Quasi-Stationary distributions for Randomly perturbed dynamical systems

Details Quasi-Stationary distributions for Randomly perturbed dynamical systems Quasi-Stationary distributions for Randomly perturbed dynamical systems

2011-01-18

arxiv.org/abs/1101.3420v2

Mathematics

Probability

32 pages

Scientific paper

We analyze quasi-stationary distributions $\mu^\epsilon$ 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 subset of $\R^k$. 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.

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