Reaction-diffusion models have been widely successful in explaining a large variety of patterning phenomena in biology ranging from embryonic development to cancer growth and angiogenesis. Firstly proposed by Alan Turing in 1952 and applied to a simple two-component system, reaction-diffusion models describe spontaneous spatial pattern formation, driven purely by interactions of the system components and their diffusion in space. Today, access to unprecedented amounts of quantitative biological data allows us to build and test biochemically accurate reaction-diffusion models of intracellular processes. However, any increase in model complexity increases the number of unknown parameters and thus the computational cost of model analysis. To efficiently characterize the behavior and robustness of models with many unknown parameters is, therefore, a key challenge in systems biology. Here, we propose a novel computational framework for efficient high-dimensional parameter space characterization of reaction-diffusion models. The method leverages the $L_p$-Adaptation algorithm, an adaptive-proposal statistical method for approximate high-dimensional design centering and robustness estimation. Our approach is based on an oracle function, which describes for each point in parameter space whether the corresponding model fulfills given specifications. We propose specific oracles to estimate four parameter-space characteristics: bistability, instability, capability of spontaneous pattern formation, and capability of pattern maintenance. We benchmark the method and demonstrate that it allows exploring the ability of a model to undergo pattern-forming instabilities and to quantify model robustness for model selection in polynomial time with dimensionality. We present an application of the framework to reconstituted membrane domains bearing the small GTPase Rab5 and propose molecular mechanisms that potentially drive pattern formation.

Ce travail de thèse est dédié à la visualisation par microscopie optique des transitions de phases, thermo- et photo-induites dans des monocristaux à transition de spin. L’étude des cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] a permis de montrer la possibilité de contrôler la dynamique de l’interface HS/BS (haut spin/bas spin) par une irradiation lumineuse appliquée sur toute la surface du cristal ou de manière localisée. Les investigations expérimentales menées sur l’effet de l’intensité de la lumière sur la température de transition ont mis en évidence d’une part l’importance du couplage entre le cristal et le bain thermique, et d’autre part le rôle de la diffusion de la chaleur dans le monocristal. En parallèle, un modèle basé sur une description de type Ginzburg-Landau, a permis de mettre sur pied une description de type réaction diffusion des effets spatio-temporels accompagnant la transition de spin dans un monocristal. Celui-ci a permis d’identifier et de comprendre le rôle des paramètres pertinents entrant en jeu dans le contrôle du mouvement de l’interface HS/BS. Les résultats obtenus sont très encourageants et reproduisent avec une grande fidélité les données expérimentales. Cependant l’origine de l’orientation de l’interface HS/BS observée par microscopie optique dans les cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] était restée mystérieuse. Pour résoudre cette question, nous avons développé un modèle électro-élastique qui tient compte du changement de volume au cours de la transition de spin. Ce dernier nous a conduits à analyser l’effet de la symétrie du réseau cristallin et de la forme du cristal sur l’orientation de l’interface élastique. En l’appliquant au composé [{Fe(NCSe)(py)2}2(m-bpypz)], en tenant compte du caractère anisotrope du changement de la maille élémentaire lors du passage HSBS, nous avons réussi à retrouver quantitativement l’orientation du front observée expérimentalement en microscopie optique. Ceci confirme bien le rôle primordial de l’élasticité dans le comportement des matériaux à transition de spin. Des études sous lumière à très basse température nous ont donné la possibilité de suivre en temps réel, l’effet LIESST (Light Induced Excited Spin State Trapping), la re-laxation coopérative du cristal ainsi que l’instabilité photo-induite LITH (Light Induced Thermal Hysteresis). Un monde fascinant est apparu autour de cette dernière, avec la présence de comportements totalement inédits. Ainsi, et pour la première fois, nous avons mis en évidence l’existence de phénomènes d’auto-organisation et de comportements autocatalytiques du front de transition. Cette physique non-linéaire dénote un comportement actif du cristal, par suite d’une subtile préparation autour d’un état instable. Ces comportements rappellent les structures dissipatives de Turing et ouvrent des perspectives fascinantes pour cette thématique, tant sur le plan expérimental que théorique. / This thesis work is devoted to visualization by optical microscopy of thermo- and photo-induced phase transitions, in switchable spin transition single crystals. The study of crystals of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] showed the possibility to control reversibly the dynamics of the HS/LS interface through a photo-thermal