ABSTRACT. A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing. Fluid dynamics model and Hyperbolic equations and Phototrophic biofilms and Front propagation AMS : 92C17, 35L50, 65M06.
Abstract. In this article we propose a model to describe the inflammatory process which occurs during ischemic stroke. First, an introduction to some basic concepts about the biological phenomenon is given. Then, a detailed derivation of the model and the numerical scheme used are presented. Finally, the studies of the model robustness and sensitivity are showed and some numerical results on the time and space evolution of the process are presented and discussed.Résumé. Dans cet article, nous nous attachons à modéliser le processus inflammatoire lors d'un accident vasculaire cérébral ischémique. Tout d'abord, nous expliquons brièvement les bases biologiques de ce phénomène. Puis, nous décrivons les équations et les schémas numériques retenus pour le modéliser et l'implémenter. Nous présentons ensuite des études de robustesse et de sensibilité et montrons et discutons les premiers résultats de simulation obtenus.
In this article, we study in details the fluid dynamics system proposed in Clarelli et al. (2013) to model the formation of cyanobacteria biofilms. After analyzing the linear stability of the unique non trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases.
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