Turing–Hopf Bifurcation of Brusselator Model with Cross-Diffusion and Gene Expression Time Delay

In this paper, we are concerned with the Turing–Hopf bifurcation problem of the Brusselator model with gene expression delay and cross-diffusion. First, by analyzing the stability of nonspatial systems and spatial systems, the stable region and Turing instability region of the model are obtained. Moreover, the conditions of delay-induced Hopf bifurcation and Turing–Hopf bifurcation are established. Then, the Turing–Hopf bifurcation normal form algorithm of this model is derived by using the normal form theory and the central manifold method. Finally, numerical simulations are performed to verify the theoretical results. The results show that the occurrence of mixed spatial-temporal patterns is caused by the interaction of cross-diffusion and delay. In contrast to self-diffusion, the presence of cross-diffusion enlarges the Turing instability region. Meanwhile, instability of the positive equilibrium solution or the failure of Turing instability through Turing–Hopf bifurcation can also result from variations in the time delay.