  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.
