Spatiotemporal patterns of reaction–diffusion systems with advection mechanisms on large-scale regular networks

This paper primarily investigates the dynamics behaviour of reaction–diffusion systems with advection mechanisms on network structures. Introducing the advection mechanism from continuous space to network structure, this study uses the Laplacian matrix of a directed network to describe it and sets up a network-organized reaction–diffusion system with directed migration mechanism based on this. Firstly, we derived the Turing instability conditions of the system under certain restrictions. We then focus on the Turing dynamical of the system on a type of composite circular networks and a type of composite torus networks. Furthermore, through numerical methods, we reveal the existence of travelling Turing patterns in the system under various circumstances, including the significant case where the system only has diffusion behaviours but no directed migration. Simultaneously, the result of numerical simulations shows that on the network structures presented in this paper, increasing the global directed migration strength promotes homogeneity among the connected nodes; increasing the difference of directed migration strengths between two groups is conducive to the occurrence of Turing instability.