  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. 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.

Dynamical Systems