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Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model

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  • Leonid Shaikhet

    (Department of Mathematics, Ariel University, Ariel 40700, Israel)

  • Andrei Korobeinikov

    (School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China)

Abstract

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. 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, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations.

Suggested Citation

  • Leonid Shaikhet & Andrei Korobeinikov, 2024. "Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model," Mathematics, MDPI, vol. 12(10), pages 1-8, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1588-:d:1397613
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    References listed on IDEAS

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    1. Vadillo, Fernando, 2019. "Comparing stochastic Lotka–Volterra predator-prey models," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 181-189.
    2. Rudnicki, Ryszard, 2003. "Long-time behaviour of a stochastic prey-predator model," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 93-107, November.
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