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We describe the resulting spatiotemporal dynamics when a homogeneousequilibrium loses stability in a spatially extended system. More precisely, weconsider reaction-diffusion systems, assuming only that the reaction kineticsundergo a transcritical, saddle-node, or supercritical pitchfork bifurcation asa parameter passes through zero. We construct traveling front solutions whichdescribe the invasion of the now-unstable state by a nearby stable state. Weshow that these fronts are marginally spectrally stable near the bifurcationpoint, which, together with recent advances in the theory of front propagationinto unstable states, establishes that these fronts govern the dynamics oflocalized perturbation to the unstable state. Our proofs are based onfunctional analytic tools to study the existence and eigenvalue problems forfronts, which become singularly perturbed after a natural rescaling.

Front propagation close to the onset of instability