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

Sonner, Stefanie; Efendiev, Messoud; Eberl, Hermann J.

doi:10.1002/mma.1475

Cited by 16 publications

(29 citation statements)

References 15 publications

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“…In and , the behavior of solutions was studied in numerical simulations and the existence of solutions was established, but the question of uniqueness of solutions remained unanswered in both cases. The first uniqueness result for multispecies models was obtained in .…”

Section: Multicomponent Biofilm Modelsmentioning

confidence: 99%

“…Using the known results for the prototype model, a different approach than for the previous multicomponent models was developed in , which led to a uniqueness result for the solutions. The following theorem states the well‐posedness of the quorum sensing model (Theorems 3.5 and 3.11; ). Theorem Let the initial data satisfy

X_{0}, Y_{0}, A_{0} \in H_{0}^{1} (normalΩ)

, S 0 ∈ H 1 (Ω) such that S 0 | ∂ Ω =1, and

\begin{matrix} 0 \leq S_{0}, X_{0}, Y_{0}, A_{0} \leq 1 in normalΩ, ∥ X_{0} + Y_{0} ∥_{L^{\infty} (normalΩ)} < 1 . \end{matrix}

Then, there exists a unique global solution of the quorum sensing model ,

\begin{matrix} A, S, X, Y \in C ([0, \infty); L^{2} (normalΩ)) \cap L^{\infty} (normalΩ \times [0, \infty)), \\ A, S \in L^{2} ((0, \infty); H^{1} (normalΩ)), \\ D (M) ▿ X, D (M) ▿ Y \in L^{2} ((0, \infty); L^{2} (normalΩ; R^{n})), \end{matrix}

the functions A , S , X and Y are nonnegative and satisfy system in distributional sense.…”

Section: Multicomponent Biofilm Modelsmentioning

confidence: 99%

“…We aim to show that the solution theory developed in can be extended to other multicomponent biofilm models.…”

Section: Introductionmentioning

confidence: 99%

“…Taking advantage of the known results for the prototype model, the existence and uniqueness of solutions and the continuous dependence of solutions on initial data could be established in [5]. It is the first uniqueness result for multispecies reaction-diffusion models of biofilms that extend the single-species model [8].We aim to show that the solution theory developed in [5] can be extended to other multicomponent biofilm models. In Section 2, for the convenience of the reader, we introduce the prototype biofilm growth model and summarize important results on which our work is based.…”

mentioning

confidence: 99%

“…In [4] and [12], the behavior of solutions was studied in numerical simulations and the existence of solutions was established, but the question of uniqueness of solutions remained unanswered in both cases. The first uniqueness result for multispecies models was obtained in [5].where we use the same notations as in (1). The additional constants 1 , 2 and d B in (2) are positive and have the following meaning:d B Diffusion coefficient of antibiotics 1 Antibiotics consumption rate 2 Inert biomass production rate Apart from the diffusion of the dissolved substrates and the death, growth and spatial spreading of biomass, the disinfection mechanism is included in the model.…”

mentioning

confidence: 99%

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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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Section: Multicomponent Biofilm Modelsmentioning

confidence: 99%

X_{0}, Y_{0}, A_{0} \in H_{0}^{1} (normalΩ)

, S 0 ∈ H 1 (Ω) such that S 0 | ∂ Ω =1, and

\begin{matrix} 0 \leq S_{0}, X_{0}, Y_{0}, A_{0} \leq 1 in normalΩ, ∥ X_{0} + Y_{0} ∥_{L^{\infty} (normalΩ)} < 1 . \end{matrix}

Then, there exists a unique global solution of the quorum sensing model ,

\begin{matrix} A, S, X, Y \in C ([0, \infty); L^{2} (normalΩ)) \cap L^{\infty} (normalΩ \times [0, \infty)), \\ A, S \in L^{2} ((0, \infty); H^{1} (normalΩ)), \\ D (M) ▿ X, D (M) ▿ Y \in L^{2} ((0, \infty); L^{2} (normalΩ; R^{n})), \end{matrix}

the functions A , S , X and Y are nonnegative and satisfy system in distributional sense.…”

Section: Multicomponent Biofilm Modelsmentioning

confidence: 99%

“…We aim to show that the solution theory developed in can be extended to other multicomponent biofilm models.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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Simulating cyclic voltammetry under advection for electrochemical cantilevers

Adesokan

Evgrafov

Sørensen

2014

Math Methods in App Sciences

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We present a mathematical model describing an electrochemical system involving electrode–electrolyte interaction. The model is governed by a system of advection–diffusion equations with a nonlinear reaction term at the boundary. Our calculations based on such model demonstrate the dynamics of ionic currents in the electrolyte. The model allows us to predict the effect of varying flow rates, scan rates, and electrolyte concentration of the electrochemical system. Copyright © 2014 John Wiley & Sons, Ltd.

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Simulation Based Exploration of Bacterial Cross Talk Between Spatially Separated Colonies in a Multispecies Biofilm Community

Zarva

Eberl

2020

Lecture Notes in Computer Science

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

Cited by 16 publications

References 15 publications

On the well‐posedness of mathematical models for multicomponent biofilms

On the well‐posedness of mathematical models for multicomponent biofilms

Simulating cyclic voltammetry under advection for electrochemical cantilevers

Simulation Based Exploration of Bacterial Cross Talk Between Spatially Separated Colonies in a Multispecies Biofilm Community

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