Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

No description available.

Cette thèse porte sur différents modèles de propagation en épidémiologie évolutive. L'objectif est d'en faire une analyse mathématique rigoureuse puis d'en tirer des enseignements biologiques. Dans un premier temps nous envisageons le cas d'une population d'hôtes répartis de manière homogène dans un espace linéaire, dans laquelle se propage un pathogène pouvant muter entre deux phénotypes plus ou moins virulents. Ce phénomène de mutation est à l'origine d'une interaction entre les dynamiques évolutive et épidémiologique du pathogène. Nous étudions la vitesse de propagation de l'épidémie et l'existence de fronts progressifs, ainsi que l'influence sur la vitesse de différents facteurs biologiques, comme des effets stochastiques liés à la taille de la population d'hôtes (explorations numériques). Dans un deuxième temps nous envisageons une hétérogénéité spatiale périodique dans la population d'hôtes, et l'existence de fronts pulsatoires pour le système de réaction-diffusion (non-coopératif) associé. Enfin nous considérons un pathogène pouvant muter vers un grand nombre de phénotypes différents et étudions l'existence de fronts potentiellement singuliers, modélisant ainsi une concentration sur un trait optimal. / In this thesis we consider several models of propagation arising in evolutionary epidemiology. We aim at performing a rigorous mathematical analysis leading to new biological insights. At first we investigate the spread of an epidemic in a population of homogeneously distributed hosts on a straight line. An underlying mutation process can shift the virulence of the pathogen between two values, causing an interaction between epidemiology and evolution. We study the propagation speed of the epidemic and the influence of some biologically relevant quantities, like the effects of stochasticity caused by the hosts' finite population size (numerical explorations), on this speed. In a second part we take into account a periodic heterogeneity in the hosts' population and study the propagation speed and the existence of pulsating fronts for the associated (non-cooperative) reaction-diffusion system. Finally, we consider a model in which the pathogen is allowed to shift between a large number of different phenotypes, and construct possibly singular traveling waves for the associated nonlocal equation, thus modelling concentration on an optimal trait.

Molecular recognition plays an essential role in the self-assembly and self-organisation of biological and chemical systems alike—allowing individual components to form complex interconnected networks. Within these systems, the nature of the recognition and reactive processes determines their functional and structural properties, and even small changes in their identity or orientation can exert a dramatic effect on the observed properties. The rapidly developing field of systems chemistry aims to move away from the established paradigm in which molecules are studied in isolation, towards the study of networks of molecules that interact and react with each other. Taking inspiration from complex natural systems, where recognition processes never operate in isolation, systems chemistry aims to study chemical networks with the view to examining the system-level properties that arise from the interactions and reactions between the components within these systems. The work presented in this thesis aims to advance the nascent field of systems chemistry by bringing together small organic molecules that can react and interact together to form interconnected networks, exhibiting complex behaviour, such as self-replication, as a result. Three simple building blocks are used to construct a network of two structurally similar replicators and their kinetic behaviour is probed through a comprehensive kinetic analysis. The selectivity for one of the recognition-mediated reactive processes over another is examined within the network in isolation as well as in a scenario where the network is embedded within a pool of exchanging components. The interconnected, two-replicator network is examined under far-from-equilibrium reaction-diffusion conditions, showing that chemical replicating networks can exhibit signs of selective replication—a complex phenomenon normally associated with biological systems. Finally, a design of a well-characterised replicator is exploited for the construction of a network integrating self-replication with a another recognition-directed process, leading to the formation of a mechanically-interlocked architecture—a [2]rotaxane.

