Cette thèse est dédiée à l’étude des propriétés de propagation de systèmes de réaction – diffusion issus de la dynamique des populations. Dans la première partie, on étudie la limite de forte compétition de systèmes à deux espèces. À l’aide de la ségrégation spatiale, on détermine le signe de la vitesse de l’onde progressive bistable. La généralisation aux ondes pulsatoires bistables en milieu spatialement périodique est ensuite envisagée afin d’étudier le rôle de l’hétérogénéité spatiale. Après avoir donné une condition suffisante pour l’existence de telles ondes ainsi qu’une condition suffisante pour l’existence d’états stationnaires stables susceptibles au contraire de bloquer l’invasion, on suppose qu’une famille d’ondes pulsatoires existe et on prouve un résultat semblable à celui obtenu en milieu homogène. Dans la seconde partie, des systèmes de type KPP à un nombre arbitraire d’espèces sont considérés. On étudie l’existence d’états stationnaires et d’ondes progressives, les propriétés qualitatives de ces solutions ainsi que la vitesse asymptotique de propagation de certaines solutions du problème de Cauchy. Cela résout des questions ouvertes sur les systèmes de mutation – compétition – diffusion, qui constituent le prototype de système de type KPP. Dans la troisième partie, on revient aux systèmes à deux espèces. Considérant cette fois-ci le cas monostable, on étudie les vitesses asymptotiques de propagation de certaines solutions du problème de Cauchy et, ce faisant, on montre l’existence de solutions décrivant l’invasion d’un territoire inhabité par un compétiteur faible mais rapide suivie de l’invasion de ce territoire par un compétiteur fort mais lent. / This thesis is dedicated to the study of propagation properties of various reaction–diffusion systems coming from population dynamics. In the first part, we study the strong competition limit of competition–diffusion systems with two species. Thanks to the spatial segregation, we determine the sign of the speed of the bistable traveling wave. The generalization to bistable pulsating fronts in spatially periodic media is then considered in order to study the role of spatial heterogeneity. We find a condition sufficient for the existence of such fronts as well as a condition sufficient for the existence of stable steady states which might on the contrary block the propagation. Then we show that whenever a family of strongly competing pulsating fronts exists, we can establish a result very similar to the one obtained in homogeneous media. In the second part, systems of KPP type with any number of species are considered. We study the existence of steady states and traveling waves, the qualitative properties of these solutions as well as the asymptotic speed of spreading of certain solutions of the Cauchy problem. This settles several open questions on the prototypical KPP systems that are mutation–competition–diffusion systems. In the third part, we go back to competition–diffusion systems with two species. Considering this time the monostable case, we study the asymptotic speeds of spreading of certain solutions of the Cauchy problem. By so doing, we show the existence of propagating terraces describing the invasion of an uninhabited territory by a weak but fast competitor followed by the invasion by a strong but slow competitor.

This thesis deals with a newly-developed model for the spatiotemporal calcium dynamics within presynaptic terminals. The model is based on single-protein kinetics and has been used to successfully describe different neuron types such as pyramidal neurons in the rat neocortex and the Calyx of Held of neurons from the rat brainstem. A limited number of parameters had to be adjusted to fluorescence measurements of the calcium concentration. These values can be interpreted as a prediction of the model, and in particular the protein densities can be compared to independent experiments. The contribution of single proteins to the total calcium dynamics has been analysed in detail for voltage-dependent calcium channel, plasma-membrane calcium ATPase, sodium-calcium exchanger, and endogenous as well as exogenous buffer proteins. The model can be used to reconstruct the unperturbed calcium dynamics from measurements using fluorescence indicators. The calcium response to different stimuli has been investigated in view of its relevance for synaptic plasticity. This work provides a first step towards a description of the complete synaptic transmission using single-protein data.

