Global stability of a reaction-diffusion predator–prey model with a nonlocal delay

In this paper, a reaction-diffusion predator–prey system with nonlocal delay due to the gestation of the predator and homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each boundary steady state is established. The existence of Hopf bifurcations at the positive steady state is also discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the proposed problem by using the method of upper–lower solutions and its associated monotone iteration scheme, respectively. Numerical simulations are carried out to illustrate the main results.