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Abstract. We present a continuous time/discrete space model of biofilm growth, starting from the semi-discrete master equation. The probabilities of biomass movement into neighboring sites depend on the local biomass density and on the biomass density in the target site such that spatial movement only takes place if (i) locally not enough space is available to accommodate newly produced biomass and (ii) the target site has capacity to accommodate new biomass. This mimics the rules employed by Cellular Automata models of biofilms. Grid refinement leads formally to a degenerate parabolic equation. We show that a set of transition rules can be found such that a previously studied ad hoc density-dependent diffusion-reaction model of biofilm formation is approximated well.

We present a mathematical model for growth and control of facultative anaerobic bacterial biofilms in nutrient rich environments. The growth of the microbial population is limited by protonated lactic acids and the local pH value, which in return are altered as the microbial population changes. The process is described by a non-linear parabolic system of three coupled equations for the dependent variables biomass density, acid concentration and pH. While the equations for the dissolved substrates are semi-linear, the equation for bacterial biomass shows two non-linear diffusion effects, a power law degeneracy as the dependent variable vanishes and a singularity in the diffusion coefficient as the dependent variable approaches its a priori known threshold. The interaction of both effects describes the spatial spreading of the biofilm. The interface between regions where the solution is positive and where it vanishes is the biofilm/bulk interface. We adapt a numerical method to explicitly track this interface in x -t space, based on the weak formulation of the biofilm model in a moving frame. We present numerical simulations of the spatio-temporal biofilm model, applied to a probiotic biofilm control scenario. It is shown that in the biofilm neighbouring regions co-exist in which pathogenic bacterial biomass is produced or killed, respectively. Furthermore, it is illustrated how the augmentation of the bulk with probiotic bacteria leads to an accelerated decay of the pathogenic biofilm.

We present a mathematical model and computer simulations for the control of a pathogenic biofilm by a probiotic biofilm. This is a substantial extension of a previous model of control of a pathogenic biofilm by microbial control agents that are suspended in the aqueous bulk phase (H. Khassehkhan and H.J. Eberl, Comp. Math. Meth. Med, 9(1) (2008), pp. 47-67). The mathematical model is a system of double-degenerate diffusion-reaction equations for the microbial biomass fractions probiotics, pathogens and inert bacteria, coupled with convection-diffusion-reaction equations for two growth controlling substrates, protonated lactic acids and hydrogen ions (pH). The latter are produced by the bacteria and become detrimental at high concentrations. In simulation studies, we find that the site of attachment of probiotics in the flow channel is crucial for success and efficacy of the probiotic control mechanism.

We study a previously introduced mathematical model of amensalistic control of the foodborne pathogen Listeria monocytogenes by the generally regarded as safe lactic acid bacteria Lactococcus lactis in a chemostat setting under nutrient rich growth conditions. The control agent produces lactic acids and thus affects pH in the environment such that it becomes detrimental to the pathogen while it is much more tolerant to these self-inflicted environmental changes itself. The mathematical model consists of five nonlinear ordinary differential equations for both bacterial species, the concentration of lactic acids, the pH and malate. The model is algebraically too involved to allow a comprehensive, rigorous qualitative analysis. Therefore, we conduct a computational study. Our results imply that depending on the growth characteristics of the medium in which the bacteria are cultured, the pathogen can survive in an intermediate flow regime but will be eradicated for slower flow rates and washed out for higher flow rates.