Diffusion-Driven Instability of a fourth order system

TL;DR

This paper investigates how replacing the Laplacian with a combination of bi-Laplacian and Laplacian operators affects diffusion-driven instability in reaction-diffusion systems, revealing new phenomena and stability criteria.

Contribution

It introduces a novel analysis of Turing instability using a mixed fourth and second order diffusion operator, expanding understanding of pattern formation mechanisms.

Findings

Identifies new phenomena when bi-Laplacian and Laplacian compete.

Characterizes Turing space based on system parameters.

Provides criteria for instability related to domain size and tension.

Abstract

We analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.