The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Introduction to 'New trends in pattern formation and nonlinear dynamics of extended systems'

Pattern formation is a widespread phenomenon observed in physical, chemical and biological systems. During the past decades, powerful experimental and theoretical tools for the investigation of that phenomenon have been developed. Also, the set of pattern forming system was diversified. The present issue presents a panorama of pattern formation and other nonlinear dynamical phenomena in different natural and engineering systems. Special attention is paid to nonlinear dynamics of fluids. Non-equilibrium phenomena in chemical and quantum systems are also considered. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.