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Modèles et analyses mathématiques pour les mouvements collectifs de cellules

Abstract : We investigate some mathematical models describing the collective motion of a population of cells, interacting through a chemical signal. We highlight the parabolic model of Patlak-Keller-Segel, and also the kinetic model of Othmer-Dunbar-Alt.

The first part consists in studying several variants of the classical PKS model, including for instance nonlinear diffusion of the cells, or a chemical diffusion law based on a logarithmic Green kernel. Next we tackle the global existence problem for the full parabolic-parabolic PKS system in the whole space $\mathbb{R}^2$. Independently we complexify the basic model by adding a second reactant, modifying henceforth the system's homogeneity. Finally we are able to weaken the previous global existence assumptions for the kinetic ODA model with delocalization effects.

In the second part we apply the critical mass instability to model phenomenologically a remarkable pattern formation issue in the human brain, namely the Baló's Concentric Sclerosis. A suitable coupling between a front propagation and the PKS model exhibits the concentric rings appearing in this disease in a reasonable way.

In the third part we adopt the recent optimal transportation viewpoint to analyse the one dimensional PKS model obtained previously (which captures the key features of the 2D PKS). Although the energy functional is not displacement convex, we prove that solutions converge to a unique stationary state, as soon as it exists. This interpretation is performed numerically: a discrete gradient flow adapted to the Wasserstein distance is analysed and simulated in one dimension of space.

Several appendices complete this work, among which one gathers all numerical aspects of this thesis.
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Submitted on : Thursday, February 14, 2008 - 10:28:55 AM
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Vincent Calvez. Modèles et analyses mathématiques pour les mouvements collectifs de cellules. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2007. Français. ⟨tel-00255811⟩

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