Stability analysis and parameter classification of a reaction-diffusion model on non-compact circular geometries

TL;DR

This paper investigates how the size of a non-compact annular domain affects pattern formation in an activator-depleted reaction-diffusion system, deriving spectral conditions and verifying them through numerical simulations.

Contribution

It provides a closed-form spectral expression, bifurcation conditions dependent on domain size, and numerical verification for pattern formation in reaction-diffusion systems on annular geometries.

Findings

Spectral expression for Laplace operator on annular domain derived.

Bifurcation conditions depend on domain size and reaction parameters.

Numerical simulations confirm analytical predictions on pattern formation.

Abstract

This work explores the influence of domain size of a non-compact two dimensional annular domain on the evolution of pattern formation that is modelled by an \textit{activator-depleted} reaction-diffusion system. A closed form expression is derived for the spectrum of Laplace operator on the domain satisfying a set of homogeneous conditions of Neumann type both at inner and outer boundaries. The closed form solution is numerically verified using the spectral method on polar coordinates. The bifurcation analysis of \textit{activator-depleted} reaction-diffusion system is conducted on the admissible parameter space under the influence of two bounds on the parameter denoting the thickness of the annular region. The admissibility of Hopf and transcritical bifurcations is proven conditional on the domain size satisfying a lower bound in terms of reaction-diffusion parameters. The admissible parameter space is partitioned under the proposed conditions corresponding to each case, and in turn such conditions are numerically verified by applying a method of polynomials on a quadrilateral mesh. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived conditions on the domain size for all types of admissible bifurcations.