Morphological Instability and Dynamics of Fronts in Bacterial Growth Models with Nonlinear Diffusion

TL;DR

This paper investigates the stability and dynamics of bacterial growth fronts with nonlinear diffusion, extending stability analysis methods to singular fronts where diffusion vanishes, and applies findings to biological and physical interface problems.

Contribution

It introduces a novel stability analysis for nonlinear diffusion fronts where the diffusion coefficient vanishes, and derives an interfacial approximation applicable to various interface formation scenarios.

Findings

Mapped stability regions in the D-k-plane for bacterial growth fronts

Developed an extended stability analysis for singular nonlinear diffusion fronts

Derived an interfacial approximation for nonlinear diffusion-driven interfaces

Abstract

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.