Spatiotemporal Patterns in a General Networked Activator-Substrate Model

Abstract For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one with the change of the connection probability of the nodes, this is different from the pattern formation in the continuous media systems.