Bifurcations, stability switches and chaos in a diffusive predator-prey model with fear response delay

Recent studies demonstrate that the reproduction of prey is suppressed by the fear of predators. However, it will not respond immediately to fear, but rather reduce after a time lag. We propose a diffusive predator-prey model incorporating fear response delay into prey reproduction. Detailed bifurcation analysis reveals that there are three different cases for the effect of the fear response delay on the system: it might have no effect, both stabilizing and destabilizing effect, or destabilizing effect on the stability of the positive equilibrium, respectively, which are found by numerical simulations to correspond to low, intermediate or high level of fear. For the second case, through ordering the critical values of Hopf bifurcation, we prove the existence of stability switches for the system. Double Hopf bifurcation analysis is carried out to better understand how the fear level and delay jointly affect the system dynamics. Using the normal form method and center manifold theory, we derive the normal form of double Hopf bifurcation, and obtain bifurcation sets around double Hopf bifurcation points, from which all the dynamical behaviors can be explored, including periodic solutions, quasi-periodic solutions and even chaotic phenomenon.