effect generated by an irradiation of the whole crystal or using a spatially localized light spot on the crystal surface. The investigations of the effect of the light intensity on the transition temperature have highlighted the importance of the coupling between the crystal and the thermal bath in these experiments. Concomitantly, we developped a reaction diffusion model allowing to describe and iden-tify the relevant physical parameters involved in the control of the movement of HS/LS interface. The obtained results are very encouraging and reproduce the main features of the experimental data. However the origin of the interface orientation observed by the optical microscopy in the crystal of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] re-mained mysterious, and needed an elastic approach to be handled. At this end, an electro-elastic model including the volume change at the spin transition was developed. By taking into account for the anisotropy of the unit cell deformation at the transition, we were able to reproduce quantitatively the experimental HS/LS interface orientation. This result confirms the crucial role of the lattice symmetry and its elastic properties in the emergence of a stable interface orientation. The last part of the thesis is devoted to the investigation of photo-induced effects at very low temperatures (~10K). There, we visualized for the first time the real time transformation of a single crystal under LIESST (Light Induced Excited Spin State Trapping) effect as well as its subsequent relaxation at higher temperatures. We have also studied the light induced instabilities through investigation on the LITH (Light Induced Thermal Hysteresis) loops. Around the latter, a fascinating world made of nonlinear effects, and patterns formation emerged, recalled the well known Turing structures. These results lead to new horizons that will give access to new theories and original experimental observations that will enrich the topics opening the new avenues to study of nonlinear phenomena in spin crossover solids.

Ce travail est dédié à la modélisation multi-échelle des phénomènes liés à la transition de spin dans des composés du Fe(II). Le développement d'un modèle macroscopique type réaction-diffusion pour la transition de phase à partir de l'Hamiltonien d'Ising a permis l'étude théorique des aspects spatio-temporels de la fraction haut-spin lors de la transition de phase du premier ordre dans des monocristaux commutables. La comparaison à l'expérience a conduit à de très bons accords pour le comportement du front de transition, ce qui a permis de mieux comprendre les mesures de microscopie optique. Ce travail a été étendu à l'étude des effets photo-thermiques qui causent l'échauffement du cristal par la lumière du microscope conduisant à un système d'équations différentielles couplées tenant compte du couplage thermique avec le bain.Ces équations prédisent des comportements non-linéaires du cristal dans son domaine bistable, tels que l’existence d’effets autocatalytiques, dont les conditions d'émergence ont été précisées. La dernière partie de la thèse est consacrée à une extension du modèle électro-élastique. Ici on démontre que la frustration élastique est à l'origine de la transition de spin en deux étapes et des transitions incomplètes. Ceci nous a amené aussi à prédire l'organisation de structures complexes de la fraction haut-spin dans les phases intermédiaires. Plusieurs types d'auto-organisation ont été révélés dont des structures modulées de la fraction haut-spin. Ce type de comportements a été observé expérimentalement très récemment dans les composés à transition de spin. / This work is devoted to the multiscale modeling of the spin transition phenomena in Fe(II) spin crossover compounds. The development of a macroscopic reaction-diffusion-like model for the phase transition from the Ising-like Hamiltonian allowed the theoretical study of the spatio-temporal behavior of the high-spin fraction accompanying the first-order phase transition in switchable spin crossover single crystals. The comparison to experiments led to an excellent agreement for the dynamics of the high-spin/low-spin interface which improved the understanding of the optical microscopy measurements. Next, this work was extended to the study of photothermic effects due to the crystal heating by the light of the microscope leading to a coupled system of differential equations accounting for the thermal coupling with the bath temperature. These equations predict nonlinear behaviors for crystals in the bistable region, such as the autocatalytic effects, for which we established the conditions of their emergence. The last part of this thesis is devoted to an extension of the electro-elastic model. Here we prove that the elastic frustration is at the origin of the existence of two-step and of incomplete spin crossover transitions. Furthermore, this model allowed us to predict structures of complex patterns in high-spin fractions for intermediate phases. Several types of self-organisation were revealed such as the spatially-modulated structures of the high-spin fractions. Some of these behaviors have been experimentally observed, very recently, in spin crossover compounds.

