Analysis of a degenerate biofilm model with a nutrient taxis term
Hermann J. Eberl Messoud A. Efendiev Dariusz Wrzosek Anna Zhigun
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 99-119 doi: 10.3934/dcds.2014.34.99
We introduce and analyze a prototype model for chemotactic effects in biofilm formation. The model is a system of quasilinear parabolic equations into which two thresholds are built in. One occurs at zero cell density level, the second one is related to the maximal density which the cells cannot exceed. Accordingly, both diffusion and taxis terms have degenerate or singular parts. This model extends a previously introduced degenerate biofilm model by combining it with a chemotaxis equation. We give conditions for existence and uniqueness of weak solutions and illustrate the model behavior in numerical simulations.
keywords: singular quasilinear parabolic equation compactness method. biofilm Chemotaxis fast diffusion degenerate diffusion
On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis
Messoud Efendiev Anna Zhigun
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 651-673 doi: 10.3934/dcds.2018028

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function.

keywords: Attractor biofilm chemotaxis degenerate diffusion longtime dynamics
On a coupled SDE-PDE system modeling acid-mediated tumor invasion
Sandesh Athni Hiremath Christina Surulescu Anna Zhigun Stefanie Sonner
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2339-2369 doi: 10.3934/dcdsb.2018071

We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

keywords: Multiscale model intra-and extracellular proton dynamics tumor acidity stochastic differential equations partial differential equations pH-taxis

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