Turing–Hopf Bifurcation Coinduced by Diffusion and Delay in Gierer–Meinhardt Systems

This paper investigates the impact of gene expression time delay and diffusion on the dynamic behavior of a class of Gierer–Meinhardt systems under Neumann boundary conditions. It provides necessary and sufficient conditions for the emergence of Hopf, Turing, and Turing–Hopf bifurcations. Utilizing the normal form of the Turing–Hopf bifurcation, the spatiotemporal dynamics close to the bifurcation point are categorized into six types, encompassing spatially inhomogeneous and homogeneous periodic solutions, as well as spatially homogeneous and inhomogeneous steady states, along with their transitions. Notably, it’s observed that the systems may lack stable spatially inhomogeneous periodic solutions under specific parameters, despite the emergence of Turing–Hopf bifurcation. These findings are supported by numerical simulations.