Waves and distributions connected to systems of interacting populations

TL;DR

This paper explores wave phenomena and probability distributions in models of interacting populations, deriving exact solutions for wave migration and stationary distributions, with implications for understanding population dynamics under various fluctuation regimes.

Contribution

It introduces a modified method of simplest equation to find exact traveling-wave solutions and derives stationary distributions for population densities considering both deterministic and stochastic effects.

Findings

Exact traveling-wave solutions for systems with 1 or 3 populations.

Derived stationary distributions for population densities.

Calculated expected extinction times for populations.

Abstract

We discuss two cases that can be connected to the dynamics of interacting populations: (I.) density waves for the case or negligible random fluctuations of the populations densities, and (II.) probability distributions connected to the model equations for of spatially averaged populations densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density. For the case (I.) we consider model equations containing polynomial nonlinearities. Such nonlinearities can 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). In the both cases we can apply the modified method of simplest equation to obtain exact traveling-wave solutions connected to migration of population members. Such solutions are obtained for systems consisting of 1 or 3 populations respectively. For the case (II.) we discuss model equations of the Fokker-Planck kind for the evolution of the statistical distributions of population densities. We derive several stationary distributions for the population density and calculate the expected exit time connected to the extinction of the studied population.