Logarithmic Lipschitz norms and diffusion-induced instability

TL;DR

This paper demonstrates that contractive ODE systems maintain their contractiveness when diffusion is introduced, preventing Turing-type diffusive instabilities, with an application to a biochemical system.

Contribution

It proves that diffusion does not induce instability in contractive systems, extending the understanding of stability in reaction-diffusion models.

Findings

Contractive ODE systems remain stable with diffusion.

Diffusive instabilities like Turing patterns cannot occur in such systems.

A biochemical system satisfies the conditions for stability.

Abstract

This paper proves that contractive ordinary differential equation systems remain contractive when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems. An important biochemical system is shown to satisfy the required conditions.