In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.

When the finite element method is used to solve boundary value problems, the corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters. There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation. Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment. In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown. Numerical experiments for the equilibrated residual method, for the hierarchical error estimator and for the adaptive algorithm confirm the theory. The adaptive algorithm shows its potential by creating the anisotropic mesh for the problem with the boundary layer starting with a very coarse isotropic mesh.

Epidermal wound healing requires the coordinated involvement of complex cellular and biochemical processes. In the case of epidermal wounds associated with burns, the healing process may be less than optimal and may take a significant amount of time, possibly resulting in infection and scarring. An innovative method to assist in the repair of the epidermis (the outer layer of skin) is to use an aerosolised apparatus. This method involves taking skin cells from an area of the patient's undamaged skin, culturing the cells in a laboratory, encouraging them to rapidly proliferate, then harvesting and separating the cells from each other. The cells are then sprayed onto the wound surface. We investigate this novel treatment strategy for the healing of epidermal wounds, such as burns. In particular, we model the application of viable cell colonies to the exposed surface of the wound with the intent of identifying key factors that govern the healing process. Details of the evolution of the colony structure are explored in this two-dimensional model of the wound site, including the effect of varying the initial population cluster size and the initial distribution of cell types with different proliferative capacities. During injury, holoclones (which are thought to be stem cells) have a large proliferative capacity while paraclones (which are thought to be transient amplifying cells) have a more limited proliferative capacity. The model predicts the coverage over time for cells that are initially sprayed onto a wound. A detailed analysis of the underlying mathematical models yields novel mathematical results as well as insight into phenomena of healing processes under investigation. Two one-dimensional systems that are simplifications of the full model are investigated. These models are significant extensions of Fisher's equation and incorporate the mixed clonal population of quiescent and active cells. In the first model, an active cell type migrates and proliferates into the wound and undergoes a transition to a quiescent cell type that neither migrates nor proliferates. The analysis yields the identification of the key parameter constraints on the speed of the healing front of the cells on this model and hence the rate of healing of epidermal wounds. Approximations for the maximum cell densities are also obtained, including conditions for a less than optimal final state. The second model involves two active cell types with different proliferative capacity and a quiescent cell type. This model exhibits two distinct behaviours: either both cell types coexist or one of them dies out as the wound healing progresses leaving the other cell type to fill the wound space. Conditions for coexistence are explored.

