Mean exit time and escape probability for the stochastic logistic growth model with multiplicative α-stable Lévy noise

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 ${\lambda} {\sqrt2}$, 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.