Global attractivity of delayed and nonlocal diffusive logistic models with variable coefficients

Abstract

In this paper, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients which models the evolution of a single species. We establish a sharp threshold dynamic result: there exists a critical value λ ( a ) such that if λ ( a ) < 0 the positive steady state solution of the equation is globally attractive, while if λ ( a ) ≥ 0 the trivial steady state is globally attractive. To this end, we analyze the ω -limit set of the equation and prove that it is a singleton. Moreover, we apply our method to obtain global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.