Made available in DSpace on 2015-03-26T13:35:24Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1414332 bytes, checksum: 6e2f42018f3e3dcdf9e8cbccab567e7a (MD5) Previous issue date: 2006-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Pattern, structure and emergent collective properties are ubiquitous in systems with many units (alive or inanimated) coupled through nonlinear interactions. Within this context, the study of cooperative phenomena in population dynamics of ecological interest has attracted the attention of the mathematicians and physicists since Lotka and Volterra in the 1920s. Thenceforth, in addition to differential equations, theoretical ecology has continuously incorporated powerful and well-established techniques of contacts processes, cellular automata models and others, developed in the fields of condensed matter physics, statistical physics and computational physics. In the present work, a predator-prey model with pursuit and escape triggered by alarm pheromones is proposed and studied through analytical methods and computer simulations. Such models can show oscillatory behavior of the population density, phase transitions that belong to distinct universality classes and rich stationary phase diagrams. Two distinct levels of description were used. In a first approach, we consider a model of cellular automata in which predators and preys walk on a square lattice, according specific rules for each species, in a homogeneous environment and with periodic boundary conditions. The second part of our study is based on the analysis of partial differential equations that also describes the dynamics of a prey-predator system with the same characteristics above. Both, spatially uniform or mean field like and explicit spatio-temporal partial differential equations were considered. These models can represent relevant tools to design better strategies of biological control of pests by predators. In successful cases, the pests and its predators must persist in stable interactions at a low level of pest density. / Padrões, estruturas, propriedades coletivas emergentes são ubíquas em sistemas com muitas unidades (vivas ou inanimadas) acopladas por meio de interações não-lineares. Dentro desse contexto, o estudo de fenômenos cooperativos em dinâmica de populações de interesse ecológico tem atraído a atenção de físicos e matemáticos desde os anos de 1920 com Lotka e Volterra. Portanto, além de equações diferenciais, a teoria ecológica tem continuamente incorporado poderosas e bem-estabelecidas técnicas dos processos de contatos, modelos de autômatos celulares e outros, desenvolvidos no campo de física da matéria condensada, física estatística e física computacional. No presente trabalho, um modelo presa-predador com perseguição e fuga mediada por um feromônio de alarme é proposto e estudado através de métodos analíticos e simulações computacionais. Tais modelos podem exibir comportamentos oscilatórios da densidade de população, transições de fases que pertencem a classes de universalidade distintas e um diagrama de fases rico. Duas abordagens distintas de descrição foram usadas. Numa primeira abordagem, propomos um modelo de Autômato Celular (AC) onde predadores e presas se movimentam, segundo regras específicas para cada espécie, num ambiente homogêneo e com condições de contorno periódicas. A outra parte do nosso estudo é baseado na análise de EDP s que também descrevem a dinâmica de um sistema presa-predador com as mesmas características citadas acima. É feito um estudo considerando as equações sem termos espaciais, isto é, tipo campo médio e depois considerando esses termos. Esses modelos podem representar ferramentas relevantes para o estudo das melhores estratégias para o controle biológico de pragas por predadores. Em casos bem sucedidos, as pestes e seus predadores devem persistir em interações estáveis e com uma baixa densidade da população de pragas.

In this thesis, parabolic-pseudoparabolic equations are derived coupling chemical reactions, diffusion, flow and mechanics in a heterogeneous medium using the framework of mixture theory. The weak solvability in 1-D of the obtained models is studied. Furthermore, it is numerically illustrated that approximate solutions according to the Rothe method exhibit expected realistic behaviour. For a simpler model formulation, the periodic homogenization in higher space dimensions is performed. / <p>Research is funded by the Netherlands Organisation of Scientific Research (NWO) with MPE-grant 657.000.004, and a research stay at Karlstads Universitet is funded by NWO cluster Nonlinear Dynamics in Natural Systems (NDNS+).</p>

