Dynamics analysis of a predator-prey model incorporating fear effect in prey species

Fear effects, as spontaneous inherent phenomena among species, ubiquitously manifest in natural ecosystems. In this study, we investigate a Leslie-Gower type predator-prey system incorporating a Holling Type Ⅲ functional response and fear effects under no-flux boundary conditions. For temporal dynamical behaviors, we investigate the Hopf bifurcation for the ordinary differential equations (ODEs). It is found that the system will admit the stable or unstable periodic solution due to supercritical or subcritical Hopf bifurcation based on the first Lyapunov coefficient technique. When we add the diffusion into the system, we rigorously establish the stability conditions of the positive equilibrium. In this manner, the precise existence interval of the Turing instability is obtained so that the spatial profiles of the system can be analyzed. Furthermore, we systematically analyze the existence of Hopf bifurcation in the reaction-diffusion system and characterize the stability of bifurcating periodic solutions by employing normal form theory. The theoretical framework is supported by comprehensive numerical simulations. The research highlights of this paper include the following: (1) The existence of the supercritical and subcritical Hopf bifurcation for the temporal and spatiotemporal predator-prey systems are confirmed; (2) the Turing instability of the spatiotemporal system is found via strict theoretical derivations so that the pattern formation can be observed.