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

Pattern formation in reaction-diffusion information propagation model on multiplex simplicial complexes

“…So the research on the reaction-diffusion system comes into being [6,[31][32][33][34][35]. Inspired by Turing's ideas, Turing pattern on the network has developed rapidly [36][37][38]. One of the pioneering works is that Nakao and Mikhailov study Turing pattern under large random networks and found multiple steady-state coexistences and hysteresis effects [36].…”