Analyzing Spatiotemporal Dynamics of the Delayed Gierer–Meinhardt Model on a Network

The Gierer–Meinhardt (G–M) model can be used to describe how cells rely on the interactions of chemical substances to form complex morphological structures during the development of an organism. In this paper, the chemical reaction molecules within the organism are treated as network nodes, and the interaction pathways between these molecules are regarded as edges of the network, based on which an activator–inhibitor G–M model is constructed on the network. Taking into account time delays and self-diffusion, the stability of the delayed G–M model on the diffusion network, Hopf bifurcation and its stability and direction, as well as the conditions for Turing bifurcation through linear stability analysis, bifurcation theory, and partial functional differential equation theory are investigated. These conditions are related to time delays, diffusion coefficients, and network connection probability. The research results indicate that when certain conditions are met, regardless of the value of the network connection probability, the stability of the system changes with the variation of time delay. When the time delay is minimal, changes in the network structure can influence the occurrence of Turing bifurcation. Additionally, we have identified regions of Turing instability related to network connection probability and self-diffusion. Finally, numerical simulations are presented to validate the theoretical results.