Population dynamics in presence of state dependent fluctuations

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 Itô kind; or (ii) to integrals of Stratonovich kind. The systems of stochastic differential equations are reduced to the corresponding Fokker–Planck equations. For the Itô case and for the case of 1 population analytic results are obtained for the stationary p.d.f. of the population density. For the case of more than one population and for both the Itô case and Stratonovich case the detailed balance conditions are not satisfied. As a result the exact analytic solutions of the corresponding Fokker–Planck equations for the stationary p.d.f.s for the population densities are not known. We obtain approximate solutions for this case by the method of adiabatic elimination.