Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

Miti, Filippo (2016) Mathematical models for cellular aggregation: the chemotactic instability and clustering formation. [Laurea magistrale], Università di Bologna, Corso di Studio in Matematica [LM-DM270]
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Abstract

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Abstract
Tipologia del documento
Tesi di laurea (Laurea magistrale)
Autore della tesi
Miti, Filippo
Relatore della tesi
Scuola
Corso di studio
Indirizzo
Curriculum A: Generale e applicativo
Ordinamento Cds
DM270
Parole chiave
chemotaxis parabolic model mean-field approximation degenerate diffusivity Cattaneo's law hydrodynamic model friction force spectral analysis clusters network patterns Jeans instability
Data di discussione della Tesi
28 Ottobre 2016
URI

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