  The memory-based diffusion systems have wide applications in practice. Hopf
bifurcations are observed from such systems. To meet the demand for computing
the normal forms of the Hopf bifurcations of such systems, we develop an
effective new algorithm where the memory delay is treated as the perturbation
parameter. To illustrate the effectiveness of the algorithm, we consider a
diffusive predator-prey system with memory-based diffusion and Holling type-II
functional response. By employing this newly developed procedure, we
investigate the direction and stability of the delay-induced mode-1 and mode-2
Hopf bifurcations. Numerical simulations confirm our theoretical findings, that
is the existence of stable spatially inhomogeneous periodic solutions with
mode-1 and mode-2 spatial patterns, and the transition from the unstable mode-2
spatially inhomogeneous periodic solution to the stable mode-1 spatially
inhomogeneous periodic solution.
