Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term

We study reaction-diffusion equations with a spatially inhomogeneous reacti on term. If the coefficient of these reaction term is much larger than the diffusion coefficient, a sharp interface appears between two different phas es. We show that the equation of motion of such an interface involves a dri ft term despite the absence of drift in the original diffusion equations. I n particular, we show that the same rich spatial patterns observed for a ch emotaxis-growth model can be realized by a system without a drift term.