  We consider a Fisher-KPP equation with density-dependent diffusion and
advection, arising from a chemotaxis-growth model. We study its behavior as a
small parameter, related to the thickness of a diffuse interface, tends to
zero. We analyze, for small times, the emergence of transition layers induced
by a balance between reaction and drift effects. Then we investigate the
propagation of the layers. Convergence to a free-boundary limit problem is
proved and a sharp estimate of the thickness of the layers is provided.
