The diffusive logistic equation on periodically evolving domains

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) under bar (0), while the species become extinct if it is above the critical value (D) over bar (0). Finally, we analyze the effect of domain evolution rate on the persistence of a species. Precisely, it depends on the average value (rho(-2)) over bar, where p(t) is the domain evolution rate, and (rho(-2)) over bar = 1/T integral(T)(0) 1/rho(2)(t)dt, If (rho(-2)) over bar > 1, the periodical domain evolution has a negative effect on the persistence of a species. If (rho(-2)) over bar < 1, the periodical domain evolution has a positive effect on the persistence of a species. If <(rho(-2))over bar> = 1, the periodical domain evolution has no effect on the persistence of a species. Numerical simulations are also presented to illustrate the analytical results. (C) 2017 Elsevier Inc. All rights reserved.