Minimum nonlinearity for pattern-forming Turing instability in a mathematical autocatalytic model

TL;DR

This paper mathematically analyzes the conditions under which Turing instability can occur in reaction-diffusion models, concluding that a minimum nonlinearity degree of three is necessary for pattern formation.

Contribution

It establishes a theoretical lower bound on the nonlinearity degree required for Turing pattern formation in autocatalytic models.

Findings

Turing instability requires nonlinearity degree ≥ 3

Mathematical constraints limit pattern formation mechanisms

Provides criteria for reaction-diffusion model design

Abstract

Pattern formation is ubiquitous in nature and the mechanism widely-accepted to underlay them is based on the Turing instability, predicted by Alan Turing decades ago. This is a non-trivial mechanism that involves nonlinear interaction terms between the different species involved and transport mechanisms. We present here a mathematical analysis aiming to explore the mathematical constraints that a reaction-diffusion dynamical model should comply in order to exhibit a Turing instability. The main conclusion limits the existence of this instability to nonlinearity degrees larger or equal to three.