Cette thèse est consacrée à l'étude des équations et systèmes de réaction-diffusion dans des milieux hétérogènes. Elle est divisée en deux parties. La première est dédiée à l'étude des équations de réaction-diffusion dans des milieux périodiques. Nous nous intéressons en particulier aux équations posées dans des domaines qui ne sont pas l'espace entier $\mathbb{R}^{N}$, mais des domaines périodiques, avec des "obstacles". Dans un premier chapitre, nous étudions l'effet de la géométrie du domaine sur la vitesse d'invasion des solutions. Après avoir dérivé une formule de type Freidlin-Gartner, nous construisons des domaines où la vitesse d'invasion est strictement inférieure à la vitesse critique des fronts. Nous donnons également des critères géométriques qui garantissent l'existence de directions où l'invasion se produit à la vitesse critique. Dans le chapitre suivant, nous donnons des conditions nécessaires et suffisantes pour garantir que l'invasion ait lieu, après quoi nous construisons des domaines où des phénomènes intermédiaires (blocage, invasion orientée) se produisent. La deuxième partie de cette thèse est consacrée à l'étude de modèles décrivant l'influence de lignes à diffusion rapide (une route, par exemple) sur la propagation d'espèces invasives. Il a en effet été observé que certaines espèces, dont le moustique-tigre, envahissent plus rapidement que prévu certaines zones proches du réseau routier. Nous étudions deux modèles : le premier décrit l'influence d'une route courbe sur la propagation. Nous nous intéressons en particulier au cas de deux routes non-parallèles. Le second modèle décrit l'influence d'une route sur une niche écologique, en présence d'un changement climatique. Le résultat principal est que l'effet de la route est ambivalent : si la niche est stationnaire, alors l'effet de la route est délétère. Cependant, si la niche se déplace, suite à un changement climatique, nous montrons que la route peut permettre à une population de survivre. Pour étudier ce second modèle, nous développons une notion de valeur propre principale généralisée pour des systèmes de type KPP, et nous dérivons une inégalité de Harnack, qui est nouvelle pour ce type de systèmes. / This thesis is dedicated to the study of reaction-diffusion equations and systems in heterogeneous media. It is divided into two parts. The first one is devoted to the study of reaction-diffusion equations in periodic media. We pay a particular attention to equations set on domains that are not the whole space $\mathbb{R}^{N}$, but periodic domains, with "obstacles". In a first chapter, we study how the geometry of the domain can influence the speed of invasion of solutions. After establishing a Freidlin-Gartner type formula, we construct domains where the speed of invasion is strictly less than the critical speed of fronts. We also give geometric criteria to ensure the existence of directions where the invasion occurs with the critical speed. In the second chapter, we give necessary and sufficient conditions to ensure that invasion occurs, and we construct domains where intermediate phenomena (blocking, oriented invasion) occur. The second part of this thesis is dedicated to the study of models describing the influence of lines with fast diffusion (a road, for instance) on the propagation of invasive species. Indeed, it was observed that some species, such as the tiger mosquito, invade faster than expected some areas along the road-network. We study two models : the first one describes the influence of a curved road on the propagation. We study in particular the case of two non-parallel roads. The second model describes the influence of a road on an ecological niche, in presence of climate change. The main result is that the effect of the road is ambivalent: if the niche is stationary, then effect of the road is deleterious. However, if the niche moves, because of a shifting climate, the road can actually help the population to persist. To study this model, we introduce a notion of generalized principal eigenvalue for KPP-type systems, and we derive a Harnack inequality, that is new for this type of systems.

Cette thèse est consacrée à l’étude de phénomènes de propagation et de concentration dans des modèles d’équations intégro-différentielles venant de la écologie. On étudie certaines équations de réaction-diffusion non locales apparaissant en dynamique de populations, ainsi que des modèles représentant l’évolution Darwinienne avec un mode de reproduction sexué.Dans une première partie, nous étudions la propagation spatiale pour une équation de réaction-diffusion ou la dispersion opère via un noyau de convolution à queue lourde. Nous mesurons de manière précise l’accélération du front de propagation de la solution. Nous proposons également une échelle adaptée pour mesurer les «petites» mutations. Dans les deux cas nous utilisons le formalisme des équations de Hamilton-Jacobi.Dans un second temps nous étudions un modèle de génétique quantitative, avec un mode de reproduction sexuée. Un petit paramètre mesure la déviation entre le trait des descendants est la moyenne des traits des parents. Dans le régime où ce paramètre est petit nous étudions l’existence de solutions stationnaires, puis le problème de Cauchy lié à ce modèle. Les solutions se concentrent autour des optima de sélection, sous la forme de perturbations de distributions Gaussiennes avec petite variance fixée par le paramètre. Notre analyse généralise le cas linéaire de la reproduction asexuée en utilisant des outils d’analyse perturbative. Enfin dans une dernière partie nous fournissons des simulations numériques et des méthodes mathématiques pour étudier la dynamique interne des équilibres dans le régime de petite variance, pour les deux modes de reproduction : asexué et sexué. / This manuscript tackles propagation and concentration phenomena in different integro-differential equations with a background in ecology. We study non local reaction-diffusion equations from population dynamics, and models for Darwinian evolution with a sexual or asexual mode of reproduction, with a preference for the former.In a first part, we study spatial propagation for a reaction diffusion equation where dispersion acts through a fat tailed kernel. We measure accurately the acceleration of the propagation front of the population. We propose as well a scaling well adapted to “small mutations” when we consider the model in the context of adaptative dynamics. This scaling is very natural following the previous spatial investigation. In both cases we look at the long time behavior and we use the Hamilton-Jacobi framework. Then we turn our attention towards a quantitative genetics model, with a sexual mode of reproduction, imposed by the “infinitesimal operator”. In this non-linear setting, a small parameter tunes the deviation between the phenotypic trait of the offspring and the mean of the traits of the parents. In the regime where this parameter is small, we prove existence of stationary solutions, and their local uniqueness. We also provide an example of non-uniqueness in the case where the selection function admits several extrema. We prove that the solution concentrates around the points of minimum of the selection function. The analysis is carried by the small perturbations of special profiles : Gaussian distributions with small variance fixed by the parameter.We then study the stability of the Cauchy problem associated to the previous model. This time we prove that at all times, for a well prepared initial data, the solutions is arbitrary close to a Gaussian distribution with small variance. The proof follows the framework of the previous : we use perturbative analysis tools, but this time an even more precise description of the correctors is needed and we linearize the equation to obtain it. In a final part we show numerical simulations and different mathematical approaches to study inside dynamics of phenotypic lineages in the regime of small variance, with a moving environement.

