Stochastic Persistence in Infinite Dimensions

TL;DR

This paper develops a general criterion for stochastic persistence in infinite-dimensional models like reaction-diffusion SPDEs, extending finite-dimensional results and analyzing ecological, epidemic, and turbulence models.

Contribution

It introduces a novel average Lyapunov function approach for stochastic persistence in infinite dimensions and proves new well-posedness and nonnegativity results for reaction-diffusion SPDEs.

Findings

Coexistence in Lotka-Volterra models depends on invasion rates.

The criteria apply to ecological, epidemic, and turbulence models.

Extended known results in reaction-diffusion SPDE well-posedness.

Abstract

Motivated by infinite-dimensional ecological and biological models such as reaction-diffusion SPDEs and stochastic functional differential equations, we develop a general criteria for stochastic persistence (coexistence) in terms of an average lyapunov function, which was previously known only in finite dimensions. To apply our results to SPDEs we analyze the projective process, and we employ a combination of mild (stochastic convolution) and variational (lyapunov function) techniques. Our analysis also requires some nontrivial well-posedness and nonnegativity results for reaction-diffusion SPDEs, which we state and prove in great generality, extending the known results in the literature. Finally, we present several examples including ecological models (Lotka-Volterra), an epidemic model (SIR), and a model for turbulence. Notably we show that, as in the SDE case, coexistence in the Lotka-Volterra model is determined by the invasion rates.