2014/09/22 by Mathieu Fauré, Sebastian J. Schreiber, Faure, Mathieu +1
Biochemistry, Genetics and Molecular Biology · Medicine · Social Sciences · #37C70 #60Jxx #92D25 #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR)
paper · doi:10.48550/arxiv.1409.6122
openalex publication_date 2014/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Generalized Polya urn models have been used to model the establishment dynamics of a small founding population consisting of k different genotypes or strategies. As population sizes get large, these population processes are well-approximated by a mean limit ordinary differential equation whose state space is the k simplex. We prove that if this mean limit ODE has an attractor at which the temporal averages of the population growth rate is positive, then there is a positive probability of the population not going extinct (i.e. growing without bound) and its distribution converging to the attractor. Conversely, when the temporal averages of the population growth rate is negative along this attractor, the population distribution does not converge to the attractor. For the stochastic analog of the replicator equations which can exhibit non-equilibrium dynamics, we show that verifying the conditions for convergence and non-convergence reduces to a simple algebraic problem. We also apply these results to selection-mutation dynamics to illustrate convergence to periodic solutions of these population genetics models with positive probability.