The Annals of Applied Probability

Generalized URN models of evolutionary processes

Michel Benaïm, Sebastian J. Schreiber, and Pierre Tarrès

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 3 (2004), 1455-1478.

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736292

Digital Object Identifier
doi:10.1214/105051604000000422

Mathematical Reviews number (MathSciNet)
MR2071430

Zentralblatt MATH identifier
1051.60072

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 92D25: Population dynamics (general)

Keywords
Markov chains random genetic drift urn models replicator equations

Citation

Benaïm, Michel; Schreiber, Sebastian J.; Tarrès, Pierre. Generalized URN models of evolutionary processes. Ann. Appl. Probab. 14 (2004), no. 3, 1455--1478. doi:10.1214/105051604000000422. https://projecteuclid.org/euclid.aoap/1089736292


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