L'identité des organes floraux est définie par l’expression de gènes homéotiques appartenant à la famille des MADS-box au début du développement floral. Un de ces gènes, AGAMOUS (AG), est responsable de l’identité des étamines et des carpelles chez Arabidopsis thaliana. Dans ce manuscrit, je tente de comprendre les propriétés spatiales et temporelles de l’expression d’AG en cherchant à connaître les mécanismes impliqués dans le bon établissement de la dynamique d’expression d’AG pendant les jeunes stades du développement floral.Je débute par développer un modèle de réaction-diffusion qui prend en compte la croissance de la fleur pendant les stades d’intérêt, ainsi que quelques facteurs de transcriptions clefs impliqués dans la régulation d’AG. Ensuite j’ai imagé en direct et en 4D la croissance des fleurs pour quantifier l’activation de l’expression d’AG de son initiation à son patron d’expression stable. Je montre que son expression se déroule en deux phases: une phase de faible expression, et une phase de forte expression. Bien que toutes les cellules du dôme central de la fleur présentent un profil d’activation d’AG similaire, le temps précis au cours du développement où AG est activé est différent pour chacunes d’entre elles et est à l’origine de la stochasticité du patron d’expression. Avec l’aide du modèle, je propose quatres nouvelles hypothèses relatives à la régulation d’AG :AG est capable de maintenir sa propre activation en se liant directement à son second intron au travers d’un complexe protéique contenant au moins deux molécule d'AG, créant ainsi un seuil d'auto-activation.AP2 influence la valeur de ce seuil, restreint l’expression d’AG dans le dôme central de la fleur et produit un retard dans l’activation complète d’AG.LFY et WUS sont nécessaire à l’accumulation des protéines d’AG dans les cellules pour pouvoir atteindre le seuil d’auto-activation et obtenir une expression complète d’AG.Le mouvement d’AG est nécessaire pour obtenir l’expression d’AG dans toutes les cellules du dôme central. Pour prouver ces hypothèses, j’ai réalisé différentes expériences. En premier, utilisant une expérience de FRET-FLIM dans les protoplastes, nous proposons qu’AG est capable de s’associer en homodimer dans les cellules végétales. Néanmoins, sur-exprimer AG pour aider les cellules à atteindre le seuil d’auto-activation plus tôt que dans la plante sauvage ne semble pas modifier la dynamique d’expression de l’AG endogène. En deuxième, j’ai testé le rôle précis de LFY au cours des différentes phases et transitions de la dynamique d’expression d’AG en mutant les sites d'interactions spécifiques pour LFY au sein des séquences de régulation d’AG. Ces mutations retardent l’expression l’expression d’AG et modifient légèrement son patron d’expression. Je montre que seulement d’important retards dans l’activation d’AG induit des modifications phénotypiques. Ensuite, pour tester le rôle de la répression par AP2 dans la dynamique d’expression d’AG, j’analyse le rapporteur d’AG dans le contexte d’un mutant fort d’ap2. Dans ce mutant, l’expression d’AG s’étend à une région plus large et le retard entre l’initiation de l’expression d’AG et la transition entre les phases de faible et forte expressions est diminué. Ces résultats correspondent aux simulations du modèle. Finalement, pour comprendre l’importance du mouvement d’AG d’une cellule à l’autre dans sa propre dynamique, je bloque cette capacité de bouger en utilisant un tag de localisation nucléaire. Bien que cela induit un retard dans l’activation de quelques cellules au stade 3 au moment où toutes les cellules du dôme centrale de la fleur expriment AG dans la plante sauvage, ce retard n’a pas d’effets visible sur le phénotype. / The identity of flower organs is defined by the expression of homeotic genes during early development that belongs to the MADS-box family. One of these genes, AGAMOUS (AG), is responsible for the identity of the stamens and the carpels in Arabidopsis thaliana. In this manuscript, I attempt to fully understand the spatial and temporal properties of AG expression by investigating the mechanisms underlying the proper establishment of AG expression dynamics during the early stages of flower development. I start by developing a reaction-diffusion model that takes into account the growth of the flower at the relevant stages, as well as the few key transcription factors involved in AG regulation. Next I used real-time 4D imaging on growing flowers to quantify the activation of AG expression from its onset to the stable pattern. I show that the AG expression occurs in two phases: a low-expression phase and a high-expression phase. Thus although all cells of the central dome of the flower present similar profiles of AG activation, the precise developmental time at which AG is activated is different in each case, and is the origin of the initial stochastic pattern. With the aid of the model, I also propose four new hypotheses to explain AG regulation: AG is able to maintain its own activation by directly binding its own second intron through a protein complex containing at least two molecules of AG leading to the creation of an auto-activation threshold.AP2 influences the value of this threshold, restraining AG expression to the central dome of the flower and producing a delay in complete AG activation.LFY and WUS are necessary to accumulate AG proteins in cells in order to reach the auto-activation threshold and obtain a full expression of AG.AG movement is necessary to obtain expression of AG in every cell of the central dome. To prove these hypotheses, I have carried out various experiments, using FRET-FLIM in protoplast cells, we suggest that AG is able to form homo-dimers in plant cells. However, overexpressing AG to help cells reach the auto-activation threshold earlier than in the wild-type does not appear to alter the endogenous AG dynamics of expression. Secondly, I test the precise role of LFY in the different phases and transitions in the AG expression dynamics by mutating specific interaction sites for LFY within AG regulatory sequences. These mutations appear to delay AG expression and slightly modify its pattern of expression. I show that only important delays in AG activation induce phenotypic differences. Then, to test the role of AP2 repression in AG expression dynamics, I analyse the AG reporter in the context of a strong ap2 mutant. In these mutants, AG expression spreads to a wider region and reduces the delay between the onset of AG expression and the transition from low- to high-expression. These results match with simulations of the model. Lastly, to understand the importance of AG cell-to-cell movement in AG dynamics, I block its ability to move using a nuclear localisation tag. Although this induces a delay in the activation of few cells at stage 3, when all cells of the central dome of the flower express AG in the WT. This delay has no visible effects on the phenotype.

Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification. / This thesis is divided into two main parts. In the first part, we consider an example of reaction-diffusion-taxis system (Pε), which is a haptotaxis model - a mechanism about the spread of cancer cells. The main result concerns the convergence of the solution of System (Pε) to the solution of a free boundary problem (P0), where system (P0) is well-posed. In the second part, we consider a general class of Hénon type elliptic equations : −∆u = |x|^{α} f(u) in Ω ⊂ R^Nwith α > −2. We investigate two classical cases f(u) = e^u, |u|^{p−1} u and two others cases f(u) = u^{p}_{+} , f(u) is a general function. By studying the solutions which are stable outside a compact set (in particular, stable solutions and finite Morse index solutions) with different methods, we establish some classification results.

