Quasi-stationary distributions for randomly perturbed dynamical systems

TL;DR

This paper studies the behavior of quasi-stationary distributions in randomly perturbed dynamical systems, showing they concentrate on attractors of the underlying deterministic map, with applications in biology.

Contribution

It establishes the connection between quasi-stationary distributions of perturbed Markov chains and attractors of the deterministic map under large deviations assumptions.

Findings

Quasi-stationary distributions concentrate on positive attractors.

Results apply to biological models like metapopulations and evolutionary games.

Weak* limit points are supported by attractors of the deterministic system.

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.