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We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

We investigate a nonlinear reaction-diffusion system coupled with convection- diffusion system. This combined system corresponds to physical description of heteroge- neous catalysis when the flow of bulk-constituents is driven by a given stationary velocity field; diverse mechanisms between bulk- and surface-parts of the model-domain are de- scribed by Langmuir-Hinshelwood absorption kinetics; and the irreversible reactions on the catalytic walls meets the law of mass action with quadratic rate. The first part of the thesis is focused on analytical results; in Chapter 2 we prove existence and unique- ness of a mild solution for so-called near-by problem using nonlinear semigroup theory; in Chapter 3 we investigate the weak formulation of the problem. We prove an existence of a weak solution for little modified problem which, under an assumption, coincides with the original problem. In the second part of the thesis (Chapter 4) we numerically investigate the evolution of the bio-diesel microreactor. We compute numerical solutions using several methods and we test the results by analytical and physical conditions; with the aim to find the most efficient way to compute precise and physically correct solution. Keywords: heterogeneous catalysis, coupled reaction-diffusion/convection-diffusion system, nonlinear...

La compréhension du cycle de la matière organique du sol (MOS) est un outil majeur dans la lutte contre le réchauffement climatique, la préservation de la biodiversité ainsi que dans la consolidation de la sécurité alimentaire. Dans ce contexte, cette thèse porte sur la modélisation et l'analyse mathématique de modèles de la dynamique du carbone organique dans le sol.Dans le chapitre 2, nous avons étudié la robustesse et les propriétés mathématiques d'un modèle non linéaire (MOMOS). Nous avons montré que si les données sont périodiques nous obtenons l'existence d'une solution périodique attractive. Le chapitre3 est consacré à la validation mathématique d'un modèle spatialisé basé sur les équations de MOMOS, auxquels nous avons ajouté des opérateurs de diffusion et de transport. L'effet de l'hétérogénéité spatiale sur ce modèle est étudié dans le chapitre4 en utilisant des techniques d'homogénéisation. Suivant la méthodologie de Bosattaet Agren, nous dérivons un autre modèle à qualité continue, qui prend en compte l'effet de l'âge sur la décomposition de la MOS. Le chapitre 5 contient la validation mathématique et expérimentale du modèle. Enfin, nous considérons dans les chapitres6 et 7, un modèle incluant l'effet de la chemotaxie. Nous montrons l'existence, la positivité et l'unicité des solutions dans des domaines suffisamment réguliers de dimension inférieure ou égale à 3. / Understanding the soil organic matter (SOM) cycle is a major tool in the effort toreduce global warming, to preserve biodiversity and to improve food safety strategies.In this context, this thesis is about modelling and mathematical analysis of thedynamics of the organic carbon in soil.In chapter 2, we validate mathematically a nonlinear soil organic carbon model(MOMOS) and we prove that, if data is periodic, then there is a unique attractiveperiodic solution. In chapter 3, we focus on the mathematical validation of a spatialmodel derived from MOMOS and where we used diffusion and transport operators.We prove also the existence of a periodic solution. In addition, the effect of soilheterogeneities on the model is studied in chapter 4 using homogenization techniques.Moreover, following the Bosatta and Agren methodology, we derive a continuousquality model taking in consideration the effect of age on the quality of SOM. Wevalidate the model mathematically and experimentally in chapter 5. Finally, weconsider in chapters 6 and 7 another model that takes into account the chemotaxismovement of soil microorganisms. We prove mainly the existence and uniqueness of apositive solution in a regular spatial domain of dimension less or equal to 3.

