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

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September 2009, 12(2): 371-388. doi: 10.3934/dcdsb.2009.12.371

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

Hassan Khassehkhan ^1, , Messoud A. Efendiev ^2, and Hermann J. Eberl ^1,

1.	Department of Mathematics and Statistics, University of Guelph, Guelph, On, N1G 2W1, Canada, Canada
2.	Institute of Biomathematics and Biometry, HelmholtzZentrum München, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany

Received September 2008 Revised December 2008 Published July 2009

Abstract
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We study a mathematical model that describes how a "good" bacterial biofilm controls the growth of a harmful pathogenic bacterial biofilm. The underlying mechanism is a modification of the local protonated acid concentration, which in turn decreases the local pH and, thus, makes growth conditions for the pathogens less favorable, while the control-agent itself is more tolerant to these changes. This system is described by a system of 5 density-dependent diffusion-reaction equations that show two nonlinear diffusion effects: porous medium degeneracy and fast diffusion. This is a multi-species expansion of a previously studied single species biofilm model. In this paper we prove the existence of solutions to this model and show in numerical simulations the effectiveness of the control mechanism.

Keywords: degeneracy, simulation., nonlinear diffusion, biofilm, probiotic.

Mathematics Subject Classification: Primary: 35K65; Secondary: 92D2.

Citation: Hassan Khassehkhan, Messoud A. Efendiev, Hermann J. Eberl. A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 371-388. doi: 10.3934/dcdsb.2009.12.371

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