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

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