  We consider a degenerate partial differential equation arising in population
dynamics, namely the porous medium equation with a bistable reaction term. We
study its asymptotic behavior as a small parameter, related to the thickness of
a diffuse interface, tends to zero. We prove the rapid formation of transition
layers which then propagate. We prove the convergence to a sharp interface
limit whose normal velocity, at each point, is that of the underlying
degenerate travelling wave.
