The Lottery Model for Ecological Competition in Nonstationary Environments

Abstract

A two-species lottery competition model with nonstationary reproduction and mortality rates of both species is studied. First, a diffusion approximation for the fraction of sites occupied by each adult species is derived as the continuum limit of a classical discrete-time lottery model. Then a nonautonomous SDE on sites occupied by the species as well as a Fokker-Planck equation on its transitional probability are developed. Existence, uniqueness, and dynamics of solutions for the resulting SDE are investigated, from which sufficient conditions for the existence of a time-dependent limiting process and coexistence of species in the sense of stochastic persistence are established. A unique classical solution to the Fokker-Planck equation is also proved to exist and shown to be a probability density. Numerical simulations are presented to illustrate the theoretical results. © 2021 Society for Industrial and Applied Mathematics