  It has been argued that there is biological and modeling evidence that a
non-linear diffusion coefficient of the type D(b) = D_0 b^{k} underlies the
formation of a number of growth patterns of bacterial colonies. We study a
reaction-diffusion system with a non-linear diffusion coefficient introduced by
Ben-Jacob et al. Due to the fact that the bacterial diffusion coefficient
vanishes when the bacterial density b -> 0, the standard linear stability
analysis for fronts cannot be used. We introduce an extension of the stability
analysis which can be applied to such singular fronts, map out the region of
stability in the D-k-plane and derive an interfacial approximation in some
limits. Our linear stability analysis and sharp interface formulation will also
be applicable to other examples of interface formation due to nonlinear
diffusion, like in porous media or in the problem of vortex motion in
superconductors.
