Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect

We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. The emergence of quasi-periodic and chaotic phenomenon may indicate that there exists complex dynamical behavior of biological system itself. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.