# Time adaptive numerical solution of a highly degenerate diffusion-reaction biofilm model based on regularisation

## TL;DR

This paper develops a regularisation-based, adaptive numerical method for solving a complex degenerate diffusion-reaction biofilm model, enabling accurate simulation of biofilm dynamics with moving interfaces and gradient blow-up.

## Contribution

It introduces a regularisation approach combined with adaptive integration to effectively solve a highly degenerate PDE system in biofilm modeling.

## Key findings

- The method accurately captures biofilm formation dynamics.
- Adaptive time-stepping improves computational efficiency.
- Signal diffusion impacts quorum sensing induction.

## Abstract

We consider a quasilinear degenerate diffusion-reaction system that describes biofilm formation. The model exhibits two non-linear effects: a power law degeneracy as one of the dependent variables vanishes and a super diffusion singularity as it approaches unity. Biologically relevant solutions are characterized by a moving interface and gradient blow-up there. Discretisation of the PDE in space by a standard Finite Volume scheme leads to a singular system of ordinary differential equations. We show that regularisation of this system allows the application of error controlled adaptive integration techniques to solve the underlying PDE. This overcomes the major limitation of existing methods for this type of problem which work with fixed time-steps. We apply the resulting numerical method to study the effect of signal diffusion in the aqueous phase on quorum sensing induction in a biofilm colony.