Quasi-stationary distributions for randomly perturbed dynamical systems

Abstract

We analyze quasi-stationary distributions $\{{\mu }^{\epsilon }{\}}_{\epsilon >0}$ of a family of Markov chains $\{X^{\epsilon }{\}}_{\epsilon >0}$ that are random perturbations of a bounded, continuous map $F:M\to M$, where $M$ is a closed subset of ${\mathbb{R}}^k$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_0\subset M$ such that $F(M_0)=M_0$ and $F(M\setminus M_0)=M\setminus M_0$. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for $F$ (i.e., an attractor for $F$ in $M\setminus M_0$), then the weak* limit points of ${\mu }_{\epsilon }$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.

Published in at http://dx.doi.org/10.1214/13-AAP923 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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