From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ

Abstract

<jats:p>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.</jats:p>

Turing Patterning in Stratified Domains

“…Of course, in the absence of a homogeneous steady state, one must develop new methods for the analysis of pattern-forming instabilities. This has been done recently for heterogeneous steady states (Krause et al 2020), but extending such an analysis to these coupled geometries is nontrivial. Mathematically, the limit of η → ∞ can be thought of as a step function heterogeneity, as explored in Kozák et al (2019), so that the systems studied here are also in some sense a generalization of piecewiseconstant reaction-diffusion problems, providing another perspective on heterogeneous reaction-diffusion systems.…”

“…Examples that deviate even further from the classical case are growing domains (Crampin et al 1999;Plaza et al 2004;Krause et al 2019;Sánchez-Garduño et al 2019) and spatially heterogeneous reaction-diffusion processes (Benson et al 1998;Page et al 2003Page et al , 2005Haim et al 2015;Kolokolnikov and Wei 2018), for which the canonical approach does not work. In such cases, novel approaches to pattern-forming instabilities have recently been developed for growth (Madzvamuse et al 2010;) and heterogeneity (Krause et al 2020) under certain simplifications, but such analyses are quite different to the classical case. In a similar direction, as part of our objective in exploring the Turing instability for layered reaction-diffusion systems, we will aim to demonstrate a much richer diversity of structure in the resulting dispersion relations (and hence instability conditions), compared to classical counterparts.…”

“…2019 ) and heterogeneity (Krause et al. 2020 ) under certain simplifications, but such analyses are quite different to the classical case. In a similar direction, as part of our objective in exploring the Turing instability for layered reaction–diffusion systems, we will aim to demonstrate a much richer diversity of structure in the resulting dispersion relations (and hence instability conditions), compared to classical counterparts.…”

Spike density distribution for the Gierer–Meinhardt model with precursor

“…We now proceed as in [15] to derive an equivalent representation of (2.18a) that is based only on nearest neighbor interactions. To do so, we first write (2.18) compactly in matrix form as 20) where G and P are defined in terms of the Green's function by…”

“…As a result, a conventional Turing stability approach is not applicable and the initial development of small amplitude patterns must be analyzed through either a slowly-varying assumption or from full numerical simulations (cf. [13], [22], [23], [20]).…”