Effect of geometry on the positioning of a single spot in reaction-diffusion systems

TL;DR

This paper investigates how the shape of a curved surface influences the orientation and formation of localized reaction-diffusion spots, analyzing eigenmodes on ellipsoids and comparing theoretical predictions with numerical simulations.

Contribution

It provides a detailed analysis of how geometry affects spot positioning and eigenmode alignment in reaction-diffusion systems on curved surfaces, including small deformations from spherical shape.

Findings

Eigenfunction aligns along the symmetry axis for prolate ellipsoids.

Eigenfunction aligns perpendicular to the symmetry axis for oblate ellipsoids.

Numerical simulations confirm the theoretical predictions of eigenfunction orientation.

Abstract

We consider the formation of a single spot (localized solution) in reaction-diffusion (RD) equation on a curved manifold. Specifically, we study the direction (alignment) of the normal to interface between maxima and minima of concentration in the steady-state on a prolate and on an oblate ellipsoid. We further analyse the effect of shape asymmetry on l = 1 eigenmode of the sphere by assuming a small deformation from the spherical geometry. Our analysis shows that the eigenfunction corresponding to highest eigenvalue align along the symmetry axis for a prolate ellipsoid, and perpendicular to the symmetry axis for an oblate ellipsoid. Finally, we compare the direction of variation of the most unstable mode (eigenfunction with highest growth rate) in the system obtained by assuming a small deformation from the sphere and the alignment of interface normal obtain from the numerical simulations.