  In this article, we consider a class of bi-stable reaction-diffusion
equations in two components on the real line. We assume that the system is
singularly perturbed, i.e. that the ratio of the diffusion coefficients is
(asymptotically) small. This class admits front solutions that are
asymptotically close to the (stable) front solution of the `trivial' scalar
bi-stable limit system $u_t = u_{xx} + u(1-u^2)$. However, in the system these
fronts can become unstable by varying parameters. This destabilization is
either caused by the essential spectrum associated to the linearized stability
problem, or by an eigenvalue that exists near the essential spectrum. We use
the Evans function to study the various bifurcation mechanisms and establish an
explicit connection between the character of the destabilization and the
possible appearance of saddle-node bifurcations of heteroclinic orbits in the
existence problem.
