  In this paper, we consider a diffusive predator-prey system with spatial
memory and predator-taxis. Since in this system, the memory delay appears in
the diffusion term, and the diffusion term is nonlinear, the classical normal
form of Hopf bifurcation in the reaction-diffusion system with delay can't be
applied to this system. Thus, in this paper, we first derive an algorithm for
calculating the normal form of Hopf bifurcation in this system. Then in order
to illustrate the effectiveness of our newly developed algorithm, we consider
the diffusive Holling-Tanner model with spatial memory and predator-taxis. The
stability and Hopf bifurcation analysis of this model are investigated, and the
direction and stability of Hopf bifurcation periodic solution are also
researched by using our newly developed algorithm for calculating the normal
form of Hopf bifurcation. At last, we carry out some numerical simulations, two
stable spatially inhomogeneous periodic solutions corresponding to the mode-1
and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis
results.
