Parameter spaces for cross-diffusive-driven instability in a reaction-diffusion system on an annular domain

TL;DR

This paper analyzes how geometry and domain size influence pattern formation in a reaction-diffusion system on an annular domain, deriving conditions for various instabilities and validating findings with finite element simulations.

Contribution

It provides a comprehensive analytical and numerical study of pattern formation driven by cross-diffusion on non-convex annular domains, including instability conditions and pattern classification.

Findings

Cross-diffusion relaxes conditions for pattern formation.

Domain size bounds determine instability onset.

Patterns resemble hypoxic tumour ring structures.

Abstract

In this work, the influence of geometry and domain size on spatiotemporal pattern formation is investigated to establish parameter spaces for a cross-diffusive reaction-diffusion model on an annulus. By applying linear stability theory, we derive conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on non-convex geometries e.g. annulus. Hence, the key research methodology and outcomes of our studies include: a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction-diffusion system; a linear stability analysis to characterise the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through lower and upper bounds of the domain size; the full classification of the model parameters, and a demonstration of how cross-diffusion relaxes the general conditions for the reaction-diffusion system to exhibit pattern formation. To validate theoretical findings and predictions, we employ the finite element method to reveal spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction-diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumours. Specifically, the cross-section of an actively invasive region in a hypoxic tumour can be effectively approximated by an annulus.