Verhulst model with Levy white noise excitation

TL;DR

This paper analyzes the Verhulst population model under non-Gaussian Levy white noise, deriving exact probability distributions and relaxation times for various noise types, revealing complex transition behaviors and nonmonotonic relaxation dynamics.

Contribution

It provides exact solutions for the Verhulst model with non-Gaussian Levy noise, including probability distributions and relaxation times, extending understanding of stochastic population dynamics.

Findings

Exact probability distributions for population density under Levy noise.

Transition from trimodal to bimodal distribution induced by Levy noise.

Nonmonotonic nonlinear relaxation time as noise intensity varies.

Abstract

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Levy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise, (ii) noise with a probability density of increments expressed in terms of Gamma function, and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Levy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.