Stochastic Extinction, An Average Lyapunov Function Approach

TL;DR

This paper develops a general theory of average Lyapunov functions to analyze the stability and extinction of Markov processes, with applications to epidemic, ecological, and dynamical systems, often improving existing criteria.

Contribution

It introduces a unified approach using average Lyapunov functions and Lyapunov exponents to determine extinction in stochastic systems, enhancing previous methods.

Findings

Provides new criteria for extinction based on Lyapunov exponents.

Applies the theory to diverse stochastic models including epidemics and ecological systems.

Improves existing results by removing unnecessary assumptions or sharpening conditions.

Abstract

We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. In many examples we improve existing results by removing unnecessary assumptions or providing sharper criteria for the extinction.