Dynamics of Gierer-Meinhardt System on Network: Influence of Time Delay

This paper extends the classical Gierer-Meinhardt system by incorporating both gene expression time delay and network diffusion, providing a comprehensive model of complex biological phenomena. The model is represented by a system of equations that combine discrete space and continuous time. We establish the stability and instability conditions and examine how gene expression time delay induces Hopf bifurcation. Using the central manifold theory, we derive the expressions to determine the stability and direction of the Hopf bifurcation. Furthermore, we explore Turing instability, which is caused by the gene expression time delay and network diffusion. Numerical simulations validate our theoretical results, illustrating the detailed dynamic behaviors arising from these effects.