A computational approach for mode isolation for reaction-diffusion systems on arbitrary geometries

TL;DR

This paper introduces a numerical framework to identify spatial patterns in reaction-diffusion systems on arbitrary geometries by computing eigenpairs of the Laplacian, enabling pattern prediction beyond analytically solvable domains.

Contribution

The authors develop a method to numerically compute Laplacian eigenpairs on arbitrary domains, extending pattern analysis techniques to complex geometries.

Findings

Mode isolation is effective for low eigenvalues.

Multiple eigenvalues can produce linear combinations of eigenfunctions.

Pattern formation is robust across similar surfaces regardless of boundary presence.

Abstract

In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.