The diffusive logistic equation on periodically evolving domains

Abstract

A diffusive logistic equation on n-dimensional periodically and isotropically evolving domains is investigated. We first derive the model and present the eigenvalue problem on evolving domains. Then we prove that the species persists if the diffusion rate d is below the critical value D_0, while the species become extinct if it is above the critical value D‾0. Finally, we analyze the effect of domain evolution rate on the persistence of a species. Precisely, it depends on the average value ρ−2‾, where ρ(t) is the domain evolution rate, and ρ−2‾=1T∫0T1ρ2(t)dt. If ρ−2‾>1, the periodical domain evolution has a negative effect on the persistence of a species. If ρ−2‾<1, the periodical domain evolution has a positive effect on the persistence of a species. If ρ−2‾=1, the periodical domain evolution has no effect on the persistence of a species. Numerical simulations are also presented to illustrate the analytical results.