Traveling waves and statistical distributions connected to systems of interacting populations

We discuss the following two issues from the dynamics of interacting populations: •(I) density waves for the case or negligible random fluctuations of the population densities,•(II) probability distributions connected to the model equations for spatially averaged population densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density. For the case of issue (I) we consider model equations containing polynomial nonlinearities. Such nonlinearities arise as a consequence of interaction among the populations (for the case of large population densities) or as a result of a Taylor series expansion (for the case of small density of interacting populations). By means of the modified method of the simplest equation we obtain exact traveling-wave solutions of the model equations and these solution. For the case of issue (II) we discuss model equations of the Fokker–Planck kind for the evolution of the statistical distributions of population densities. We derive a few stationary distributions for the population density and calculate the expected exit time associated with the extinction of the studied population.