Pattern formation driven by cross--diffusion in a 2D domain

TL;DR

This paper investigates how cross-diffusion in a 2D reaction-diffusion system with Lotka-Volterra kinetics leads to various spatial patterns through Turing bifurcation, using stability analysis and amplitude equations.

Contribution

It provides a detailed analysis of pattern formation mechanisms driven by cross-diffusion, including classification of bifurcation types and derivation of amplitude equations for different pattern types.

Findings

Cross-diffusion induces Turing bifurcation leading to pattern formation.

Different bifurcation types (regular, degenerate, resonant) are characterized.

The system supports diverse patterns such as rolls, squares, and hexagons.

Abstract

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction-diffusion system with nonlinear diffusion terms and the competitive Lotka-Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, hexagonal patterns.