Domain-dependent stability analysis and parameter classification of a reaction-diffusion model on spherical geometries

TL;DR

This paper analyzes how the size of a spherical domain influences diffusion-driven instabilities in a reaction-diffusion system, providing explicit eigenvalue expressions, bifurcation classification, and numerical verification.

Contribution

It offers a domain-dependent stability analysis with explicit eigenvalue formulas and a comprehensive bifurcation classification for reaction-diffusion models on disks.

Findings

Necessary conditions on domain size for instabilities

Explicit eigenvalues and eigenfunctions derived

Numerical verification of analytical results

Abstract

In this work an activator-depleted reaction-diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using chebyshev non-periodic grid for the radial variable and Fourier periodic grid on the angular variable to verify the nodal lines and eigen-surfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Furthermore, analytical results are found relating the area of (disk-shape) domain with reaction-diffusion rates in the form of necessary conditions for the different types of bifurcations. These results are on one hand presented in the form of mathematical theorems with rigorous proofs, and, on the other hand using finite element method, each claim of the corresponding theorems are verified by obtaining the theoretically predicted behaviour of the dynamics in the numerical simulations.