The role of spatial dimension in the emergence of localised radial patterns from a Turing instability

TL;DR

This paper proves the existence of localized radial patterns in reaction-diffusion systems across arbitrary spatial dimensions, revealing how pattern profiles depend explicitly on dimension via Bessel functions.

Contribution

It extends the theoretical understanding of Turing patterns by establishing existence results for higher dimensions and linking pattern profiles to Bessel functions.

Findings

Existence of localized radial patterns in arbitrary dimensions.

Explicit dependence of pattern profiles on spatial dimension.

Connection between pattern formation and Bessel functions.

Abstract

The emergence of localised radial patterns from a Turing instability has been well studied in two and three dimensional settings and predicted for higher spatial dimensions. We prove the existence of localised $(n+1)$-dimensional radial patterns in general two-component reaction-diffusion systems near a Turing instability, where $n>0$ is taken to be a continuous parameter. We determine explicit dependence of each pattern's radial profile on the dimension $n$ through the introduction of $(n+1)$-dimensional Bessel functions, revealing a deep connection between the formation of localised radial patterns in different spatial dimensions.