Hexagonal patterns and resonant Turing–Turing bifurcation in the Brusselator model with cross-diffusion on a 2D domain

Besides self-diffusion, cross-diffusion also plays an important role in biochemical reactions. This paper focuses on the 2D Brusselator model with cross-diffusion, and investigates the effects of cross-diffusion on spatial pattern formation (e.g., hexagonal patterns arising from resonant Turing–Turing bifurcation). Firstly, with the aid of linear stability analysis, we establish the conditions for the occurrence of (equivariant) Turing/Hopf bifurcations and their interactions (e.g., (equivariant) Turing-(equivariant) Turing bifurcation, (equivariant) Turing–Hopf bifurcation), and finally determine the first (Turing/Hopf) bifurcation curves and stability regions of the unique coexistence equilibrium. Roughly speaking, the emergence of cross-diffusion renders the coexistence equilibrium more prone to losing stability through (equivariant) Turing bifurcation, but exerts no influence on the occurrence of Hopf bifurcation. Then, with the aid of the normal form method and center manifold theory, we establish the third-order normal form of resonant Turing–Turing bifurcation for the 2D Brusselator model with cross-diffusion, and then investigate spatial hexagonal pattern formation arising from the interacting modes (0,2)/(2,1), and finally theoretically predict and numerically display bistable spatial hexagonal patterns, their coexistence with uniform/stripe patterns, as well as transient flipped hexagonal patterns.