Hopf bifurcation for general network-organized reaction-diffusion systems and its application in a multi-patch predator-prey system

Abstract

For decades, the network-organized reaction-diffusion models have been widely used to study ecological and epidemiological phenomena in discrete space. However, the high dimensionality of these nonlinear systems places a long-standing restriction to develop the normal forms of various bifurcations. In this paper, we take an important step to present a rigorous procedure for calculating the normal form associated with the Hopf bifurcation of the general network-organized reaction-diffusion systems, which is similar to but can be much more intricate than the corresponding procedure for the extensively explored PDE systems. To show the potential applications of our obtained theoretical results, we conduct the detailed Hopf bifurcation analysis for a multi-patch predator-prey system defined on any undirected connected underlying network and on the particular non-periodic one-dimensional lattice network. Remarkably, we reveal that the structure of the underlying network imposes a significant effect on the occurrence of the spatially nonhomogeneous Hopf bifurcations.