Application of variational inequalities to the moving-boundary problem in a fluid model for biofilm growth

We consider a moving-boundary problem associated with the fluid model for biofilm growth proposed by J. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM J. Appl. Math. 62 (3) (2001) 853–869. Notions of classical, weak, and variational solutions for this problem are introduced. Classical solutions with radial symmetry are constructed, and estimates for their growth given. Using a weighted Baiocchi transform, the problem is reformulated as a family of variational inequalities, allowing us to show that, for any initial biofilm configuration at time t=0$t=0$ (any bounded open set), there exists a unique weak solution defined for all t≥0$t\ge 0$.     

Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models

The work presents an analysis of solutions to a free boundary value problem for a multispecies biofilm growth model in one space dimension. The mathematical model consists of a system of nonlinear partial differential equations and a free boundary. It is quite general and can include a large variety of special situations. An existence and uniqueness theorem is discussed and properties of solutions are given. As a numerical application, simulations for a heterotrophic–autotrophic competition are developed by the method of characteristics.     

3D finite element analysis of biofilm growth considering nutrient diffusion

There exist a lot of models that are able to describe the behavior of biofilm systems. They can be classified into (1) analytical models, (2) pseudo-analytical models, and (3) numerical models. Based on the numerous advantages of numerical models in this work a 3D mechanical biofilm model is discussed and implemented into a finite element program in order to simulate growth effects coupled with the nutrient diffusion through the biofilm. Based on three-dimensional biofilm structures, numerical examples are presented in order to show the ability of the new modeling approach. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

3‐D spatio–temporal structures of biofilms in a water channel

We develop a numerical predictive tool for multiphase fluid mixtures consisting of biofilms grown in a viscous fluid matrix by implementing a second‐order finite difference discretization of the multiphase biofilm model developed recently on a general purpose graphic processing unit. With this numerical tool, we study a 3‐D biomass–flow interaction resulting in biomass growth, structure formation, deformation, and detachment phenomena in biofilms grown in a water channel in quiescent state and subject to a shear flow condition, respectively. The numerical investigation is limited in the viscous regime of the biofilm–solvent mixture. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, the so‐called mushroom shape growth pattern. The simulated biomass growth both in density and thickness matches very well with the experimentally grown biofilm in a water channel. When shear is imposed at a boundary, our numerical studies reproduce wavy patterns, pinching, and streaming phenomena observed in biofilms grown in a water channel. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical simulation of biofilm growth in porous media

Biofilms are very important in controlling pollution in aquifers. The bacteria may either consume the contaminant or form biobarriers to limit its spread. In this paper we review the mathematical modeling of biofilm growth at the microscopic and macroscopic scales, together with a scale-up technique. At the pore-scale, we solve the Navier-Stokes equations for the flow, the advection-diffusion equation for the transport, together with equations for the biofilm growth. These results are scaled up using network model techniques, in order to have relations between the amount and distribution of the biomass, and macroscopic properties such as permeability and porosity. A macroscopic model is also presented. We give some results.     

Moving boundary problem for the detachment in multispecies biofilms

The work presents the qualitative analysis of the free boundary value problem related to the detachment process in multispecies biofilms. In the framework of continuum approach to one-dimensional mathematical modelling of multispecies biofilm growth, we consider the system of nonlinear hyperbolic partial differential equations governing the microbial species growth, the differential equation for the biomass velocity, the differential equation that governs the free boundary evolution and also accounts for detachment, and the elliptic system for substrate dynamics. The characteristics are used to convert the original moving boundary equation into a suitable differential equation useful to solve the mathematical problem. We also provide another form of the same equation that could be used in numerical applications. Several properties of the solutions to the free boundary problem are shown, such as positiveness of the functions that describe the microbial concentrations and estimates on the characteristic functions. Uniqueness and existence of solutions are proved by introducing a suitable system of Volterra integral equations and using the fixed point theorem.

     

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

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 well‐posedness 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. Copyright © 2011 John Wiley & Sons, Ltd.     

Continuum approach to mathematical modelling of multispecies biofilms

The work presents a contribution to the mathematical modelling of formation and growth of multispecies biofilms in the framework of continuum approach, without claiming to be complete. Mathematical models for biofilms often lead to consider free boundary value problems for nonlinear PDEs. The emphasis is on the qualitative analysis, uniqueness and existence of solutions and their main properties. Biofilm life is a complex biological process formed by several phases from the formation, development of colonies, attachment and detachment of microbial mass from (to) biofilm to (from) bulk liquid. Most of these processes are modelled and discussed. Moreover, some problems of interest for engineering and biological applications are considered. Indeed, we discuss the free boundary value problem related to biofilm reactors extensively used in wastewater treatment, and the invasion of new species into an already constituted biofilm with the successive colonizations. The main mathematical methodology used is the method of characteristics. The original differential problem is converted to integral equations. Then, the fixed point theorem is applied.     

