Mean Exit Time and Escape Probability for the Stochastic Logistic Growth Model with Multiplicative {\alpha}-Stable L\'evy Noise

TL;DR

This paper investigates the stochastic logistic growth model for fish populations under Gaussian and non-Gaussian noise, analyzing extinction times, escape probabilities, and the effects of various parameters through PDEs and numerical methods.

Contribution

It introduces a comprehensive analysis of mean exit times and escape probabilities in a stochastic logistic model with multiplicative {\

Findings

MET is finite if Gaussian noise intensity is below a threshold

Higher stability index {\

Numerical solutions reveal parameter effects on extinction probabilities

Abstract

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker-Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness, and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker-Planck equation as growth rate r, carrying capacity K, the intensity of Gaussian noise ${\lambda}$, noise intensity ${\sigma}$ and stability index ${\alpha}$ vary. The MET from the interval (0,1) at the right boundary is finite if ${\lambda} <{\sqrt2}$. For ${\lambda} > {\sqrt2}$, the MET from (0,1) at this boundary is infinite. A larger stability index ${\alpha}$ is less likely to lead to the extinction of the fish population.