Integration of a degenerate diffusion-reaction system with an application to cross-talk in biofilms

Starting point will be a quasilinear system of diffusion-reaction equations, one of which (i) degenerates if the dependent variable vanishes, and (ii) attains a super-diffusion
singularity if the dependent variable reaches an a priori known critical value. We introduced this type of equations many years ago to model bacterial biofilms. Both  effects (i) and (ii) introduce numerical (and analytical) challenges: (i), like the porous medium equation, leads to the formation of sharp interfaces and gradient blow-up. (ii) is characterised by blow-up of the density-dependent diffusion coefficient, which prevents the application of  standard arguments to show boundedness of solutions.

After semi-discretisation in space one obtains a very stiff system of ordinary differential equations that, owing to (ii), a priori does not satisfy a Lipschitz condition. We explore this system using regularistaion techniques. We are able to prove that (for practically
reasonable assumptions on initial and boundary conditions) the numerical solutions remain bounded by unity (as they ought to) and that traditional ODE integrators can be safely applied. We present a small application of this method that is concerned with quorum sensing cross-talk in a biofilm community.

This is joint work with Maryam Ghasemi (now at the University of Waterloo).

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