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

Xun Cao and Weihua Jiang

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 248, issue C, 42-69

Abstract: 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.

Keywords: Brusselator model; Cross-diffusion; 2D Hexagonal patterns; Resonant Turing–Turing bifurcation; The third-order normal form (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378475426001552
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:248:y:2026:i:c:p:42-69

DOI: 10.1016/j.matcom.2026.04.013

Access Statistics for this article

Mathematics and Computers in Simulation (MATCOM) is currently edited by Robert Beauwens

More articles in Mathematics and Computers in Simulation (MATCOM) from Elsevier
Bibliographic data for series maintained by Catherine Liu ().