Cette thèse est consacrée à l'étude de phénomènes de propagation dans des modèles d’EDP venant de la biologie. On étudie des équations cinétiques inspirées par le déplacement de colonies de bactéries ainsi que des équations de réaction-diffusion importantes en écologie afin de reproduire plusieurs phénomènes de dynamique et d'évolution des populations. La première partie étudie des phénomènes de propagation pour des équations cinétiques. Nous étudions l'existence et la stabilité d'ondes progressives pour des modèles ou la dispersion est donnée par un opérateur hyperbolique et non par une diffusion. Cela fait entrer en jeu un ensemble de vitesses admissibles, et selon cet ensemble, divers résultats sont obtenus. Dans le cas d'un ensemble de vitesses borné, nous construisons des fronts qui se propagent à une vitesse déterminée par une relation de dispersion. Dans le cas d'un ensemble de vitesses non borné, on prouve un phénomène de propagation accélérée dont on précise la loi d'échelle. On adapte ensuite à des équations cinétiques une méthode basée sur les équations de Hamilton-Jacobi pour décrire des phénomènes de propagation. On montre alors comment déterminer un Hamiltonien effectif à partir de l'équation cinétique initiale, et prouvons des théorèmes de convergence.La seconde partie concerne l'étude de modèles de populations structurées en espace et en phénotype. Ces modèles sont importants pour comprendre l'interaction entre invasion et évolution. On y construit d'abord des ondes progressives que l'on étudie qualitativement pour montrer l'impact de la variabilité phénotypique sur la vitesse et la distribution des phénotypes à l'avant du front. On met aussi en place le formalisme Hamilton-Jacobi pour l'étude de la propagation dans ces équations de réaction-diffusion non locales.Deux annexes complètent le travail, l'une étant un travail en cours sur la dispersion cinétique en domaine non-borné, l'autre étant plus numérique et illustre l’introduction. / This thesis is devoted to the study of propagation phenomena in PDE models arising from biology. We study kinetic equations coming from the modeling of the movement of colonies of bacteria, but also reaction-diffusion equations which are of great interest in ecology to reproduce several features of dynamics and evolution of populations. The first part studies propagation phenomena for kinetic equations. We study existence and stability of travelling wave solutions for models where the dispersal part is given by an hyperbolic operator rather than by a diffusion. A set of admissible velocities comes into the game and we obtain various types of results depending on this set. In the case of a bounded set of velocities, we construct travelling fronts that propagate according to a speed given by a dispersion relation. When the velocity set is unbounded, we prove an accelerating propagation phenomena, for which we give the spreading rate. Then, we adapt to kinetic equations the Hamilton-Jacobi approach to front propagation. We show how to derive an effective Hamiltonian from the original kinetic equation, and prove some convergence results.The second part is devoted to studying models for populations structured by space and phenotypical trait. These models are important to understand interactions between invasion and evolution. We first construct travelling waves that we study qualitatively to show the influence of the genetical variability on the speed and the distribution of phenotypes at the edge of the front. We also perform the Hamilton-Jacobi approach for these non-local reaction-diffusion equations.Two appendices complete this work, one deals with the study of kinetic dispersal in unbounded domains, the other one being numerical aspects of competition models.

