The Annals of Applied Probability

Quasi-stationary distributions for randomly perturbed dynamical systems

Mathieu Faure and Sebastian J. Schreiber

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 2 (2014), 553-598.

Dates
First available in Project Euclid: 10 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1394465365

Digital Object Identifier
doi:10.1214/13-AAP923

Mathematical Reviews number (MathSciNet)
MR3178491

Zentralblatt MATH identifier
1334.60137

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]
Secondary: 92D25: Population dynamics (general) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations

Keywords
Random perturbations quasi-stationary distributions large deviations nonlinear branching process

Citation

Faure, Mathieu; Schreiber, Sebastian J. Quasi-stationary distributions for randomly perturbed dynamical systems. Ann. Appl. Probab. 24 (2014), no. 2, 553--598. doi:10.1214/13-AAP923. https://projecteuclid.org/euclid.aoap/1394465365


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