Version 1
: Received: 19 March 2024 / Approved: 20 March 2024 / Online: 20 March 2024 (10:23:40 CET)
How to cite:
Shaikhet, L.; Korobeinikov, A. Persistence and Stochastic Extinction in a Lotka-Volterra Predator-Prey Stochastically Perturbed Model. Preprints2024, 2024031205. https://doi.org/10.20944/preprints202403.1205.v1
Shaikhet, L.; Korobeinikov, A. Persistence and Stochastic Extinction in a Lotka-Volterra Predator-Prey Stochastically Perturbed Model. Preprints 2024, 2024031205. https://doi.org/10.20944/preprints202403.1205.v1
Shaikhet, L.; Korobeinikov, A. Persistence and Stochastic Extinction in a Lotka-Volterra Predator-Prey Stochastically Perturbed Model. Preprints2024, 2024031205. https://doi.org/10.20944/preprints202403.1205.v1
APA Style
Shaikhet, L., & Korobeinikov, A. (2024). Persistence and Stochastic Extinction in a Lotka-Volterra Predator-Prey Stochastically Perturbed Model. Preprints. https://doi.org/10.20944/preprints202403.1205.v1
Chicago/Turabian Style
Shaikhet, L. and Andrei Korobeinikov. 2024 "Persistence and Stochastic Extinction in a Lotka-Volterra Predator-Prey Stochastically Perturbed Model" Preprints. https://doi.org/10.20944/preprints202403.1205.v1
Abstract
The classical Lotka-Volterra predator-prey model is globally stable and uniformly persistent. However, in real-life biosystems extinction of species due to stochastic effects is possible. In this paper, we consider the classical Lotka-Volterra predator-prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and the precise necessary conditions for the stochastic system persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type, and, in order to establish the sufficient conditions for extinction, we found the conditions for the stabilization.
The analytical results are illustrated by numerical simulations.
{\bf Keywords:} stochastic perturbations, white noise, Ito's stochastic differential equation, the Lyapunov functions method,
stability in probability, stabilization by noise, stochastic extinction, persistence.
Keywords
Ito's stochastic differential equation; the Lyapunov functions method; stability in probability; stabilization by noise; stochastic extinction; persistence
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
