## The Annals of Applied Probability

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

### Generalized URN models of evolutionary processes

Michel Benaïm, Sebastian J. Schreiber, and Pierre Tarrès

#### 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