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August 2004 Generalized URN models of evolutionary processes
Michel Benaïm, Sebastian J. Schreiber, Pierre Tarrès
Ann. Appl. Probab. 14(3): 1455-1478 (August 2004). DOI: 10.1214/105051604000000422


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.


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Michel Benaïm. Sebastian J. Schreiber. Pierre Tarrès. "Generalized URN models of evolutionary processes." Ann. Appl. Probab. 14 (3) 1455 - 1478, August 2004.


Published: August 2004
First available in Project Euclid: 13 July 2004

zbMATH: 1051.60072
MathSciNet: MR2071430
Digital Object Identifier: 10.1214/105051604000000422

Primary: 60J10
Secondary: 92D25

Keywords: Markov chains , random genetic drift , replicator equations , urn models

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.14 • No. 3 • August 2004
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