Turing patterns on a two-component isotropic growing system. Part 2: Conditions based on a potential function for exponential growth/shrinkage

TL;DR

This paper establishes conditions for Turing pattern formation in growing or shrinking domains based on a potential function approach, extending classical criteria to dynamic systems and validating with numerical simulations.

Contribution

It introduces a potential function-based framework for analyzing Turing patterns in exponentially growing or shrinking domains, generalizing traditional fixed-domain conditions.

Findings

Conditions recover classical Turing criteria in fixed domains

Numerical simulations validate the theoretical conditions

Patterns' amplitude and wavenumber evolve predictably over time

Abstract

We propose conditions for the emergence of Turing patterns in a domain that changes in size by homogeneous growth/shrinkage. These conditions to determine the bifurcation are based on considering the geometric change of a potential function whose evolution determines the stability of the trajectories of all the Fourier modes of the perturbation. For this part of the work we consider the situation where the homogeneous state of the system are constant concentrations close to its stationary value, as occurs for exponential growth/decrease. This proposal recovers the traditional Turing conditions for two-component systems in a fixed domain and is corroborated against numerical simulations of increasing/decreasing domains of the Brusselator reaction system. The simulations carried out allowed us to understand some characteristics of the pattern related to the evolution of its amplitude and wavenumber and allow us to anticipate which features strictly depend on its temporal evolution.