Turing patterns on a two-component isotropic growing system. Part 3: Time dependent conditions and linear growth

TL;DR

This paper establishes general conditions for Turing pattern formation in a growing or shrinking domain with time-dependent homogeneous states, supported by numerical simulations of the Brusselator system.

Contribution

It extends Turing pattern theory to include time-dependent homogeneous states and provides numerical validation for linear growth and shrinkage scenarios.

Findings

Patterns depend on domain growth rate and time-dependent parameters.

Amplitude and wave number of patterns vary with domain size.

Time acts as a bifurcation parameter influencing pattern emergence.

Abstract

We propose general conditions for the emergence of Turing patterns in a domain that changes size through homogeneous growth/shrinkage based on the qualitative changes of a potential function. For this part of the work, we consider the most general case where the homogeneous state of the system depends on time. Our hypotheses for the Turing conditions are corroborated with numerical simulations of increasing/decreasing domains of the Brusselator system for the linear growth/shrinking case. The simulations allow us to understand the characteristics of the pattern, its amplitude, and wave number, in addition to allowing us to glimpse the role of time as a bifurcation parameter.