ArticleJournal of the Royal Society, Interface2020
From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ.
Article in Journal of the Royal Society, Interface, 2020. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 17 papers.
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17 citing papers in PubMed.
- Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation.Fundamental research · 2026Article
- Persistent Homology Classifies Parameter Dependence of Patterns in Turing Systems.Bulletin of mathematical biology · 2025Article
- Utilizing Causal Network Markers to Identify Tipping Points ahead of Critical Transition.Advanced science (Weinheim, Baden-Wurttemberg, Germany) · 2025Article
- Characterizing symmetry transitions in systems with dynamic morphology.Mathematical biosciences · 2025Article
- Article
- Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model.Bulletin of mathematical biology · 2024Article
- Turing Instabilities are Not Enough to Ensure Pattern Formation.Bulletin of mathematical biology · 2024Article
- Novel Aspects in Pattern Formation Arise from Coupling Turing Reaction-Diffusion and Chemotaxis.Bulletin of mathematical biology · 2023Article
- Of Turing and zebras: Turing diffusion inspires applications in nature and beyond: Turing diffusion inspires applications in nature and beyond.EMBO reports · 2023Article
- Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems.Bulletin of mathematical biology · 2023Article
- Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems.Bulletin of mathematical biology · 2022Article
- Modern perspectives on near-equilibrium analysis of Turing systems.Philosophical transactions. Series A, Mathematical, physical, and engineering sciences · 2021Review
- Isolating Patterns in Open Reaction-Diffusion Systems.Bulletin of mathematical biology · 2021Article
- Bespoke Turing Systems.Bulletin of mathematical biology · 2021Article
- Landscape-induced spatial oscillations in population dynamics.Scientific reports · 2021Article
- Turing conditions for pattern forming systems on evolving manifolds.Journal of mathematical biology · 2021Article
- Turing Patterning in Stratified Domains.Bulletin of mathematical biology · 2020Article
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4 authors.
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Abstract
Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial 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.
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