Instability and bifurcation analysis of a diffusive predator–prey model with fear effect and delay.

In this paper, a delayed diffusive predator–prey model with fear effect under Neumann boundary conditions is considered. For the system without diffusion and delay, the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues. Then, the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed, respectively. Moreover, the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients. Furthermore, the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory. Finally, some numerical simulations are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]