Convergence of generalized urn models to non-equilibrium attractors

TL;DR

This paper proves conditions under which generalized urn models converge to non-equilibrium attractors, linking stochastic population dynamics with deterministic mean limit equations and applying results to population genetics models.

Contribution

It establishes a rigorous connection between stochastic urn models and their mean limit ODEs, especially regarding convergence to non-equilibrium attractors and their biological implications.

Findings

Positive probability of population growth when mean limit has a positive attractor.

Non-convergence when the average growth rate is negative.

Application to population genetics showing convergence to periodic solutions.

Abstract

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.