Localized slow patterns in singularly perturbed 2-component reaction-diffusion equations

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 talk we consider the existence and stability of localized slow 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 linear and thus cannot exhibit homoclinic pulse or heteroclinic front solutions; (ii) the slow manifolds occurring in the literature are typically trivial, or ‘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 talk is motivated by several explicit ecosystem models (of singularly perturbed reaction-diffusion type) that do give rise to nontrivial 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 type 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 4 critical asymptotically small eigenvalues. Through this somewhat subtle 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 heteroclinic fronts and/or homoclinic pulses. Finally, we show that heteroclinic orbits may correspond to stable (slow) interfaces (in 2-dimensional space), while the homoclinic pulses must be unstable as localized stripes –even when they are stable in 1 space dimension.