From One Pattern into Another: Analysis of Turing Patterns in Heterogeneous Domains via WKBJ

TL;DR

This paper uses WKBJ asymptotics to analyze Turing instabilities in heterogeneous reaction-diffusion systems, revealing localized unstable modes and complex pattern formation that extend classical Turing theory to more realistic biological scenarios.

Contribution

It introduces a local version of Turing conditions for heterogeneous media and demonstrates how spatial localization of modes explains complex patterning phenomena.

Findings

Unstable modes are spatially localized in heterogeneous systems.

Localized modes have different growth rates and spatial regions.

Numerical simulations confirm the theoretical predictions.

Abstract

Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.