  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.
