Dynamics of a delayed diffusive predator-prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey-predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction-diffusion model without time delay.