Dans différents domaines de recherche, la modélisation est devenue un outil efficace pour étudier et prédire l’évolution possible d’un système, en particulier en épidémiologie. En raison de la mondialisation et de la mutation génétique de certaines maladies ou vecteurs de transmission, plusieurs épidémies sont apparues dans des régions non encore concernées ces dernières années. Dans cette thèse, un modèle décrivant la transmission de l’épidémie de chikungunya à la population humaine est étudié. Ce modèle prend en compte la mobilité spatiale des humains, ce qui est nouveau. En effet, c’est un facteur intéressant qui a influencé la réapparition de plusieurs maladies épidémiques. Le déplacement des moustiques est omis puisqu’il est limité à quelques mètres. Le modèle complet (modèle EDOs-EDPs) est alors composé d’un système à réaction-diffusion (prenant la forme d’équations différentielles partielles (EDPs) paraboliques semi-linéaires) couplé à des équations différentielles ordinaires (EDOs). Nous démontrons pour ce modèle, d’abord l’existence et l’unicité de la solution globale, sa positivité et sa bornitude, puis nous donnons quelques simulations numériques. Dans ce modèle, certains paramètres ne sont pas directement accessibles à partir des expériences et doivent être estimés numériquement. Cependant, avant de rechercher leurs valeurs, il est essentiel de vérifier l’identifiabilité des paramètres pour déterminer si l’ensemble des paramètres inconnus peut être déterminé de manière unique à partir des données. Cette étude permettra de s’assurer que les procédures numériques peuvent être couronnées de succès. Si l’identifiabilité n’est pas assurée, certaines données supplémentaires doivent être ajoutées. En fait, une première étude d’identifiabilité a été effectuée pour le modèle EDOs en considérant que le nombre d’œufs peut être facilement compté. Toutefois, après avoir discuté avec les chercheurs épidémiologistes, il apparaît que c’est le nombre de larves qui peut être estimé semaines par semaines. Ainsi, nous ferons une étude d’identifiabilité pour le nouveau modèle EDOs-EDPs avec cette hypothèse. Grâce à l’intégration de l’une des équations du modèle, on obtient des équations plus faciles reliant les entrées, les sorties et les paramètres, ce qui simplifie vraiment l’étude d’identifiabilité. A partir de l’étude d’identifiabilité, une méthode et une procédure numérique sont proposés pour estimer les paramètres sans en avoir connaissance. / In different fields of research, modeling has become an effective tool for studying and predicting the possible evolution of a system, particularly in epidemiology. Due to the globalization and the genetic mutation of certain diseases or transmission vectors, several epidemics have appeared in regions not yet concerned in the last years. In this thesis, a model describing the transmission of the chikungunya epidemic to the human population is studied. As a novelty, this model incorporates the spatial mobility of humans. Indeed, it is an interesting factor that has influenced the re-emergence of several epidemic diseases. The displacement of mosquitoes is omitted since it is limited to a few meters. The complete model (ODEs-PDEs model) is then composed of a reaction-diffusion system (taken the form of semi-linear parabolic partial differential equations (PDEs)) coupled with ordinary differential equations (ODEs). We prove the existence, uniqueness, positivity and boundedness of a global solution of this model at first and then give some numerical simulations. In such a model, some parameters are not directly accessible from experiments and have to be estimated numerically. However, before searching for their values, it is essential to verify the identifiability of parameters in order to assess whether the set of unknown parameters can be uniquely determined from the data. This study will insure that numerical procedures can be successful. If the identifiability is not ensured, some supplementary data have to be added. In fact, a first identifiability study had been done for the ODEs model by considering that the number of eggs can be easily counted. However, after discussing with epidemiologist searchers, it appears that it is the number of larvae which can be estimated weeks by weeks. Thus, we will do an identifiability study for the novel ODEs-PDEs model with this assumption. Thanks to an integration of one of the model equations, some easier equations linking the inputs, outputs and parameters are obtained which really simplify the study of identifiability. From the identifiability study, a method and numerical procedure are proposed for estimating the parameters without any knowledge of them.

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling periodic wave speeds for stage-structured populations in heterogeneous landscapes. The model includes structured populations of two age groups, juveniles and adults, in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape. Secondly, we use asymptotic techniques to find an explicit formula for the traveling periodic wave speed and to calculate the spread rates for structured populations in heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the approximation and the exact expression. We find an excellent agreement between the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups. Lastly, we use a finite difference method to explore the numerical solutions for the juveniles-adults model. We compare numerical solutions to analytic solutions and explore population dynamics in non-linear models, where the numerical solution for the time-dependent problem converges to a steady state. We apply our theory to study various aspects of marine protected areas (MPAs). We develop a model of two age groups, juveniles and adults, in which only adults can be harvested and only outside MPAs, and recruitment is density dependent and local inside MPAs and fishing grounds. We include diffusion coefficients in density matching conditions at interfaces between MPAs and fishing grounds, and examine the effect of fish mobility and bias movement on yield and fish abundance. We find that when the bias towards MPAs is strong or the difference in diffusion coefficients is large enough, the relative density of adults inside versus outside MPAs increases with adult mobility. This observation agrees with findings from empirical studies.