Bifurcation and stability of a diffusive predator-prey model with the fear effect and time delay

The predator-prey system can induce wealth properties with fear effects. In this paper, we propose a diffusive predator-prey model where the influence of fear effects and time delay is considered, under the Dirichlet boundary condition. It follows from the Lyapunov-Schmidt reduction method that there exists a non-homogeneous steady-state solution of the system and the specific expressions are also given. By the aid of bifurcation theory and eigenvalue theory, we also investigate the existence/non-existence and the stability of Hopf bifurcation under three different conditions of bifurcation parameters. Furthermore, the effects of the fear on population density, stability, and Hopf bifurcation are also considered and the results show that the increase of fear effects will reduce the population density, and Hopf bifurcation is more likely difficult to undergo as k increases under some conditions.