Made available in DSpace on 2016-06-02T20:27:39Z (GMT). No. of bitstreams: 1 3744.pdf: 1323892 bytes, checksum: 71a7a7180d61db167b8cbec4db2bbe8b (MD5) Previous issue date: 2011-05-30 / Universidade Federal de Sao Carlos / We address the question of existence of stationary stable solutions to a class of reaction-diffusion equations with spatial dependence in 2 and 3-dimensional bounded domains. The approach consists of proving the existence of local minimizer of the corres-ponding energy functional. For existence, it was enough to give sufficient conditions on the diffusion coefficient and on the reaction term to ensure the existence of isolated mi¬nima of the Γlimit functional of the energy functional family. In the second part we take the techniques developed in the first part to minimize functional in 2 and 3-dimensional rectangles, with and without constraint, solving in a more general form this problem, which was originaly proposed in 1989 by Robert Kohn and Peter Sternberg. / Na primeira parte deste trabalho, abordamos a existência de soluções estacioná-rias estáveis para uma classe de equações de reação-difusão com dependência espacial em domínios limitados 2 e 3-dimensionais. Esta abordagem foi feita via existência de míni¬mos locais dos funcionais de energia correspondentes. Para tal, foi suficiente encontrar condições no coeficiente de difusão e no termo de reação que garantam existência de míni¬mos isolados do funcional Γlimite da família de funcionais de energia. Na segunda parte, aproveitamos as técnicas desenvolvidas na primeira parte para minimizar funcionais em retângulos e paralelepípedos, com e sem vínculo, resolvendo de forma bem mais geral este problema, originalmente proposto em 1989 por Robert Kohn e Peter Sternberg.

O estudo dos mecanismos que desencadeiam as reações alérgicas é um tema de grande interesse científico na atualidade. A anafilaxia, reação alérgica sistêmica severa, tem ocupado um lugar de destaque nas pesquisas. Diferentes experimentos em laboratório, tanto in vivo quanto in vitro, assim como diferentes modelos matemáticos baseados nos resultados experimentais, têm procurado investigar a participação ou não dos mastócitos nesse mecanismo. No entanto, os resultados obtidos não são conclusivos, dividindo a comunidade científica em dois grupos: os que consideram determinante o papel dos mastócitos responsáveis pela liberação de histamina e os que consideram que a histamina não é o neurotransmissor determinante na reação anafilática. Trabalhos anteriores propuseram modelos diferenciais para simular processos relacionados com a reação anafilática na escala celular para o mecanismo de geração de potencial na membrana das células. Mais recentemente foi proposto, a nível de tecido, um modelo probabilístico para simular a resposta in vitro a antígenos. A nível de organismo têm sido propostos modelos de multi compartimentos para a cinética da histamina no fluido sanguíneo. Contudo, nenhum trabalho até o momento abordou a construção de um modelo capaz de descrever os processos que participam no mecanismo de reação anafilática nas diversas escalas. Neste trabalho propomos um modelo que integra as escalas celular e do tecido, que permite modelar experimentos in vitro, e que pode ser estendido para escala do organismo incluindo o fluxo sanguíneo para modelar experimentos in vivo. O modelo proposto integra o mecanismo de resposta elétrica a nível celular com o processo de reação-difusão da histamina e dos antígenos no tecido, considerando o mecanismo de reação mediado por mastócitos. Para integrar as duas escalas propomos uma relação constitutiva baseada em resultados experimentais da resposta mecânica (contração do tecido) a estímulos elétricos. Este modelo permite o desenho de novos experimentos especificamente direcionados ao estudo da reação anafilática, indicando os parâmetros a serem estimados. Utilizando-se o modelo proposto, foram realizadas simulações numéricas para uma ampla faixa de variação dos parâmetros visando identificar domínios com diferentes comportamentos do modelo. Uma análise dos resultados obtidos baseada em parâmetros adimensionais é apresentada. / The study of the mechanisms that set off allergic reactions is being a subject of great scientific interest. Anaphylaxis, severe systemic allergic reaction, occupies a prominence place in researches. Different laboratory experiments, in vivo as well as in vitro, and also different mathematical models based on experimental results, tries to investigate if mast cells takes part in those mechanisms or not. However, the obtained results are inconclusive, dividing the scientific community in two groups: one considering that mast cells have a prime role in releasing histamine, and another one which considers that histamine is not the determinative neurotransmitter in the anaphylactic reaction. Previous works proposed differential models to simulate processes related to anaphylactic reactions in the cellular scale for the cell membrane potential generation mechanism. More recently, it has been proposed a probabilistic model, in the tissue scale, to simulate an in vitro antigen response. In the organism level scale, multi-compartimental models have been proposed for the kinetics of histamine in the blood. Nevertheless, no work, until now, has proposed the construction of a model that is able to describe the processes that participate in the mechanism of anaphylactic reaction in different scales. In this work, a model is proposed that integrates the cellular and the tissue scales, allowing to model in vitro experiments, being capable to be extended to the organism scale by the inclusion of the blood flow to model in vivo experiments. The proposed model couples the electric response in the cellular level with the reaction-diffusion of histamine and antigens in the tissue, considering the reaction mechanism mediated by the mast cells. To integrate these two scales, it is proposed here a constitutive relation based on experimental results for the mechanical response (tissue contraction) to electric stimulus. This model allows to design experiments specifically related to the anaphylaxis reaction, indicating the parameters that should be estimated. With this model, numerical simulations have been performed for a wide variation range of the parameters to identify the different domains of the model. A dimensionless parameter based analysis is presented for the obtained results.

