A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions

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.

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“…in [4,5,6,8]. The pecularity of that model lies in the form of the nonlinear diffusion coefficient D(u), namely…”

Section: An Inverse Problemmentioning

confidence: 99%

“…Moreover, the right hand side of (9) blows up for u = 1. We introduce the following regularization with small parameter > 0, which was already used in the analysis of the biofilm model (6), (7) in [6,8].…”

Section: An Inverse Problemmentioning

confidence: 99%

A Nonlinear Master Equation for a Degenerate Diffusion Model of Biofilm Growth

Khassehkhan

Hillen

Eberl

2009

Lecture Notes in Computer Science

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“…Compared with other biofilm models with several particulate substances, such as [11,12], the particularity of the quorum-sensing model (2) is that we recover the single-species biofilm growth model for the total biomass fraction M and the nutrient concentration S. Indeed, adding the equations for the biomass fractions X and Y in system (2) we obtain…”

Section: Preliminariesmentioning

confidence: 99%

“…In [10] it was shown that for every (sufficiently small) > 0 there exists a unique solution (S ,M ) of system (13) and the solutions are uniformly bounded with respect to the regularization parameter . Moreover, if the initial data belong to class (12), the solution M is separated from the singularity. To be more precise, there exists a constant ∈ (0, 1), which is independent of , such that M (x, t) < 1− holds for all x ∈ ,t > 0 (cf.…”

Section: Existencementioning

confidence: 99%

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On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities

Sonner

Efendiev

Eberl

2011

Math. Meth. Appl. Sci.

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We analyze a system of reaction-diffusion equations that models quorum-sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium-type degeneracy and super diffusion. We prove the wellposedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations.

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On the well‐posedness of mathematical models for multicomponent biofilms

Sonner

Efendiev

Eberl

2015

Math Methods in App Sciences

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Abstract. Bacterial biofilms are microbial depositions on immersed surfaces. Their mathematical description leads to degenerate diffusion-reaction equations with two non-Fickian effects:(i) a porous medium equation like degeneracy where the biomass density vanishes, and (ii) a super-diffusion singularity if the biomass density reaches its threshold density. In the case of multi-species interactions, several such equations are coupled, both in the reaction terms and in the nonlinear diffusion operator. In this paper we generalize previous work on existence and uniqueness of solutions of this type of models and give a general, relatively easy to apply criterion for well-posedness. The use of the criterion is illustrated in several examples from the biofilm modeling literature.

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A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions

Cited by 17 publications

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A Nonlinear Master Equation for a Degenerate Diffusion Model of Biofilm Growth

A Nonlinear Master Equation for a Degenerate Diffusion Model of Biofilm Growth

On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities

On the well‐posedness of mathematical models for multicomponent biofilms

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