On the extinction-extinguishing dichotomy for a stochastic Lotka-Volterra type population dynamical system

TL;DR

This paper analyzes a stochastic two-species population model to determine conditions under which one or both species go extinct, revealing nuanced probabilistic extinction behaviors influenced by their interactions.

Contribution

It provides sharp conditions for extinction or extinguishing of species in a stochastic Lotka-Volterra model with complex jump processes.

Findings

Conditions for almost sure extinction of Y.

Conditions for almost sure extinguishing of Y.

Probabilistic coexistence or extinction scenarios.

Abstract

We study a two-dimensional process $(X, Y)$ arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive L\'evy random measures. Both processes $X$ and $Y$ can be identified as continuous-state nonlinear branching processes where the evolution of $Y$ is negatively affected by $X$. Assuming that process $X$ extinguishes, i.e. it converges to $0$ but never reaches $0$ in finite time, and process $Y$ converges to $0$, we identify rather sharp conditions under which the process $Y$ exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.