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La présente thèse est divisée en deux parties. La première partie concerne l'analyse mathématique et la contrôlabilité exacte à zéro pour une catégorie de systèmes structurés décrivant la dynamique d'une population d'insectes. La seconde partie est consacrée à l'étude de la stabilité de la conductivité d'un système de réaction diffusion modélisant l'activité électrique du coeur.Dans le chapitre 2, on considère que la population d'adultes se diffuse dans la vignoble,la fonction de la croissance des individus à chaque stade dépend des variations climatiques et de la variété des raisins. En utilisant la méthode de point fixe, on obtient l'existence et l'unicité des solutions du modèle. On démontre ensuite l'existence d'un attracteur global pour le système dynamique. Enfin, on utilise la théorie des opérateurs compacts et le théorème de point fixe de Krasnoselskii pour prouver l'existence des états stationnaires.Dans le chapitre 3, on traite le problème de contrôlabilité exacte du modèle de Lobesia Botrana, lorsque la fonction de croissance est égale à 1. On suppose que les quatre sous-catégories de ce système sont dans une phase statique. On obtient que la population d'oeufs peut être contrôlée à zéro. Ce résultat est basé sur des estimations à priori combinées avec un théorème de point fixe.Lorsque les papillons adultes se dispersent spatialement, on introduit un contrôle sur la population d'oeufs, de larves et de femelles dans une petite région du vignoble. On montre alors la contrôlabilité exacte à zéro pour les femelles.Dans la deuxième partie de cette thèse, on analyse la stabilité des coefficients de diffusion d'un système parabolique qui modélise l'activité électrique du coeur. On établit une estimation de Carleman pour le système de réaction-diffusion. En combinant cette estimation avec des estimations d'énergie avec poids on obtient le résultat de stabilité. / This thesis is divided into two parts.One is mainly devoted to make a qualitative analysis and exact null controlfor a class of structured population dynamical systems, and the other concernsstability of conductivities in an inverse problem of a reaction-diffusion systemarising in electrocardiology.In the first part, we study the dynamics ofEuropean grape moth, which has caused serious damages on thevineyards in Europe,North Africa, and even some Asian countries.To model this grapevine insect, physiologically structured multistage population systems are proposed.These systemshave nonlocal boundary conditions arising in nonlocal transition processes in ecosystem.We consider the questions of spatial spread of the populationunder physiological age and stage structures,and show global dynamical properties for the model.Furthermore, we investigate the control problem for this Lobesia botrana modelwhen the growth function is equal to $1$.For the case that four subclasses of this system are all in static station,we conclude that the population of eggs can be controlled to zero at acertain moment by acting on eggs.While the adult moths can disperse,we describe a control by a removal of egg and larvapopulation, and also on female moths in a small region of the vineyard.Then the null controllability for female mothsin a nonempty open sub-domain at a given time is obtained.In the second part, a reaction-diffusion system approximating a parabolic-elliptic systemwas proposed tomodel electrical activity in the heart. We are interested inthe stability analysis of an inverse problem for this model.Then we use the method of Carleman estimates and certain weight energyestimatesfor the identification of diffusion coefficients for the parabolicsystem to draw the conclusion.