  We describe the resulting spatiotemporal dynamics when a homogeneous
equilibrium loses stability in a spatially extended system. More precisely, we
consider reaction-diffusion systems, assuming only that the reaction kinetics
undergo a transcritical, saddle-node, or supercritical pitchfork bifurcation as
a parameter passes through zero. We construct traveling front solutions which
describe the invasion of the now-unstable state by a nearby stable state. We
show that these fronts are marginally spectrally stable near the bifurcation
point, which, together with recent advances in the theory of front propagation
into unstable states, establishes that these fronts govern the dynamics of
localized perturbation to the unstable state. Our proofs are based on
functional analytic tools to study the existence and eigenvalue problems for
fronts, which become singularly perturbed after a natural rescaling.
