Pattern formation from spatially heterogeneous reaction–diffusion systems

Abstract

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Turing instability and pattern formation on directed networks

“…In Section 2.1 we not did diagonalise, as the resulting expressions depended only on the eigenvalues (and not the eigenbasis) and as such one does "need" to diagonalise. However, similar to [36], the expression that we obtain here does explicitly depend on the eigenbasis. Let us now return to the matrix P .…”

“…Moreover, in the case of a continuum, the considered systems are not allowed to depend on their spatial coordinates. These issues were addressed, again by Van Gorder, in [36]. Although networks themselves do not possess any 'spatial' coordinates one can think of an analogous scenario in a system of reaction-diffusion equations with global reaction kinetics.…”

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

“…More recently, Kozak et al [109] employed this approach to study systems with piecewise constant reaction kinetics, finding that one can broadly use the intuitive homogeneous Turing conditions away from the discontinuity in the domain (with a base state that itself is piecewise homogeneous). Finally we remark that [190], in this Theme Issue, provides a vast generalisation of this Galerkin approach to a range of systems with spatial heterogeneity in kinetics, diffusion coefficients, and boundary conditions.…”

Modern perspectives on near-equilibrium analysis of Turing systems

“…More recently, Kozak et al [ 122 ] employed this approach to study systems with piecewise constant reaction kinetics, finding that one can broadly use the intuitive homogeneous Turing conditions away from the discontinuity in the domain (with a base state that itself is piecewise homogeneous). Finally we remark that [ 123 ], in this theme issue, provides a significant generalization of this Galerkin approach to a range of systems with spatial heterogeneity in kinetics, diffusion coefficients, and boundary conditions.…”