An Error Controlled Time Adaptive Numerical Scheme for Nonlinear Degenerate Diffusion-reaction Biofilm Models with Applications in Microbiology and Bioengineering
We consider a class of mathematical models that describe biofilm formation. These models are quasilinear degenerate diffusion-reactions equations which exhibit three nonlinear diffusion effects: (i) a power law degeneracy as dependent variable biomass density vanishes, (ii) a super-diffusion singularity as it approaches unity and (iii) cross-diffusion. Discretizing the PDE model in space by a standard Finite Volume scheme results in a singular system of ordinary differential equations. We first consider the single species biofilm model with (i) and (ii) but not (iii) and show by regularisation that the solution of the corresponding singular ODE system does not attain the singularity. The resulting regularity allows the application of error-controlled adaptive time integration methods. This will overcome the difficulty of small time-step that methods with fixed time step experience. Then we use the proposed method to solve multi-component biofilm models to answer questions arising in microbiology and bioengineering more specifically in the context of quorum sensing (QS) and Membrane Biofilm Reactors (MBfR). We introduce a model with several biomass fractions that describes the biofilm behavior in the exposure to antibiotic in a system in which quorum sensing can trigger increased resistance to antibiotics. The resulting system has nonlinear diffusion effects (i) and (ii) and is solved by the introduced time-adaptive method. The results suggest that an adaptive, quorum sensing controlled, mechanism to switch between modes of fast growth with little protection and protective modes of slow growth may confer benefits to biofilm populations. Whereas quorum sensing inhibitors can delay the onset of increased resistance, their advantage is lost after induction occurs. This emphasizes the importance of timing for treatment of biofilms with antibiotics. In the context of bioengineering, we consider a model for multi-component biofilm with cross-diffusion effect (iii), to simulate the MBfR performance. The main objective in this study is to investigate the effect of initial inoculation and ammonium concentration on the compositions and mass transfer behavior of biofilms. We show that the initial coverage of the membrane affects the MBfR performance and in some scenarios results in biofilms with a heterogeneous layered structure. This highlights the importance and advantage of higher dimensional modelling over 1D models, which are not capable of describing the heterogeneous structure of biofilms.