Analysis of a bacterial model with nutrient‐dependent degenerate diffusion

We introduce and study a degenerate reaction‐diffusion system that can serve as a model prototype for the pattern formation of a bacterial multicellular community where the bacteria produce biofilm, grow and spread in the presence of a nutrient. Under proper conditions on the reaction terms, we prove the global existence and the uniqueness of solutions and illustrate the possible model behaviour in numerical simulations for a two‐dimensional setting. Copyright © 2014 John Wiley & Sons, Ltd.     

Free boundary problem for an initial cell layer in multispecies biofilm formation

The initial attached cell layer in multispecies biofilm growth is considered. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The method of characteristics is used to convert the differential system to Volterra integral equations for which an existence and uniqueness theorem is proved. Subsequently, we show that the free boundary is an increasing function of time and biomass concentrations are positive in agreement with the biological process.     

Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.     

Numerical analysis of a parabolic variational inequality system modeling biofilm growth at the porescale

In this article, we consider a system of two coupled nonlinear diffusion–reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions.     

A finite element model for biofilm volume growth

In this work, we present a continuum-based approach for biofilm volume growth. The deformation gradient will be multiplicatively decomposed into two parts: a growth part due to bacteria formation and an elastic part due to the interaction with the environment. In order to define the growth behaviour of biofilms, we use the Monod approach that depends non-linearly on the substrate concentration. The substrate concentration in the biofilm is computed by means of a diffusion process, which includes substrate consumption, together with the mechanical behaviour as part of a coupled problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of the power law logistic population model with slowly varying coefficients

We apply a multiscale method to construct general analytic approximations for the solution of a power law logistic model, where the model parameters vary slowly in time. Such approximations are a useful alternative to numerical solutions and are applicable to a range of parameter values. We consider two situations—positive growth rates, when the population tends to a slowly varying limiting state; and negative growth rates, where the population tends to zero in infinite time. The behavior of the population when a transition between these situations occurs is also considered. These approximations are shown to give excellent agreement with the numerical solutions of test cases. Copyright © 2012 John Wiley & Sons, Ltd.     

A three-dimensional steady-state tumor system

The growth of tumors can be modeled as a free boundary problem involving partial differential equations. We consider one such model and compute steady-state solutions for this model. These solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions. Linear and nonlinear stability for these solutions are determined numerically.     

含公共开支的R&D经济增长模型(英文)

本文讨论了含公共开支的经济增长模型 ,避免了对生产函数的不恰当的假设 ,生产函数的形式是很一般的 ,因此经济系统是复杂的 ,但通过精巧的数学方法 ,得到确定的均衡点 ,并且给出解为正的充分条件。最后 ,分析了系统的动态性质 ,给出了经济沿稳定流形收敛于均衡点的条件     

Interpolation solution in generalized stochastic exponential population growth model

In this paper, first we consider model of exponential population growth, then we assume that the growth rate at time t is not completely definite and it depends on some random environment effects. For this case the stochastic exponential population growth model is introduced. Also we assume that the growth rate at time t depends on many different random environment effect, for this case the generalized stochastic exponential population growth model is introduced. The expectations and variances of solutions are obtained. For a case study, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals in each year.     

多重嵌套的可计算非线性动态投入产出模型及其平衡增长解

给出多重嵌套的可计算非线性动态投入产出模型,并给出相应的价格、利润率、产出结构、增长率的平衡增长解计算公式.从本文给出的模型可以看出非线性投入产出模型可以有无穷多种不同形式,其中只有1种为众所周知的列昂惕夫线性投入产出模型.本文模型的重要意义在于它是线性投入产出模型与CGE（可计算一般均衡）模型的统一.它既克服了线性投入产出模型资本与劳动不可替代的缺点,又可方便地求解动态CGE的平衡增长解.     

First integrals and exact solutions of the SIRI and tuberculosis models

The first integrals and exact solutions of mathematical models of epidemiology: a susceptible‐infected‐recovered‐infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first‐order nonlinear ordinary differential equations, and this system is replaced by one which contains a second‐order ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic behavior of an unstirred chemostat model with internal inhibitor

This paper deals with a chemostat model with an internal inhibitor. First, the elementary stability and asymptotic behavior of solutions of the system are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which measures the effect of the inhibitor, plays a very important role in deciding the stability and longtime behavior of solutions of the system. The results show that if μ is sufficiently large, this model has no coexistence solution and one of the semitrivial equilibria is a global attractor when the maximal growth rate a of the species u lies in certain range; but when a belongs to another range, all positive solutions of this model are governed by a limit problem, and two semitrivial equilibria are bistable. The main tools used here include monotone system theory, degree theory, bifurcation theory and perturbation technique.