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The memory-based diffusion systems have wide applications in practice. Hopfbifurcations are observed from such systems. To meet the demand for computingthe normal forms of the Hopf bifurcations of such systems, we develop aneffective new algorithm where the memory delay is treated as the perturbationparameter. To illustrate the effectiveness of the algorithm, we consider adiffusive predator-prey system with memory-based diffusion and Holling type-IIfunctional response. By employing this newly developed procedure, weinvestigate the direction and stability of the delay-induced mode-1 and mode-2Hopf bifurcations. Numerical simulations confirm our theoretical findings, thatis the existence of stable spatially inhomogeneous periodic solutions withmode-1 and mode-2 spatial patterns, and the transition from the unstable mode-2spatially inhomogeneous periodic solution to the stable mode-1 spatiallyinhomogeneous periodic solution.

The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system