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

Morphological instability and dynamics of fronts in bacterial growth models with nonlinear diffusion