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

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).

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

## Altri metadati

Tipologia del documento

Tesi di laurea
(NON SPECIFICATO)

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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