Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces.

Area of Science:

• Computational Mathematics
• Mathematical Modeling
• Chemical Kinetics

Background:

• Reaction-diffusion systems are fundamental to modeling pattern formation in biological and chemical systems.
• Cross-diffusion introduces complex dynamics not captured by classical models.
• Simulating these systems on evolving domains presents significant computational challenges.

Purpose of the Study:

• To present the first application of the finite element method (FEM) for reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.
• To investigate pattern formation arising from these systems under cross-diffusion effects.
• To demonstrate the robustness and versatility of the proposed FEM approach.

Main Methods:

• Development of a FEM framework for reaction-diffusion systems incorporating linear cross-diffusion.
• Approximation of evolving domains and surfaces using triangulations.
• Definition of a finite element space for functions on these triangulated domains/surfaces.
• Numerical computation of patterns using parameter values specific to the cross-diffusion space.

Main Results:

• Successful application of FEM to solve reaction-diffusion systems with cross-diffusion on evolving domains/surfaces.
• Generation of novel patterns driven by cross-diffusion effects, distinct from classical reaction-diffusion patterns.
• Demonstration of the methodology's ability to handle complex domain/surface evolution, including isotropic, anisotropic, and concentration-driven growth.

Conclusions:

• The presented FEM is a robust, flexible, and versatile tool for analyzing reaction-diffusion systems with cross-diffusion on dynamic geometries.
• Cross-diffusion plays a crucial role in pattern formation, enabling phenomena not observed in standard models.
• The methodology provides a powerful framework for studying complex spatio-temporal dynamics in evolving systems.