On systems of interacting populations influenced by multiplicative white noise

TL;DR

This paper models interacting populations affected by multiplicative white noise, deriving analytical and approximate solutions for their stationary distributions using stochastic differential equations and Fokker-Planck equations.

Contribution

It introduces a stochastic model with polynomial nonlinearities and multiplicative noise, providing new analytical and approximate solutions for population density distributions.

Findings

Analytic stationary PDF for single population case.

Approximate solutions for multi-population cases.

Reduction to Fokker-Planck equations and use of adiabatic elimination.

Abstract

We discuss a model of a system of interacting populations for the case when: (i) the growth rates and the coefficients of interaction among the populations depend on the populations densities: and (ii) the environment influences the growth rates and this influence can be modelled by a Gaussian white noise. The system of model equations for this case is a system of stochastic differential equations with: (i) deterministic part in the form of polynomial nonlinearities; and (ii) state-dependent stochastic part in the form of multiplicative Gaussian white noise. We discuss both the cases when the formal integration of the stochastic differential equations leads: (i) to integrals of Ito kind; or (ii) to integrals of Stratonovich kind. The systems of stochastic differential equations are reduced to the corresponding Fokker-Planck equations. For the Ito case and for the case of 1 population am analytic results is obtained for the stationary PDF of the the population density. For the case of more than one population and for the both Ito case and Stratonovich case the detailed balance conditions are not satisfied and because of this exact analytic solutions of the corresponding Fokker-Plank equations for the stationary PDFs for the population densities are not known. We obtain approximate solutions for this case by the methodology of the adiabatic elimination.