<p>A wide variety of phase evolution phenomena observed in solids such as intermetallic growth at the junction between two metals subjected to high temperature, growth of oxide on metal surfaces due to atmospheric exposure and void evolution induced by electromigration in microelectronic devices for example, can be classified as being driven by reaction-diffusion processes. These phase evolution phenomena have a significant impact on material reliability for critical applications, and therefore, there is a requirement for modeling such reaction-diffusion driven phase evolution phenomena. It is difficult to analyze these due to the complexity of modeling the evolving interface between solid phases. Additional complexity is  due to the multi-physics nature of the diffusive and reactive processes. Diffusion in solids is driven by a variety of stimuli such as current, temperature and stress, in addition to the chemical potential. Therefore, there is a need for a model that accounts for the influence of such factors on phase evolution. In this thesis,  a generalized continuum based reaction-diffusion theory for phase and void evolution in solid state is developed. The derivation starts off with generalized interface balance laws for mass, momentum and energy. The thermodynamic entropy inequality for irreversible phase growth is derived for arbitrary anisotropic and inhomogeneous surface stress. These interface relations are combined with governing relations in the material bulk for the temperature, stress, electrical and concentration fields, to develop a general model capable of analyzing and describing phase evolution in solids. This theory is then applied to a variety of intermetallic phase and void evolution phenomena observed in microelectronics.</p> <p><br></p> <p>Electromigration induced voiding in thin metal films is an example of phase evolution that is an important reliability concern in microelectronics. Studies have reported that the electromigration induced void growth rate is inversely related to the adhesion of metal thin films with the base and capping layers. Electromigration experiments are performed on fabricated test devices with Cu thin films with SiNx and TiN capping layers. The observations from electromigration experiments on thin Cu metal films at a range of temperatures indicate that the contribution of interface adhesion strength to electromigration resistance decreases with an increase in temperature. The generalized reaction-diffusion theory developed here is modified to develop an expression to account for the effect of base and passivation layer adhesion and temperature on electromigration resistance of metal thin films. The void growth rates measured in the experiments are analyzed with the expression for void growth rate to estimate the interface adhesion strength for the Cu-TiN and Cu-SiNx interfaces. </p> <p><br></p> <p>Demand for increased bandwidth, power efficiency and performance requirements have resulted in a trend of reduction in size and pitch of Cu pillar-Solder micro-bump interconnects used in heterogeneously integrated packages. As the size of micro-bumps reduce, reliability challenges due to voiding in the solder joint and the growth of Cu-Sn intermetallics are observed. The underlying reaction-diffusion mechanisms responsible for Cu-Sn intermetallic growth and voiding in solder joints are unclear at this stage and require further investigation. The current practice of material characterization in micro-bumps involve destructive cross-sectioning and polishing of the micro-bumps after testing. These processes result in loss of continuity in the samples used for the experiments, and material removal due to abrasive polishing might result in a loss of critical information. Therefore, a novel test device capable of non-destructive characterization of Cu-Sn intermetallic growth and voiding in sub-30 micron size micro-bumps is designed and fabricated in this work. The fabricated test devices are subjected to thermal aging for over 1000 h and the underlying reaction-diffusion mechanisms behind the intermetallic phase and void evolution are investigated. </p> <p><br></p> <p>A reaction-diffusion mechanism is proposed explaining the evolution of  various Cu-Sn intermetallic phases and solder joint void observed from experiments. Using the reaction-diffusion mechanism inferred from the thermal aging experiments and the generalized reaction-diffusion theory for phase evolution developed in this thesis, a sharp interface model is developed for the evolution of Cu-Sn intermetallic phases and solder joint void. The diffuse interface phase field equivalent equations for the sharp interface model governing equations are developed using matched formal asymptotic analysis. The evolution of Cu-Sn intermetallic phase and voids in the solder joint are simulated for different temperatures and current density to demonstrate the validity of the phase field and sharp interface models.  </p> <p><br></p>

