Orientador: Roberto André Kraenkel / Banca: Gilberto Corso / Banca: Cláudia Pio Ferreira / Banca: Fernando Fagundes Ferreira / Banca: Francisco Antonio Bezerra Coutinho / Resumo: Este trabalho é composto de estudos independentes, mas seus temas são conectados entre si. Ele foi feito baseando-se no estudo de equações de reação-difusão e em reação-difusão-advecção. Vários modelos foram utilizados para representar populações e apresentam características em comum. As populações representadas por esses modelos difundem, crescem e saturam de forma semelhante a equação de Fisher-Kolmogorov e Lotka-Volterra e foram modeladas usando condições de contorno de Dirichlet. Domínios limitados e ilimitados foram usados para que melhor representassem as devidas e diferentes aplicações de dados coletados em campo e publicados em periódicos. Esse trabalho também leva em conta a aplicabilidade à habitats fragmentados, isoladas e não-isoladas. Como foco principal temos o estudo do movimento de populações que vivem nesses habitats mostrando que a qualidade e distribuição deles afeta no movimento das populações / Abstract: This thesis consists on independent studies, but its subjects are interconnected. It has been based on the study of reaction-diffusion-advection equations. Several models were used to represent populations and have some characteristics in common. The populations represented by the models spread, grow, and saturate in a way similar to that described by the Fisher-Kolmogorov and Lotka-Volterra equations, and were modeled using Dirichlet boundary conditions. Limited and unlimited domains were used to better represent the necessary applications and different data collected in the field and published in journals. This work also takes into account the applicability to fragmented habitats, isolated and not isolated. As the main focus we study the movement of populations living in these habitats, showing that the quality and distribution affects them the movement of populations / Doutor

No description available.

Traumatic injuries lead to articular cartilage lesion formation and result in the development of osteoarthritis. Recent research suggests that the early stage of mechanical injuries involve cell death (apoptosis and necrosis) and inflammation. In this thesis, we focus on building mathematical models to investigate the biological mechanism involving chondrocyte death and inflammatory responses in the process of cartilage degeneration. Chapter 1 describes the structure of articular cartilage, the process of carti- lage degeneration, and reviews of existing mathematical models. Chapter 2 presents a delay-diffusion-reaction model of cartilage lesion formation under cyclic loading. Computational methods were used to simulate the impact of varying loading stresses and erythropoietin levels. The model is parameterized with experimental results, and is therefore clinically relevant. Due to numerical limitations using delay differential equations, a new model is presented using tools for population dynamics. Chapter 3 presents an age and space-structured model of articular cartilage lesion formation un- der a single blunt impact. Age structure is introduced to represent the time delay in cytokine synthesis and cell transition. Numerical simulations produce similar tempo- ral and spatial patterns to our experimental data. In chapter 4, we extend our model under the cyclic loading setting. Chapter 5 builds a spatio-temporal model adapted from the former models, and investigates the distribution of model parameters using experimental data and statistical methods. Chapter 6 concludes.

acase@tulane.edu

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

L'étude mathématique des modèles et des méthodes de calcul en électrophysiologie des tissus cardiaques constitue la principale motivation de mes travaux de recherche en mathématiques appliquées. Ces travaux ont trouvé des applications en imagerie médicale et en bioingénierie grâce aux simulations numériques que nous avons rendues possibles. Les équations d'électrocardiologie, de type réaction-diffusion dégénérée, peuvent être discrétisées efficacement par des méthodes de volumes finis. <br />Ce mémoire synthétise l'ensemble des résultats de mes travaux dans ces domaines, c'est à dire : analyse des équations aux dérivées partielles d'électrocardiologie, expérimentation et applications numériques d'une part; introduction de nouveaux schémas et analyse numérique de méthodes de volumes finis pour des problèmes de diffusion anisotrope, de convection-diffusion et des systèmes hyperboliques linéaires d'autre part.<br />Ces travaux visent une meilleure compréhension scientifique des équations de l'électrophysiologie et plus généralement du fonctionnement électrique d'un tissu cardiaque ou du coeur entier.

Cette thèse est consacrée à l'étude des équations de réaction diffusion non-locale du type $u_(t)-(\int_(\R)J(x-y)[u(y)-u(x)]dy)=f(u)$. Ces équations non-linéaires apparaissent naturellement en physique et en biologie. On s'intéresse plus particulièrement aux propriétés (existence, unicité, monotonie) des solutions du type front progressif. Trois classes de non-linéarités $f$ (bistable, ignition, monostable) sont étudiées. L'existence dans les cas bistable et ignition est obtenue via une technique d'homotopie. Le cas monostable nécessite une autre approche. L'existence est obtenue via une approximation des équations sur des semi-intervales infinis $(-r,+\infty)$. L'unicité et la monotonie des solutions sont quand elles obtenues par méthode de glissement. Le comportement asymptotique ainsi que des formules pour les vitesses sont aussi établis.

No description available.

Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution. / eSSENCE

Mathematical models are now gaining in importance for descriptions of biological processes. In this thesis, such models have been used to identify and analyze principles that govern bacterial protein synthesis under amino acid limitation. New techniques, that are generally applicable for analysis of intrinsic fluctuations in systems of chemical reactions, are also presented. It is shown how multi-substrate reactions, such as protein synthesis, may display zero order kinetics below saturation, because an increase in one substrate pool is compensated by a decrease in another, so that the overall flow is unchanged. Under those conditions, metabolite pools display hyper sensitivity and large fluctuations, unless metabolite synthesis is carefully regulated. It is demonstrated that flow coupling in protein synthesis has consequences for transcriptional control of amino acid biosynthetic operons, accuracy of mRNA translation and the stringent response. Flow coupling also determines the choices of synonymous codons in a number of cases. The reason is that tRNA isoacceptors, cognate to the same amino acid, often read different codons and become deacylated to very different degrees when their amino acid is limiting for protein synthesis. This was demonstrated theoretically and used to successfully predict the choices of control codons in ribosome mediated transcriptional attenuation and codon bias in stress response genes. New tools for the analysis of internal fluctuations have been forged, most importantly, an efficient Monte Carlo algorithm for simulation of the Markov-process corresponding to the reaction-diffusion master equation. The algorithm makes it feasible to analyze stochastic kinetics in spatially extended systems. It was used to demonstrate that bi-stable chemical systems can display spontaneous domain separation also in three spatial dimensions. This analysis reveals geometrical constraints on biochemical memory circuits built from bistable systems. Further, biochemical applications of the Fokker-Planck equation and the Linear Noise Approximation have been explored.