Analytic Solution of a Stochastic Richards Equation driven by Brownian motion

Abstract

In this paper we present a new result on an analytical study of stochastic logistic equations of Richards-type for population growth. First, we introduce the Richards equation as a generalization of the classical Verhulst equation which allows a flexibility in the sigmoid shape of the solution curve. Next, the stochastic equation is formed by adding a multiplicative white noise in the corresponding deterministic Richards equation. Our goal is to solve the stochastic Richards equation and investigate some of the qualitative properties of the solution. As a main result, an exact expression for the solution of the stochastic Richards equation is obtained by using tools from the Itô calculus. Some qualitative aspects of the solution, such as long time behavior and noise-induced transition, will be also discussed within the framework of diffusion processes theory. We also give a simulation of the solution of the stochastic Richards equation to illustrate the role of the so-called allometric parameter.