  We study the existence and stability of standing pulse solutions to a
singularly perturbed three-component reaction diffusion system with
one-activator and two-inhibitor type. We apply the MAE (matched asymptotic
expansion) method to the construction of solutions and the SLEP (Singular Limit
Eigenvalue Problem) method to their stability properties. This approach is not
just an alternative approach to geometric singular perturbation and the
associated Evans function, but gives us two advantages: one is the
extendability to higher dimensional case, and the other is to allow us to
obtain more precise information on the behaviors of critical eigenvalues. This
implies the existence of codimension two singularity of drift and Hopf
bifurcations for the standing pulse solution and it is numerically confirmed
that stable standing and traveling breathers emerge around the singularity in a
physically-acceptable regime.
