The life cycle of Caulobacter crescentus is of interest because of the asymmetric nature of cell division that gives rise to progeny that have distinct morphology and function. One daughter called the stalked cell is sessile and capable of DNA replication, while the second daughter called the swarmer cell is motile but quiescent. Advances in microscopy combined with molecular biology techniques have revealed that macromolecules are localized in a non-homogeneous fashion in the cell cytoplasm, and that dynamic localization of proteins is critical for cell cycle progression and asymmetry. However, the molecular-level mechanisms that govern protein localization, and enable the cell to exploit subcellular localization towards orchestrating an asymmetric life cycle remain obscure. There are also instances of researchers using intuitive reasoning to develop very different verbal explanations of the same biological process. To provide a complementary view of the molecular mechanism controlling the asymmetric division cycle of Caulobacter, we have developed a mathematical model of the cell cycle regulatory network. Our reaction-diffusion models provide additional insight into specific mechanism regulating different aspects of the cell cycle. We describe a molecular mechanism by which the bifunctional histidine kinase PleC exhibits bistable transitions between phosphatase and kinase forms. We demonstrate that the kinase form of PleC is crucial for both swarmer-to-stalked cell morphogenesis, and for replicative asymmetry in the predivisional cell. We propose that localization of the scaffolding protein PopZ can be explained by a Turing-type mechanism. Finally, we discuss a preliminary model of ParA- dependent chromosome segregation. Our model simulations are in agreement with experimentally observed protein distributions in wild-type and mutant cells. In addition to predicting novel mutants that can be tested in the laboratory, we use our models to reconcile competing hypotheses and provide a unified view of the regulatory mechanisms that direct the Caulobacter cell cycle. / Ph. D.

Natural ecologies are prone to stochastic effects and changing environments that shape their dynamical behavior. Ecological systems can be modeled through relatively simple population dynamics models. There is a plethora of models describing deterministic models of ecological systems evolving in a constant environment. However, stochasticity can lead to extinction or fixation events, noise-stabilized patterns, and nontrivial correlations. Likewise, changing environments can greatly affect the behavior and ultimate fate of ecological systems. In fact, the dynamics of evolution are mostly driven by randomness and changing environments. Therefore, it is of utmost importance to develop population dynamics models that are able to capture the effects of noise and environmental drive. In this thesis, we use both theoretical tools and simulations to investigate population dynamics in the following contexts: We study the stochastic spatial Lotka-Volterra (LV) model for predator-prey interaction subject to a periodically varying carrying capacity. The LV model with on-site lattice occupation restrictions that represent finite food resources for the prey exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by spatio-temporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The stationary regime of our periodically varying LV system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity switching environments, the mean-field rate equations predict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast- and slow periodic switching. The analysis of the mean-field equations in the fast-switching regime enables a semi-quantitative description of the stationary state. The mean-field analysis of the Lotka-Volterra predator-prey model with seasonally varying carrying capacity is extended to the resonant regime. This is done by introducing a homotopy mapping from this model to another model that allows for the application of Floquet theory. The stability of the coexistence fixed point is studied and the period doubling is related to a bifurcation point in the homotopy mapping. However, we find that the predator-prey ecology's coexistence is stable for most of its parameter region. We apply a perturbative Doi–Peliti field-theoretical analysis to the stochastic spatially extended symmetric Rock-Paper-Scissors (RPS) and May–Leonard (ML) models, in which three species compete cyclically. Compared to the two-species Lotka–Volterra predator-prey (LV) model, according to numerical simulations, these cyclical models appear to be less affected by intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation-induced renormalizations in the perturbative regime, similar to the LV model. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS system, whereas the spontaneous emergence of spatio-temporal structures features prominently in the LV and the ML models. Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reactiondiffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka–Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka–Volterra model as well as its May–Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse-graining in spatially extended stochastic reactiondiffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data. / Doctor of Philosophy / Population dynamics models describe how the number of individuals of interacting species changes over time. This is used to understand the ultimate fate of ecological systems. An ecological system can exhibit long-time multi-species coexistence, the fixation of just one species (all other species go extinct), or total extinction of the system. Understanding the dynamics of the system can help predict the final state of the system from early observations, also, it can inform possible ways to steer the system into a desirable outcome. However, it is very difficult to model such systems due to their complexity. While great progress has been made in understanding well-mixed populations in constant environments, there is still much to learn about ecological systems under spatial and environmental variability. A complete understanding of ecological dynamics and how they couple to evolutionary dynamics requires models of populations that are random, and that take into account how different species might be more or less dominant in different environments. We contribute to investigating these models in the following way: Seasonal variations in temperature leads to a change in the availability of different crops. This affects the resources available for animal species to consume in one season compared to another season (e.g. summer and winter). We study a predator-prey model wherein the resource abundance available to the prey vary between two seasons. We showcase how this affects the system's coexistence regime, and spatial patterns. Cyclic models of predation are models where the food chain is cyclic, meaning that there is no "food chain" but rather a "food circle". We utilize theoretical tools to gain a better understanding of the spontaneous formation of well-known spatial patterns in cyclic predation models. The aim of population dynamics is to write a simple set of equations or models that can accurately capture the behavior of natural ecologies. This is rarely an easy task, because even if the microscopic interactions between species is known, it is very difficult to simplify this microscopic model to a simple set of macroscopic equations. We develop a technique that uses computer simulations to map microscopic interactions into simple rate equations. This work can inform better modeling of observational data.

