “attainable” attractor at which growth is expected, then growth and convergence toward this attractor occurs with positive probability. Second, the population distribution almost surely does not converge to sets where growth is not expected and almost surely does not converge to “nondegenerate” unstable equilibria or periodic orbits of the mean limit ODE. Applications to stochastic analogs of the replicator equations and fertility-selection equations of population genetics are given.