The spatial memory effect in predator and fear effect in prey are incorporated in a diffusive predator-prey model. We are interested in studying the dynamics generated by the memory effect and fear effect, and mainly study the local stability of coexisting equilibrium, the existence of Hopf bifurcation and the property of Hopf bifurcation. Through the numerical simulations, we show that increasing memory-based diffusion coefficient is not conducive to the stability of the coexisting equilibrium, and the fear effect has both stabilizing and destabilizing effect on the coexisting equilibrium under different parameters.

This paper studied a delayed diffusive predator-prey model with predator and prey harvesting terms. The existence of constant steady-state solutions and Hopf bifurcation were analyzed. Then by applying the central manifold theorem and the normal form method, the direction of the Hopf bifurcation and stability of the bifurcating period solution were studied. Numerical simulations were conducted to confirm the accuracy of the proposed theory. In addition, taking the predator-prey relationship between sharks and tuna as an example, this study investigated the impact of predator harvesting coefficients on the constant steady-state solutions of the system and the time required for the system to reach stability.