Scrutinizing critical dynamics of reaction-diffusion models on complex networks

Reaction-diffusion (RD) processes on complex networks, along with their associated critical dynamics, have attracted considerable attention and spurred intensive transdisciplinary research due to their ubiquity in both natural and manmade systems. Although bifurcation theory for general differential systems serves as a powerful tool for analyzing critical dynamics, applying it directly to RD models on networks remains challenging due to the models' ultra-high dimensionality. To address this longstanding issue, we rigorously develop a general and practical framework for computing various bifurcation normal forms. In particular, we derive the normal forms for Hopf bifurcation, steady-state bifurcation, and Turing-Hopf bifurcation. Our theoretical analysis reveals that critical dynamics in such models are jointly determined by the network structure and the local node dynamics. Specifically, we show that the ratio-dependent predator-prey model on complex networks can exhibit novel bifurcation types, giving rise to new forms of bistability and heterogeneous oscillatory behavior, whereas the FitzHugh-Nagumo model on complex brain networks only undergoes conventional bifurcations. This work provides a foundational nonlinear analysis tool for understanding and exploring the critical dynamics of RD processes on complex networks.