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
Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation
Hedia Fgaier Hermann J. Eberl
Conference Publications 2009, 2009(Special): 230-239 doi: 10.3934/proc.2009.2009.230
The onset of a typical bacterial growth curve shows a period of very slow increase in population counts. This is a period of physiological adaptation to new environmental conditions. While in mathematical biology much progress was made in recent years to describe physiologically structured populations, these models typically have too many degrees of freedom to easily allow a model identification against experimental data. Therefore, and for all practical purposes, microbiologists have proposed simpler models of physiological adaptation in the past, usually in connection with standard growth curves. In this paper we compare the performance of four such lag-time models, each of which described by a scalar differential equation, when combined with a model of a siderophore producing bacterial population under iron limitation. In each case this yields a system of five nonlinear ordinary differential equations that we compare against experimental data, by solving the associated vector optimization problem. Our main finding is that a big step in accuracy is made already by including a simple lag-time model that only introduces one additional degree of freedom in the parameter identification problem (the initial state of health of the population), and that this can be reliably improved if a further degree of freedom, describing the dynamics of the physiological recovery process, is included. The vector optimization problem is solved by scalarizing it with a linear functional and solving the resulting scalar optimization problem. The growth parameters that are identified in this procedure are found to be robust with respect to the scalarization coefficient.
keywords: Mathematical biology computation lag time parameter identification
On optimization of substrate removal in a bioreactor with wall attached and suspended bacteria
Alma Mašić Hermann J. Eberl
Mathematical Biosciences & Engineering 2014, 11(5): 1139-1166 doi: 10.3934/mbe.2014.11.1139
We investigate the question of optimal substrate removal in a biofilm reactor with concurrent suspended growth, both with respect to the amount of substrate removed and with respect to treatment process duration. The water to be treated is fed externally from a buffer vessel to the treatment reactor. In the two-objective optimal control problem, the flow rate between the vessels is selected as the control variable. The treatment reactor is modelled by a system of three ordinary differential equations in which a two-point boundary value problem is embedded. The solution of the associated singular optimal control problem in the class of measurable functions is impractical to determine and infeasible to implement in real reactors. Instead, we solve the simpler problem to optimize reactor performance in the class of off-on functions, a choice that is motivated by the underlying biological process. These control functions start with an initial no-flow period and then switch to a constant flow rate until the buffer vessel is empty. We approximate the Pareto Front numerically and study the behaviour of the system and its dependence on reactor and initial data. Overall, the modest potential of control strategies to improve reactor performance is found to be primarily due to an initial transient period in which the bacteria have to adapt to the environmental conditions in the reactor, i.e. depends heavily on the initial state of the dynamic system. In applications, the initial state, however, is often unknown and therefore the efficiency of reactor optimization, compared to the uncontrolled system with constant flow rate, is limited.
keywords: Biofilm treatment time optimization. suspended growth substrate removal
A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions
Hassan Khassehkhan Messoud A. Efendiev Hermann J. Eberl
Discrete & Continuous Dynamical Systems - B 2009, 12(2): 371-388 doi: 10.3934/dcdsb.2009.12.371
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
Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects
Blessing O. Emerenini Stefanie Sonner Hermann J. Eberl
Mathematical Biosciences & Engineering 2017, 14(3): 625-653 doi: 10.3934/mbe.2017036

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.

keywords: Quorum sensing biofilm cell dispersal density dependent diffusion existence uniqueness numerical simulation
Existence and longtime behavior of a biofilm model
Messoud A. Efendiev Sergey Zelik Hermann J. Eberl
Communications on Pure & Applied Analysis 2009, 8(2): 509-531 doi: 10.3934/cpaa.2009.8.509
A nonlinear, density-dependent system of diffusion-reaction equations describing development of bacterial biofilms is analyzed. It comprises two non-standard diffusion effects, degeneracy as in the porous medium equation and fast diffusion. The existence of a unique bounded solution and a global attractor is proved in dependence of the boundary conditions. This is achieved by studying a system of non-degenerate auxiliary approximation equations and the construction of a Lipschitz continuous semigroup by passing to the limit in the approximation parameter. Numerical examples are included in order to illustrate the main result.
keywords: biofilm global attractor Nonlinear diffusion degeneracy
Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates
Fazal Abbas Rangarajan Sudarsan Hermann J. Eberl
Mathematical Biosciences & Engineering 2012, 9(2): 215-239 doi: 10.3934/mbe.2012.9.215
We investigate the role of non shear stress and shear stressed based detachment rate functions for the longterm behavior of one-dimensional biofilm models. We find that the particular choice of a detachment rate function can affect the model prediction of persistence or washout of the biofilm. Moreover, by comparing biofilms in three settings: (i) Couette flow reactors, (ii) Poiseuille flow with fixed flow rate and (iii) Poiseuille flow with fixed pressure drop, we find that not only the bulk flow Reynolds number but also the particular mechanism driving the flow can play a crucial role for longterm behavior. We treat primarily the single species-case that can be analyzed with elementary ODE techniques. But we show also how the results, to some extent, can be carried over to multi-species biofilm models, and to biofilm models that are embedded in reactor mass balances.
keywords: Mathematical model detachment. biofilm
Mathematical analysis of an in vivo model of mitochondrial swelling
Messoud Efendiev Mitsuharu Ôtani Hermann J. Eberl
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 4131-4158 doi: 10.3934/dcds.2017176

We analyze the effect of Robin boundary conditions in a mathematical model for a mitochondria swelling in a living organism. This is a coupled PDE/ODE model for the dependent variables calcium ion contration and three fractions of mitochondria that are distinguished by their state of swelling activity. The model assumes that the boundary is a permeable 'membrane', through which calcium ions can both enter or leave the cell. Under biologically relevant assumptions on the data, we prove the well-posedness of solutions of the model and study the asymptotic behavior of its solutions. We augment the analysis of the model with computer simulations that illustrate the theoretically obtained results.

keywords: PDE/ODE coupling longtim dynamics Robin boundary conditions mitochondria partial and complete swelling

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