Hopf bifurcation in an activator-inhibitor system with network

Abstract A network is introduced to describe the activator–inhibitor system, where the network structure represents the movement directions of molecule random walk. We show that a Hopf bifurcation occurs in the activator–inhibitor system by the linear stability analysis. By an extension of the center manifold approach, we also prove that the Hopf bifurcation is stable and its direction is backward.