Cross-diffusion in Biofilms

Abstract

We propose a deterministic continuum model for mixed culture biofilms where movement of one species is affected by the presence of the other. Two derivations of this new model are presented. One derivation is based on the continuous time, discrete space master equation and the other one is based on the equations of conservation of mass and momentum. Starting from both viewpoints, we derive the same dual-species diffusion-reaction model for biofilms that comprises three non-standard diffusion effects: (i) degeneracy as the local biomass density vanishes, (ii) a super-diffusion singularity as the local biomass density approaches its {\it a priori} known maximum, and (iii) non-linear cross-diffusion. (i) describes the finite speed of propagation of the biofilm/water interface, (ii) describes volume filling effects, and (iii) describes the mixing of both biomass species. We present a numerical method for this highly nonlinear PDE model of biofilm that can tackle these three nonlinear diffusion effects. To investigate the effect of the new model feature, we study the role of the cross-diffusion terms in numerical simulations of three biofilm models: competition, allelopathy, and a mixed system formed by an aerobic and an anaerobic species. In all three systems we observe that accounting for cross-diffusion affects local biofilm structure, in particular the relative local distribution of biomass, but it does not affect overall lumped quantities such as the total amount of biomass in the system. As an application, our highly nonlinear density dependent cross-diffusion model is used in order to incorporate an experimental observation in models of disinfection of microbial biofilms. An extended reaction kinetics based on carbon consumption during disinfection is introduced. Our simulations show that the extended model captures the experimental observation, and suggest that the consumption of carbon substrates during inactivation due to antibiotics helps biofilms to survive and re-grow. Finally, as an extension of dual-species model, a generalized cross-diffusion model of $k$ interacting species is derived considering the continuous time and discrete space master equation passing to the continuous limit. Moreover, a criterion for preserving the positivity of the solution of this type of generalized cross-diffusion model is presented.