Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

In this paper, we study the delayed reaction–diffusion Schnakenberg systems with Neumann boundary conditions. Sufficient and necessary conditions for the occurrence of Turing instability are obtained, and the existence of Turing, Hopf and Turing–Hopf bifurcation for the model are also established. Furthermore, for Turing–Hopf bifurcation, the explicit formula of the truncated normal form up to third order is derived. With the aid of these formulas, we determine the regions on two parameters plane, on which a pair of stable spatially inhomogeneous steady states and a pair of stable spatially inhomogeneous periodic solutions exist, respectively. The theoretical results not only reveals the joint effect of diffusion and delay on the patterns that the model can exhibit, but also explain the phenomenon that time delay may induce a failure of Turing instability, found by Gaffney and Monk (Bull Math Biol 68(1):99–130, 2006).