Slow localized patterns in singularly perturbed two-component reaction–diffusion equations

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 travelling 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 ‘slow 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 ≡ 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 travelling localized slow 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. This result is obtained by explicitly determining the leading order approximations of four critical asymptotically small eigenvalues. By this analysis, that involves several orders of magnitude in the small parameter, we also obtain full control over the nature of the bifurcations—saddle-node, Hopf, global, etc—that determine the existence and stability of the (stationary and/or travelling) heteroclinic fronts and/or homoclinic pulses. Finally, we show that heteroclinic orbits may correspond to stable (slow) interfaces in two-dimensional space, while the homoclinic pulses must be unstable as localized stripes, even when they are stable in one space dimension.