Symmetry Analysis of the Stochastic Logistic Equation

Abstract

We apply the recently developed theory of symmetry of stochastic differential equations to stochastic versions of the logistic equation; these may have environmental or demographical noise, or both—in which case we speak of the complete model. We study all these cases, both with constant and with non-constant noise amplitude, and show that the only one in which there are nontrivial symmetries is that of the stochastic logistic equation with (constant amplitude) environmental noise. In this case, the general theory of symmetry of stochastic differential equations is used to obtain an explicit integration, i.e., an explicit formula for the process in terms of any single realization of the driving Wiener process.

Keywords

• symmetry
• stochastic differential equations
• logistic equation
• integrable systems