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

Abstract

Depending on the growth condition, bacterial colonies can exhibit different morphologies. As argued by Ben-Jacob et al. there is biological and modeling evidence that a nonlinear diffusion coefficient of the type D(b)=D(0)b(k) is a basic mechanism that underlies almost all of the patterns and generates a long-wavelength instability. We study a reaction-diffusion system with a nonlinear diffusion coefficient and find that a unique planar traveling front solution exists whose velocity is uniquely determined by k and D=D(0)/D(n), where D(n) is the diffusion coefficient of the nutrient. Due to the fact that the bacterial diffusion coefficient vanishes when b-->0, in the front solution b vanishes in a singular way. As a result the standard linear stability analysis for fronts cannot be used. We introduce an extension of the stability analysis that can be applied to singular fronts, and use the method to perform a linear stability analysis of the planar bacteriological growth front. We show that a nonlinear diffusion coefficient generates a long-wavelength instability for k>0 and D<D(c)(k). We map out the region of stability in the D-k-plane and determine the onset of stability that is given by D(c)(k). Both, for D-->0 and k--> infinity the dynamics of the growth zone essentially reduces to that of a sharp interface problem that is reminiscent of a so-called one-sided growth problem where the growth velocity is proportional to the gradient of a diffusion field ahead of the interface. The moving boundary approximation that we derive in these limits is quite accurate but surprisingly does not become a proper asymptotic theory in the strict mathematical sense in the limit D-->0, due to lack of full separation of scales on all dynamically relevant length scales. 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.