In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed. / Ph. D.

This thesis examines Turing patterns and how systems control theory can be used to steer them. The mathematical tools used in this thesis consist of linear algebra, partial differential equations (PDE), numerical simulations of PDEs and systems control theory. These topics will all be briefly introduced. Following that will be a section on the numerical study of Turing patters. The bulk of the thesis will focus on network controllability. The subjects explored are an analytical study of controllability, the simulations of specific patterns using controllability and a numerical analysis of the minimal energy control for a Turing system

La propagation d'un front de réaction-diffusion est étudiée expérimentalement dans un écoulement cellulaire multiéchelle. Le front est produit par réaction autocatalytique en solution. L'écoulement est réalisé en géométrie de Hele-Shaw par électroconvection, son caractère multiéchelle étant réalisé par l'action combinée de deux nappes d'aimants d'échelles différentes. La géométrie du front et sa vitesse moyenne de propagation sont déterminées pour une large gamme d'intensité des vortex de chaque échelle. Elles sont confortées par une simulation numérique de l'avancée du domaine brulé dans le domaine frais. L'effet de la nature multiéchelle de l'écoulement sur la vitesse moyenne du front est compris par une méthode de renormalisation dont la validation est fournie par l'obtention d'un courbe maitresse pour l'ensemble des données. / The propagation of a reaction-diffusion front is experimentally studied in a multi-scale cellular flow. The front is produced by an autocatalytic chemical reaction in an aqueous solution. The flow is generated by electroconvection and its multi-scale nature is induced by overlaying magnets of different scales. This enables an independent tune of the flow intensity at each scale. The geometry and the mean velocity of the front have been determined over a large range of scale intensities. These features are confirmed by a numerical simulation based on a burnt and fresh domain dynamics, the burnt domain expanding across the fresh one. The effect of the multi-scale nature of the flow on the mean front velocity is recovered by a renormalisation method validated by a collapse of the data onto a single curve.

