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 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. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper presents an effective algorithm for computing the normal forms of Hopf bifurcations in memory-based diffusion systems, using a diffusive predator-prey system as a case study. The research demonstrates the direction and stability of delay-induced Hopf bifurcations and confirms the existence of stable spatially inhomogeneous periodic solutions with mode-1 and mode-2 spatial patterns through numerical simulations.