Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

TL;DR

This paper extends the analysis of stochastic Lotka-Volterra food chains to include degenerate cases with species subject to random perturbations, establishing conditions for persistence, invariant measures, and convergence rates.

Contribution

It introduces new conditions for stochastic persistence and convergence rates in degenerate stochastic Lotka-Volterra models, expanding previous non-degenerate results.

Findings

Existence of a unique invariant measure with smooth density under persistence conditions.

Convergence to the invariant measure can be exponential or polynomial depending on intra-specific competition.

Degenerate cases with species subject to randomness are now characterized in terms of persistence and convergence.

Abstract

We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A. Hening and D. Nguyen. They provided conditions for stochastic persistence and extinction. In this paper, we extend their results to the degenerate situation in which the top or the bottom species is subject to random perturbations. Under the persistence condition, there exists a unique invariant probability measure supported by the interior of $\mathbb{R}_+^n$ having a smooth density. Moreover, we study a more general model, in which we give new conditions which make it possible to characterise the convergence of the semi-group towards the unique invariant probability measure either at an exponential rate or at a polynomial one. This will be used in the stochastic Lotka-Volterra food chain to see that if intra-specific competition occurs for all species, the rate of convergence is exponential while in the other cases it is polynomial.