<p>Moving boundary problems such as solidification, crack propagation, multi-body contact or shape optimal design represent an important class of engineering problems. Common to these problems are one or more moving interfaces or boundaries. One of the main challenges associated with boundary evolution is the difficulty that arises when the topology of the geometry changes. Other geometric issues such as distance to the boundary, projected point on the boundary and intersection between surfaces are also important and need to be efficiently solved. In general, the present thesis is concerned with the geometric arrangement and behavioral analysis of evolving parametric boundaries immersed in a domain. </p> <p>The first problem addressed in this thesis is the packing of ellipsoidal fillers in a regular domain and to estimate their effective physical behavior. Particle packing problem arises when one generates simulated microstructures of particulate composites. Such particulate composites used as thermal interface materials (TIMs) motivates this work. The collision detection and distance calculation between ellipsoids is much more difficult than other regular shapes such as spheres or polyhedra.  While many existing methods address the spherical packing problems, few appear to achieve volume loading exceeding 60%. The packing of ellipsoidal particles is even more difficult than that of spherical particles due to the need to detect contact between the particles. In this thesis, an efficient and robust ultra-packing algorithm termed Modified Drop-Fall-Shake is developed. The algorithm is used to simulate the real mixing process when manufacturing TIMs with hundreds of thousands ellipsoidal particles. The effective thermal conductivity of the particulate system is evaluated using an algorithm based on Random Network Model. </p> <p><br></p> <p>In problems where general free-form parametric surfaces (as opposed to the ellipsoidal fillers) need to be evolved inside a regular domain, the geometric distance from a point in the domain to the boundary is necessary to determine the influence of the moving boundary on the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge,  an algebraic procedure is presented here to find the level sets of rational parametric surfaces commonly utilized by commercial CAD systems. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bezier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain can also be used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G₀ continuity---problems where the Newton--Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively. </p> <p><br></p> <p>Next, ALS is also extended for boundary representation (B-rep) models that are popularly used in CAD systems for modeling solids. B-rep model generally contains multiple NURBS patches due to the trimming feature used to construct such models, and as a result are not ``watertight" or mathematically compatible at patch edges. A time consuming geometry clean-up procedure is needed to preprocess geometry prior to finite element mesh generation using a B-rep model, which can take up to 70% of total analysis time according to literature. To avoid the need to clean up geometry and directly provide link between CAD and CAE integration,  signed algebraic level sets using novel inner/outer bounding box strategy is proposed for point classification of B-rep model. Several geometric examples are demonstrated, showing that this technique naturally models single patch NURBS geometry as well, and can deal with multiple patches involving planar trimming feature and Boolean operation. During the investigation of algebraic level sets, a complex self-intersection problem is also reported, especially for three-dimensional surface. The self-intersection may occur within an interval of interest during implicitization of a curve or surface since the implicitized curve or surface is not trimmed and extends to infinity. Although there is no robust and universal solution the problem, two potential solutions are provided and discussed in this thesis.</p> <p><br></p> <p>In order to improve the computational efficiency of analysis in immersed boundary problems, an efficient local refinement technique for both mesh and quadrature  using the kd-tree data structure is further proposed. The kd-tree sub-division is theoretically proved to be more efficient against traditional quad-/oct-tree subdivision methods. In addition, an efficient local refinement strategy based on signed algebraic level sets is proposed to divide the cells. The efficiency of kd-tree based mesh refinement and adaptive quadrature is later shown through numerical examples comparing with oct-tree subdivision, revealing significant reduction of degrees of freedom and quadrature points.</p> <p><br></p> <p>Towards analysis of moving boundaries problems, an explicit interface tracking method termed enriched isogeometric analysis (EIGA) is adopted in this thesis. EIGA utilizes NURBS shape function for both geometry representation and field approximation. The behavior field is modeled by a weighted blending of the underlying domain approximation and enriching field, allowing high order continuity naturally. Since interface is explicitly represented, EIGA provides direct geometric information such as normals and curvatures. In addition, the blending procedure ensures strong enforced boundary conditions. An important moving boundary problem, namely, reaction-diffusion problem, is investigated using EIGA. In reaction-diffusion problems, the phase interfaces evolve due to chemical reaction and diffusion under multi-physics driven forces, such as mechanical, electrical, thermal, etc. Typical failure phenomenon due to reaction-diffusion problems include void formation and intermetallic compound (IMC) growth. EIGA is applied to study factors and behavior patterns in these failure phenomenon, including void size, current direction, current density, etc. A full joint simulation is also conducted to study the degradation of solder joint under thermal aging and electromigration. </p>

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.