Turing-Hopf bifurcation analysis and normal form of a diffusive Brusselator model with gene expression time delay

In this paper, by incorporating the gene expression time delay into a diffusive Brusselator model, a diffusive Brusselator model with gene expression time delay is proposed, and the Turing-Hopf bifurcation of the proposed model subject to homogeneous Neumann boundary condition is investigated. Firstly, the condition for the occurrence of Turing instability is deduced, and the existence of Turing bifurcation, Hopf bifurcation and Turing-Hopf bifurcation for the proposed model is also established. Then by a slight modification, the normal form on the center manifold near the Turing-Hopf singularity for the proposed model is derived by using the algorithm of calculating the normal form of Turing-Hopf bifurcation for the general reaction diffusion system with delay. With the aid of the obtained third-order truncated normal form of the proposed model, the spatiotemporal dynamics corresponding to six different regions in the parameter plane are analyzed. Moreover, by the corresponding relationships between the equilibrium points of the normal form and the solutions of original system, the dynamics of the original system can also be obtained. Finally, some numerical simulations are carried out to support the theoretical findings.