Turing–Hopf Bifurcation in Diffusive Gierer–Meinhardt Model

Abstract

Gierer–Meinhardt system is a molecularly plausible model to describe pattern formation. When gene expression time delay is added, the behavior of the Gierer–Meinhardt model profoundly changes. In this paper, we study the delayed reaction–diffusion Gierer–Meinhardt system with Neumann boundary condition. Necessary and sufficient conditions for the occurrence of Turing instability, Hopf bifurcation and Turing–Hopf bifurcation deduced by diffusion and gene expression time delay are obtained through linear stability analysis and root distribution of the characteristic equation with two transcendental terms. With the aid of the normal form Turing–Hopf bifurcation and numerical simulations, we theoretically and numerically obtain the expected solutions including stable spatially inhomogeneous steady states, stable spatially homogeneous periodic orbit and stable spatially inhomogeneous periodic orbit from Turing–Hopf bifurcation.