  In this paper, we derive the algorithm for calculating the normal form of the
double Hopf bifurcation that appears in a memory-based diffusion system via
taking memory-based diffusion coefficient and the memory delay as the
perturbation parameters. Using the obtained theoretical results, we study the
dynamical classification near the double Hopf bifurcation point in a
predator-prey system with Holling type II functional response. We show the
existence of different kinds of stable spatially inhomogeneous periodic
solutions, the transition from one kind to the other as well as the coexistence
of two types of periodic solutions with different spatial profiles by varying
the memory-based diffusion coefficient and the memory delay.
