Slow localized patterns in singularly perturbed 2-component reaction-diffusion equations

TL;DR

This paper investigates the existence and stability of slow localized patterns in singularly perturbed reaction-diffusion systems, revealing conditions under which certain patterns are stable or unstable, especially in models with non-vertical slow manifolds.

Contribution

It introduces a new analysis of slow localized patterns embedded in non-vertical slow manifolds, extending understanding beyond classical models with linear or vertical slow manifolds.

Findings

Homoclinic pulse patterns are generally unstable.

Heteroclinic fronts can be stable or unstable.

Homoclinic pulses near heteroclinic cycles may be stable.

Abstract

Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast excursions -- in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this paper we consider the existence and stability of stationary and traveling slow localized patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. These patterns have rarely been considered in the literature, for two reasons: (i) in the classical Gray-Scott/Gierer-Meinhardt type models that dominate the literature, the flow on the slow manifold is typically linear and thus cannot exhibit homoclinic pulse or heteroclinic front solutions; (ii) the slow manifolds occurring in the literature are typically `vertical' -- i.e. given by $u \equiv u_0$, where $u$ is the fast variable -- so that the stability problem is determined by a simple (decoupled) scalar equation. The present research concerns a general system of singularly perturbed reaction-diffusion equations and is motivated by several explicit ecosystem models that do give rise to non-vertical normally hyperbolic slow manifolds on which the flow may exhibit both homoclinic and heteroclinic orbits -- that correspond to either stationary or traveling localized patterns. The associated spectral stability problems are at leading order given by a nonlinear, but scalar, eigenvalue problem with Sturm-Liouville characteristics and we establish that homoclinic pulse patterns are typically unstable, while heteroclinic fronts can either be stable or unstable. However, we also show that homoclinic pulse patterns that are asymptotically close to a heteroclinic cycle may be stable.