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Generalized Polya urn models have been used to model the establishmentdynamics of a small founding population consisting of k different genotypes orstrategies. As population sizes get large, these population processes arewell-approximated by a mean limit ordinary differential equation whose statespace is the k simplex. We prove that if this mean limit ODE has an attractorat which the temporal averages of the population growth rate is positive, thenthere 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 isnegative along this attractor, the population distribution does not converge tothe attractor. For the stochastic analog of the replicator equations which canexhibit non-equilibrium dynamics, we show that verifying the conditions forconvergence and non-convergence reduces to a simple algebraic problem. We alsoapply these results to selection-mutation dynamics to illustrate convergence toperiodic solutions of these population genetics models with positiveprobability.

Convergence of generalized urn models to non-equilibrium attractors