Quasi-stationary distributions for randomly perturbed dynamical systems

Abstract

We analyze quasi-stationary distributions $\{\mu^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains $\{X^{\varepsilon}\}_{\varepsilon>0}$ 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 $\mu_{\varepsilon}$ 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.