Analysis of a degenerate biofilm model with a nutrient taxis term

Eberl, Hermann J.; Efendiev, Messoud; Wrzosek, Dariusz; Zhigun, Anna

doi:10.3934/dcds.2014.34.99

Cited by 13 publications

(10 citation statements)

References 22 publications

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“…. , N T and a given nonnegative constant v D such that v k σ = v D for all σ ∈ E ext , we define the piecewise constant in space and time function v by (13) v(x, t)…”

Section: Notation and Assumptionsmentioning

confidence: 99%

“…The existence and uniqueness of a global weak solution to (1)-( 5) was shown in [15], while the original model was analyzed in [21] by formulating it as a system of variational inequalities. The model of [14] was extended in [13] by taking into account nutrient taxis, which forces the biofilm to move up a nutrient concentration gradient. In that work, a fast-diffusion exponent a ∈ (0, 1) instead of a superdiffusive value a > 1 (like in [14]) was considered.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a finite-volume scheme for a single-species biofilm model

Helmer¹,

Jüngel²,

Zurek³

2021

Preprint

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An implicit Euler finite-volume scheme for a parabolic reaction-diffusion system modeling biofilm growth is analyzed and implemented. The system consists of a degenerate-singular diffusion equation for the biomass fraction, which is coupled to a diffusion equation for the nutrient concentration, and it is solved in a bounded domain with Dirichlet boundary conditions. By transforming the biomass fraction to an entropy-type variable, it is shown that the numerical scheme preserves the lower and upper bounds of the biomass fraction. The existence and uniqueness of a discrete solution and the convergence of the scheme are proved. Numerical experiments in one and two space dimensions illustrate, respectively, the rate of convergence in space of our scheme and the temporal evolution of the biomass fraction and the nutrient concentration.

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“…. , N T and a given nonnegative constant v D such that v k σ = v D for all σ ∈ E ext , we define the piecewise constant in space and time function v by (13) v(x, t)…”

Section: Notation and Assumptionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of a finite-volume scheme for a single-species biofilm model

Helmer¹,

Jüngel²,

Zurek³

2021

Preprint

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show abstract

“…[12,36,37] where many examples can be found. One such model which includes diffusion of the form given by (2.2) was proposed and analysed in [32]. In that model the cell density cannot exceed a pregiven threshold, extreme aggregation is avoided, and the propagation speed is finite.…”

Section: Rdt Systems With Fl Mechanisms 31 Motivationmentioning

confidence: 99%

Flux limitation mechanisms arising in multiscale modelling of cancer invasion

Zhigun¹

2021

Preprint

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Tumour invasion is an essential stage of cancer progression. Its main drivers are diffusion and taxis, a directed movement along the gradient of a stimulus. Here we review models with flux limited diffusion and/or taxis which have applications in modelling of cell migration, particularly in cancer. Flux limitation ensures control upon propagation speeds, precluding unnaturally quick spread which is typical for traditional parabolic equations. We recall the main properties of models with flux limitation effects and discuss ways to construct them, concentrating on multiscale derivations from kinetic transport equations.

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“…For other formulations of biomass growth with taxis terms, see [7]. These formulations are not explicitly included in our formulation, but can be easily obtained by a modification.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of biofilm development

Gokieli

Kenmochi

Niezgódka

2018

Nonlinear Analysis: Real World Applications

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We perform mathematical anaysis of the biofilm development process. A model describing biomass growth is proposed: It arises from coupling three parabolic nonlinear equations: a biomass equation with degenerate and singular diffusion, a nutrient tranport equation with a biomass-density dependent diffusion, and an equation of the Navier-Stokes type, describing the fluid flow in which the biofilm develops. This flow is subject to a biomass-density dependent obstacle. The model is treated as a system of three inclusions, or variational inequalities; the third one causes major difficulties for the system's solvability. Our approach is based on the recent development of the theory on Navier-Stokes variational inequalities. on the other hand u ε := ρ ε * u is the local spatial-average of u(x, t) by means of the usual mollifier ρ ε (x) (see Section 2 for details).(ii) The nutrient concentration w(x, t) is non-negative and has the threshold value 1, i.e. 0 ≤ w(x, t) ≤ 1. Also, we suppose that there is no nutrient supply from the exterior. The diffusion coefficient d(u) depends on the biomass density u and c d ≤ d(r) ≤ c d , |d(r 1 ) − d(r 2 )| ≤ L(d)|r 1 − r 2 |, ∀r 1 , r 2 ∈ R, (1.2)

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Analysis of a degenerate biofilm model with a nutrient taxis term

Cited by 13 publications

References 22 publications

Analysis of a finite-volume scheme for a single-species biofilm model

Analysis of a finite-volume scheme for a single-species biofilm model

Flux limitation mechanisms arising in multiscale modelling of cancer invasion

Mathematical modeling of biofilm development

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