Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix

TL;DR

This paper investigates how spiral wave spectra behave when the diffusion matrix is rank-deficient, revealing spectral discontinuities and predicting spectrum locations, with numerical confirmation in biological models.

Contribution

It extends spectral analysis of spiral waves to systems with rank-deficient diffusion matrices, previously only studied for positive matrices.

Findings

Essential spectrum is discontinuous as diffusion vanishes.

Locations of absolute spectrum are predicted for non-diffusing variables.

Numerical results confirm theoretical predictions in biological models.

Abstract

Spiral waves emerge in numerous pattern forming systems and are commonly modeled with reaction-diffusion systems. Some systems used to model biological processes, such as ion-channel models, fall under the reaction-diffusion category and often have one or more non-diffusing species which results in a rank-deficient diffusion matrix. Previous theoretical research focused on spiral spectra for strictly positive diffusion matrices. In this paper, we use a general two-variable reaction-diffusion system to compare the essential and absolute spectra of spiral waves for strictly positive and rank-deficient diffusion matrices. We show that the essential spectrum is not continuous in the limit of vanishing diffusion in one component. Moreover, we predict locations for the absolute spectrum in the case of a non-diffusing slow variable. Predictions are confirmed numerically for the Barkley and Karma models.