  Motivated by its application in ecology, we consider an extended Klausmeier
model, a singularly perturbed reaction-advection-diffusion equation with
spatially varying coefficients. We rigorously establish existence of stationary
pulse solutions by blending techniques from geometric singular perturbation
theory with bounds derived from the theory of exponential dichotomies.
Moreover, the spectral stability of these solutions is determined, using
similar methods. It is found that, due to the break-down of translation
invariance, the presence of spatially varying terms can stabilize or
destabilize a pulse solution. In particular, this leads to the discovery of a
pitchfork bifurcation and existence of stationary multi-pulse solutions.
