Turing–Hopf bifurcation co-induced by cross-diffusion and delay in Schnakenberg system

The dynamics of Schnakenberg model subjected to gene expression time delay and cross-diffusion is investigated. It is shown that the presence of cross-diffusion allows the enlarged Turing instability region compared with self-diffusion and the variation of time delay can either lead to destabilization of the positive equilibrium or fail Turing instability through Turing–Hopf bifurcation. A bifurcation analysis of the Schnakenberg model in question has been provided to derive the stable regime, Turing instability regime and also the condition for delay-induced Turing–Hopf bifurcation. With the aid of normal form of Turing–Hopf bifurcation, the spatiotemporal dynamics near this bifurcation point can be divided into six categories where various solutions including stable spatially homogeneous or inhomogeneous periodic solutions, stable steady states and the transition from one type of spatiotemporal patterns to another would exhibit. These spatiotemporal solutions have also been illustrated through some numerical simulations, confirming the theoretical predictions.