A New Mechanism Revealed by Cross-Diffusion-Driven Instability and Double-Hopf Bifurcation in the Brusselator System

The pattern dynamics of the Brusselator system with cross-diffusion and gene expression time delay are investigated. Conditions for cross-diffusion-driven instability are established, which reveal that the formation of patterns in the system with cross-diffusion does not follow the “short-range activation” and “long-range inhibition” mechanisms. Moreover, the conditions for the occurrence of Turing bifurcation, Hopf bifurcation, Turing–Hopf bifurcation, and double-Hopf bifurcation are obtained using eigenvalue analysis. The algorithms for calculating the normal forms of double-Hopf bifurcation are given using multiple timescale method, with the coefficients expressed explicitly in terms of the original system parameters. An essential contribution of this paper is the effect of the space element taken into account in this method, which results in that the normal forms of the double-Hopf bifurcation for the Brusselator system are divided into two categories: One is 1:3 strong resonance, and the other consists of other strong resonances, weak resonance, and non-resonance. Different types of spatiotemporal patterns are classified according to the exact partition of the parameter space. It is demonstrated that the Brusselator system can exhibit stable spatially inhomogeneous periodic and quasi-periodic orbits.