Fiorentino, Ludovica (2024) Recent advances concerning pattern formation in reaction-diffusion and fluid flow models. [Tesi di dottorato]

Documento PDF Fiorentino_Ludovica_37.pdf Visibile a [TBR] Amministratori dell'archivio Download (10MB) | Richiedi una copia

Abstract

This doctoral thesis explores recent advances in pattern formation in reaction-diffusion models and fluid flow in porous media. It is structured into three main parts: a detailed mathematical background providing fundamental definitions concerning reaction-diffusion models and convection problems, recent developments in pattern formation for some reaction-diffusion models, and the onset of thermal convection in bi-disperse porous media. The research is grounded in both linear and nonlinear stability theory, supported by numerical simulations, to deepen the understanding of spatial and temporal dynamics across some physical and biological systems. Particular attention is given to the formation of Turing patterns, driven by diffusion-induced instability, also known as Turing instability. The analysis of this phenomenon is framed within the theoretical context of reaction-diffusion models, which offer both quantitative and qualitative insights into a wide range of spatiotemporal structures. These models are particularly effective across various fields, from morphogenetic processes to oscillating chemical reactions and population dynamics. The core of the thesis delves into reaction-diffusion models that incorporate nonlinear cross-diffusion terms, leading to more intricate spatial dynamics than those found in traditional models. These terms arise by distinguishing predator populations into different behavioral phases, such as searching and handling, which can inhibit the occurrence of Turing instability. In this regard, the thesis explores how nonlinear diffusion reshapes the stability landscape, offering new insights into the suppression or enhancement of spatial pattern formation. One significant application of this framework is the study of spatial distribution in ecological systems, particularly focusing on multi-predator interactions where different species exhibit nonlinear cross-diffusion. Through linear stability analysis, the conditions for the emergence of Turing patterns are investigated, revealing the intricate interplay between species mobility and interaction rates. This analysis provides deeper insights into how spatial heterogeneity develops within ecosystems, impacting species coexistence and distribution. Another key application of Turing instability is in the modeling of infectious disease dynamics, specifically tuberculosis (TB). The interaction between infected and uninfected macrophages and Mycobacterium tuberculosis is modeled using a reaction-diffusion system with chemotaxis, which plays a pivotal role in granuloma formation. Nonlinear stability analysis highlights the conditions under which spatial patterns in bacterial and immune cell populations emerge, offering new insights into the mechanisms driving disease progression and the formation of structured immune responses. Concerning fluid flow models, the onset of thermal convection in bi-disperse porous media is analyzed. The bi-disperse porous materials are characterized by the presence of two distinct porosities: the porosity of macropores and the porosity of micropores. These materials, with their dual porosity structure, have significant applications in various fields, including geophysics, engineering, and industrial processes. The study aims to advance the understanding of how heat transfer and fluid flow behave in such complex porous structures, particularly under the influence of thermal gradients. Precisely, linear and nonlinear stability analyses of the conduction solution are outlined, and the main result is that convection in bi-disperse media occurs at higher Rayleigh numbers compared to single-porosity media, indicating a delayed onset of convective heat transfer, which has important implications for insulation and heat management in practical applications. For each mathematical model introduced, the qualitative study is followed by numerical simulations. The results offer new perspectives on the mechanisms driving pattern formation in complex systems, with applications spanning ecology, epidemiology, and industrial engineering and contribute to a deeper understanding of how spatial interactions influence system dynamics, opening avenues for future research in the mathematical modeling of biological and physical systems.

Downloads

Actions (login required)

Modifica documento