The Steady State Bifurcation for General Network-Organized Reaction-Diffusion Systems and Its Application in a Metapopulation Epidemic Model

Abstract

Because of the high dimensionality of reaction-diffusion systems on networks, exploration for the bifurcation properties of such systems has been far less thorough than their corresponding parabolic PDE systems. With the overcoming of this long-standing difficulty, we consider the steady state bifurcation for the general network-organized reaction-diffusion system and derive the algorithm of calculating the perturbation incorporated restricted system associated with the simple zero singularity, through seeking out the parameter dependent center manifolds belonging to the unique zero eigenvalue. Rigorous theoretical analysis for the third-order truncated restricted system shows that the complex underlying networks can trigger some bifurcation types differing from the usual supercritical and subcritical pitchfork bifurcations and so enrich the dynamical behaviors of reaction-diffusion process in networks. To illustrate the application for our obtained theoretical results, we conduct a strict steady state bifurcation analysis for a metapopulation epidemic system defined on four typical connected undirected underlying networks, and find that there exist some new multistability phenomena and hysteresis effects in the irregular small-word and scale-free lattice network.