Pattern formation for reactive species undergoing anisotropic diffusion

TL;DR

This paper investigates how anisotropic diffusion affects Turing pattern formation in reaction-diffusion systems, deriving conditions for instability and demonstrating pattern emergence under direction-sensitive diffusion constants.

Contribution

It introduces a mathematical framework for Turing instabilities with anisotropic diffusion and validates the results through numerical simulations.

Findings

Patterns form even when classical Turing conditions are violated along certain directions.

Instability can occur when the activator diffuses faster than the inhibitor in specific directions.

Pattern formation is influenced by the directional sensitivity of diffusion constants.

Abstract

Turing instabilities for a two species reaction-diffusion systems is studied under anisotropic diffusion. More specifically, the diffusion constants which characterize the ability of the species to relocate in space are direction sensitive. Under this working hypothesis, the conditions for the onset of the instability are mathematically derived and numerically validated. Patterns which closely resemble those obtained in the classical context of isotropic diffusion, develop when the usual Turing condition is violated, along one of the two accessible directions of migration. Remarkably, the instability can also set in when the activator diffuses faster than the inhibitor, along the direction for which the usual Turing conditions are not matched.