Dynamics of a Diffusive Leslie–Gower Model with Both-Density-Dependent Fear Effect

This paper focuses on the dynamics of a diffusive Leslie–Gower model with both-density-dependent fear effect. With regard to the diffusion coefficient and the fear response delay, the existence of Turing and Hopf bifurcations near the interior positive steady-state and the stability of the steady-state with their changes are proved. The interaction influence of the bifurcations is further considered, such as Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations, in order to investigate the complex spatiotemporal behaviors of the species. It is shown that the existence of the diffusion and the fear response delay is conducive to the generation of complex dynamical behavior of the system. Detailed numerical analysis indicates that a tiny fear intensity can stabilize the system and a slightly larger one will bring about rich spatiotemporal dynamics. Moreover, a spatial inhomogeneous quasiperiodic state is introduced by the diffusion and the delay as the fear intensity increases.