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

Yan-Xia Ren, Jie Xiong, Xu Yang and Xiaowen Zhou

Stochastic Processes and their Applications, 2022, vol. 150, issue C, 50-90

Abstract: Applying the Foster–Lyapunov type criteria and a martingale method, 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 Lévy 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.

Keywords: Continuous-state branching process; Nonlinear branching; Stochastic Lotka–Volterra population dynamics; Foster–Lyapunov type criteria; Extinction; Extinguishing (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2022.04.005

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