Mathematical and Numerical Analysis of a Modified Keller-Segel Model with General Diffusive Tensors.

Georges Chamoun,Mazen Saad,Raafat Talhouk

doi:10.11145/j.biomath.2013.12.071

Georges Chamoun, Mazen Saad + Show 1 more

Open Access

https://doi.org/10.11145/j.biomath.2013.12.071

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Abstract

This paper is devoted to the mathematical analysis of a model arising from biology, consisting of diffusion and chemotaxis with volume filling effect. Motivated by numerical and modeling issues, the global existence in time and the uniqueness of weak solutions to this model is investigated. The novelty with respect to other related papers lies in the presence of a two-sidedly nonlinear degenerate diffusion and anisotropic heterogeneous diffusion tensors, where we prove global existence and uniquenessunder further assumptions. Moreover, we introduce and we study the convergence analysis of the combined scheme applied to this anisotropic Keller-Segel model with general tensors. Finally, a numerical test is given to prove the effectiveness of the combined scheme.

Highlights

Chemotaxis, the directed movement of cells in response to chemical gradients, plays an important role in many biological fields, such as embrogenesis, immunology, cancer growth and wound healing
In order to discretize our model (1), it is wellknown that classical finite volume methods not permit to handle anisotropic tensors in the diffusive terms but it is very useful, especially the technique ”upwind”, to discretize the convective term since it does not allow instabilities in the numerical solution
Our first result is the following existence Theorem of weak solutions proved by using a technique of semidiscretization in time for the regularized nondegenerate problem associated to (1)

Summary

INTRODUCTION

Chemotaxis, the directed movement of cells in response to chemical gradients, plays an important role in many biological fields, such as embrogenesis, immunology, cancer growth and wound healing This behavior enables many living organisms to locate nutrients, avoid predators or find animals of the same species. In order to discretize our model (1), it is wellknown that classical finite volume methods not permit to handle anisotropic tensors in the diffusive terms but it is very useful, especially the technique ”upwind”, to discretize the convective term since it does not allow instabilities in the numerical solution. A numerical test will be given to illustrate the effectiveness of our combined scheme applied to the anisotropic Keller-Segel model (1)

SETTING OF THE PROBLEM

NUMERICAL SCHEME

NUMERICAL TEST

CONCLUSION