The retina is the ocular tissue responsible for the detection of light. Its extensive demand for oxygen, coupled with a concomitant elevated supply, renders this tissue prone to both hypoxia and hyperoxia. In this thesis, we construct mathematical models of the retina, formulated as systems of reaction-diffusion equations, investigating its oxygen-related dynamics in healthy and diseased states. In the healthy state, we model the oxygen distribution across the human retina, examining the efficacy of the protein neuroglobin in the prevention of hypoxia. It has been suggested that neuroglobin could prevent hypoxia, either by transporting oxygen from regions where it is rich to those where it is poor, or by storing oxygen during periods of diminished supply or increased uptake. Numerical solutions demonstrate that neuroglobin may be effective in preventing or alleviating hypoxia via oxygen transport, but that its capacity for oxygen storage is essentially negligible, whilst asymptotic analysis reveals that, contrary to the prevailing assumption, neuroglobin's oxygen affinity is near optimal for oxygen transport. A further asymptotic analysis justifies the common approximation of a piecewise constant oxygen uptake across the retina, placing existing models upon a stronger theoretical foundation. In the diseased state, we explore the effect of hyperoxia upon the progression of the inherited retinal diseases, known collectively as retinitis pigmentosa. Both numerical solutions and asymptotic analyses show that this mechanism may replicate many of the patterns of retinal degeneration seen in vivo, but that others are inaccessible to it, demonstrating both the strengths and weaknesses of the oxygen toxicity hypothesis. It is shown that the wave speed of hyperoxic degeneration is negatively correlated with the local photoreceptor density, high density regions acting as a barrier to the spread of photoreceptor loss. The effects of capillary degeneration and treatment with antioxidants or trophic factors are also investigated, demonstrating that each has the potential to delay, halt or partially reverse photoreceptor loss. In addition to answering questions that are not accessible to experimental investigation, these models generate a number of experimentally testable predictions, forming the first loop in what has the potential to be a fruitful experimental/modelling cycle.

Cette thèse porte sur l'analyse mathématique de modèles de réaction-dispersion de la forme [delta]tu=D(u) +f(x,u). L'objectif est de comprendre l'influence du terme de réaction f, de l'opérateur de dispersion D, et de la donnée initiale u0 sur la propagation des solutions de ces équations. Nous nous sommes intéressés principalement à deux types d'équations de réaction-dispersion : les équations de réaction-diffusion où l'opérateur de dispersion différentielle est D=[delta]2z et les équations intégro-différentielles pour lesquelles D est un opérateur de convolution, D(u)=J* u-u. Dans le cadre des équations de réaction-diffusion en milieu homogène, nous proposons une nouvelle approche plus intuitive concernant les notions de fronts progressifs tirés et poussés. Cette nouvelle caractérisation nous a permis de mieux comprendre d'une part les mécanismes de propagation des fronts et d'autre part l'influence de l'effet Allee, correspondant à une diminution de la fertilité à faible densité, lors d'une colonisation. Ces résultats ont des conséquences importantes en génétique des populations. Dans le cadre des équations de réaction-diffusion en milieu hétérogène, nous avons montré sur un exemple précis comment la fragmentation du milieu modifie la vitesse de propagation des solutions. Enfin, dans le cadre des équations intégro-différentielles, nous avons montré que la nature sur- ou sous-exponentielle du noyau de dispersion J modifie totalement la vitesse de propagation. / This thesis deals with the mathematical analysis of reaction-dispersion models of the form [delta]tu=D(u) +f(x,u). We investigate the influence of the reaction term f, the dispersal operator D and the initial datum u0 on the propagation of the solutions of these reaction-dispersion equations. We mainly focus on two types of equations: reaction-diffusion equations (D=[delta]2z and integro-differential equations (D is a convolution operator, D(u)=J* u-u). We first investigate the homogeneous reaction-diffusion equations. We provide a new and intuitive explanation of the notions of pushed and pulled traveling waves. This approach allows us to understand the inside dynamics the traveling fronts and the impact of the Allee effect, that is a low fertility at low density, during a colonisation. Our results also have important consequences in population genetics. In the more general and realistic framework of heterogeneous reaction-diffusion equations, we exhibit examples where the fragmentation of the media modifies the spreading speed of the solution. Finally, we investigate integro-differential equations and prove that emph{fat-tailed} dispersal kernels J, that is kernels which decay slower than any exponentially decaying function at infinity, lead to acceleration of the level sets of the solution u.

Made available in DSpace on 2015-03-04T18:51:19Z (GMT). No. of bitstreams: 1 TeseEsbel.pdf: 1480858 bytes, checksum: c16438606b97781ccf3a3353c4d9f319 (MD5) Previous issue date: 2010-02-28 / Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior / 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. / 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.

