Existence and Longtime Behaviour of a Biofilm Model

Efendiev, Messoud

doi:10.1007/978-3-0348-0615-2_5

Cited by 11 publications

(17 citation statements)

References 48 publications

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“…The estimate follows immediately from inequality (10) in the proof of Theorem 3.5 and Gronwall's Lemma.…”

Section: Proofmentioning

confidence: 89%

“…Moreover, the semigroup generated by the solutions of system (6) is Lipschitz-continuous in L 1 ( )-norm. The following result recalls Theorem 3.2 of [10].…”

Section: Theorem 32 If the Initial Data Satisfymentioning

confidence: 92%

“…The constants are independent of the initial data (S 0 ,M 0 ). For further details and the proof, see Theorem 3.1 in [10]. Moreover, the semigroup generated by the solutions of system (6) is Lipschitz-continuous in L 1 ( )-norm.…”

Section: Theorem 32 If the Initial Data Satisfymentioning

confidence: 95%

“…Moreover, it was shown in [10] that the semi-group generated by the solutions of system (6) is continuous in L 1 ( )-norm and possesses a compact global attractor.…”

Section: Proposition 34mentioning

confidence: 99%

“…A solution theory for this system was developed in [10]. Our proof of the well-posedness of the quorum-sensing model will be essentially based on the results obtained for the mono-species model.…”

Section: Preliminariesmentioning

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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“…The estimate follows immediately from inequality (10) in the proof of Theorem 3.5 and Gronwall's Lemma.…”

Section: Proofmentioning

confidence: 89%

“…Moreover, the semigroup generated by the solutions of system (6) is Lipschitz-continuous in L 1 ( )-norm. The following result recalls Theorem 3.2 of [10].…”

Section: Theorem 32 If the Initial Data Satisfymentioning

confidence: 92%

Section: Theorem 32 If the Initial Data Satisfymentioning

confidence: 95%

“…Moreover, it was shown in [10] that the semi-group generated by the solutions of system (6) is continuous in L 1 ( )-norm and possesses a compact global attractor.…”

Section: Proposition 34mentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

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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A functional partial differential equation arising in a cell growth model with dispersion

Efendiev

Brunt

Wake

et al. 2017

Math Methods in App Sciences

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In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.

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Time Adaptive Numerical Solution of a Highly Degenerate Diffusion–Reaction Biofilm Model Based on Regularisation

Ghasemi

Eberl

2017

J Sci Comput

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We consider a quasilinear degenerate diffusion-reaction system that describes biofilm formation. The model exhibits two non-linear effects: a power law degeneracy as one of the dependent variables vanishes and a super diffusion singularity as it approaches unity. Biologically relevant solutions are characterized by a moving interface and gradient blow-up there.Discretisation of the PDE in space by a standard Finite Volume scheme leads to a singular system of ordinary differential equations. We show that regularisation of this system allows the application of error controlled adaptive integration techniques to solve the underlying PDE. This overcomes the major limitation of existing methods for this type of problem which work with fixed time-steps.We apply the resulting numerical method to study the effect of signal diffusion in the aqueous phase on quorum sensing induction in a biofilm colony.

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Existence and Longtime Behaviour of a Biofilm Model

Cited by 11 publications

References 48 publications

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

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

A functional partial differential equation arising in a cell growth model with dispersion

Time Adaptive Numerical Solution of a Highly Degenerate Diffusion–Reaction Biofilm Model Based on Regularisation

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