Motivated by the existence of complex behaviors arising from interactions between chemistry and fluid dynamics in numerous research problems and every-day life situations, we theoretically investigate the dynamics resulting from the interplay between chemistry, diffusion, and fluid motions in a reactive aqueous solution. As a chemical reaction induces changes in the temperature and in the composition of the reactive medium, such a reaction can modify the properties of the solution (density, viscosity, surface tension,…) and thereby trigger convective motions, which in turn affect the reaction. Two classes of convective flows are commonly occurring in solutions open to air, namely Marangoni flows arising from surface tension gradients and buoyancy flows driven by density gradients. As both flows can be induced by compositional changes as well as thermal changes and in turn modify them, the resulting experimental dynamics are often complex. The purpose of our thesis is to gain insight into these intricate dynamics thanks to the theoretical analysis of model systems where only one type of convective flow is present. In particular, we numerically study the spatio-temporal evolution of model chemical fronts resulting from the coupling between reactions, diffusion, and convection. Such fronts correspond to self-organized interfaces between the products and the reactants, which typically have different density and surface tension. Fluid motions are therefore spontaneously induced due to these differences across the front. In this context, we first address the propagation of a model autocatalytic front in a horizontal solution layer, in the presence of pure Marangoni convection on the one hand and of pure buoyancy convection on the other hand. We evidence that, in both cases, the system attains an asymptotic dynamics characterized by a steady fluid vortex traveling with the front at a constant speed. The presence of convection results in a deformation and acceleration of the chemical front compared to the reaction-diffusion situation. However we note important differences between the Marangoni and buoyancy cases that could help differentiate experimentally between the influence of each hydrodynamic effect arising in solutions open to the air. We also consider how the kinetics and the exothermicity of the reaction influence the dynamics of the system. The propagation of an isothermal front occurring when two diffusive reactants are initially separated and react according to a simple bimolecular reaction is next studied in the presence of chemically-induced buoyancy convection. We show that the reaction-diffusion predictions established for convection-free systems are modified in the presence of fluid motions and propose a new way to classify the various possible reaction-diffusion-convection dynamics./En induisant des changements de composition et de température, une réaction chimique peut modifier les propriétés physiques (densité, viscosité, tension superficielle,…) de la solution dans laquelle elle se déroule et ainsi générer des mouvements de convection qui, à leur tour, peuvent affecter la réaction. Les deux sources de convection les plus courantes en solution ouverte à l’air sont les gradients de tension superficielle, ou effets Marangoni, et les gradients de densité. Comme ces deux sources sont en compétition et peuvent toutes deux résulter de différences de concentration ou de température, les dynamiques observées expérimentalement sont souvent complexes. Le but de notre thèse est de contribuer à la compréhension de telles dynamiques par une étude théorique analysant des modèles réaction-diffusion-convection simples. En particulier, nous étudions numériquement l’évolution spatio-temporelle de fronts chimiques résultant du couplage entre chimie non-linéaire, diffusion et hydrodynamique. Ces fronts constituent l’interface auto-organisée entre les produits et les réactifs qui typiquement ont des densités et tensions superficielles différentes. Des mouvements du fluide peuvent dès lors être spontanément initiés dus à ces différences au travers du front. Dans ce contexte, nous étudions la propagation d’un front chimique autocatalytique se propageant dans une solution aqueuse horizontale, d’une part en la seule présence d’effets Marangoni, et d’autre part en présence uniquement d’effets de densité. Nous avons montré que dans les deux cas, le système atteint une dynamique asymptotique caractérisée par la présence d’un rouleau de convection stationnaire se propageant à vitesse constante avec le front. Ce front est à la fois déformé et accéléré par les mouvements convectifs par rapport à la situation réaction-diffusion. Nous avons mis en évidence d’importantes différences entre les deux régimes hydrodynamiques qui pourraient aider les expérimentateurs à différencier les effets de tension superficielle de ceux de densité générés par la propagation de fronts chimiques en solution. Nous avons également considéré l’influence de la cinétique de réaction ainsi que de l’exothermicité sur la dynamique de ces fronts. Enfin, nous avons étudié la propagation en présence de convection d’un front de réaction impliquant deux espèces de densités différentes, initialement séparées et réagissant selon une cinétique bimoléculaire. Nous avons montré que la convection modifie les propriétés réaction-diffusion du système et nous proposons de nouveaux critères pour classifier les dynamiques réaction-diffusion-convection.

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.