Complex spatial structures in biology arise from random interactions of molecules. These molecular interactions can be studied using spatial stochastic models, such as Reaction Diffusion Master Equation (RDME), a mesoscopic model that subdivides the spatial domain into smaller, well mixed grid cells, in which the macroscopic diffusion-controlled reactions take place. While RDME has been widely used to study how fluctuations in number of molecules affect spatial patterns, simulations are computationally expensive and it requires a lower bound for grid cell size to avoid an apparent unphysical loss of bimolecular reactions. In this thesis, we propose Gaussian Reaction Diffusion Master Equation (GRDME), a novel model in the RDME framework, based on the discretization of the Laplace operator with Particle Strength Exchange (PSE) method with a Gaussian kernel. We show that GRDME is a computationally efficient model compared to RDME. We further resolve the controversy regarding the loss of bimolecular reactions and argue that GRDME can flexibly bridge the diffusion-controlled and ballistic regimes in mesoscopic simulations involving multiple species. To efficiently simulate GRDME, we develop Gaussian Next Subvolume Method (GNSM). GRDME simulated with GNSM up to six-times lower computational cost for a three-dimensional simulation, providing a significant computational advantage for modeling three-dimensional systems. The computational cost can be further lowered by increasing the so-called smoothing length of the Gassian jumps. We develop a guideline to estimate the grid resolution below which RDME and GRDME exhibit loss of bimolecular reactions. This loss of reactions has been considered unphysical by others. Here we show that this loss of bimolecular reactions is consistent with the well-established theory on diffusion-controlled reaction rates by Collins and Kimball, provided that the rate of bimolecular propensity is interpreted as the rate of the ballistic step, rather than the macroscopic reaction rate. We show that the reaction radius is set by the grid resolution. Unlike RDME, GRDME enables us to explicitly model various sizes of the molecules. Using this insight, we explore the diffusion-limited regime of reaction dynamics and discover that diffusion-controlled systems resemble small, discrete systems. Others have shown that a reaction system can have discreteness-induced state inversion, a phenomenon where the order of the concentrations differs when the system size is small. We show that the same reaction system also has diffusion-controlled state inversion, where the order of concentrations changes, when the diffusion is slow. In summary, we show that GRDME is a computationally efficient model, which enables us to include the information of the molecular sizes into the model.:1 Modeling Mesoscopic Biology 1.1 RDME Models Mesoscopic Stochastic Spatial Phenomena 1.2 A New Diffusion Model Presents an Opportunity For A More Efficient RDME 1.3 Can A New Diffusion Model Provide Insights Into The Loss Of Reactions? 1.4 Overview 2 Preliminaries 2.1 Reaction Diffusion Master Equation 2.1.1 Chemical Master Equation 2.1.2 Diffusion-controlled Bimolecular Reaction Rate 2.1.3 RDME is an Extention of CME to Spatial Problems 2.2 Next Subvolume Method 2.2.1 First Reaction Method 2.2.2 NSM is an Efficient Spatial Stochastic Algorithm for RDME 2.3 Discretization of the Laplace Operator Using Particle Strength Exchange 2.4 Summary 3 Gaussian Reaction Diffusion Master Equation 3.1 Design Constraints for the Diffusion Model in the RDME Framework 3.2 Gaussian-jump-based Model for RDME 3.3 Summary 4 Gaussian Next Subvolume Method 4.1 Constructing the neighborhood N 4.2 Finding the Diffusion Event 4.3 Comparing GNSM to NSM 4.4 Summary 5 Limits of Validity for (G)RDME with Macroscopic Bimolecular Propensity Rate 5.1 Previous Works 5.2 hmin Based on the Kuramoto length of a Grid Cell 5.3 hmin of the Two Limiting Regimes 5.4 hmin of Bimolecular Reactions for the Three Cases of Dimensionality 5.5 hmin of GRDME in Comparison to hmin of RDME 5.6 Summary 6 Numerical Experiments To Verify Accuracy, Efficiency and Validity of GRDME 6.1 Accuracy of the Diffusion Model 6.2 Computational Cost 6.3 hmin and Reaction Loss for (G)RDME With Macroscopic Bimolecular Propensity Rate kCK 6.3.1 Homobiomlecular Reaction With kCK at the Ballistic Limit 6.3.2 Homobiomlecular Reaction With kCK at the Diffusional Limit 6.3.3 Heterobiomlecular Reaction With kCK at the Ballistic Limit 6.4 Summary 7 (G)RDME as a Spatial Model of Collins-Kimball Diffusion-controlled Reaction Dynamics 7.1 Loss of Reactions in Diffusion-controlled Reaction Systems 7.2 The Loss of Reactions in (G)RDME Can Be Explained by Collins Kimball Theory 7.3 Cell Width h Sets the Reaction Radius σ∗ 7.4 Smoothing Length ε′ Sets the Size of the Molecules in the System 7.5 Heterobimolecular Reactions Can Only Be Modeled With GRDME 7.6 Zeroth Order Reactions Impose a Lower Limit on Diffusivity Dmin 7.6.1 Consistency of (G)RDME Could Be Improved by Redesigning Zeroth Order Reactions 7.7 Summary 8 Difussion-Controlled State Inversion 8.1 Diffusion-controlled Systems Resemble Small Systems 8.2 Slow Diffusion Leads to an Inversion of Steady States 8.3 Summary 9 Conclusion and Outlook 9.1 Two Physical Interpretations of (G)RDME 9.2 Advantages of GRDME 9.3 Towards Numerically Consistent (G)RDME 9.4 Exploring Mesoscopic Biology With GRDME Bibliography

Cette thèse est consacrée à l'étude de systèmes d'équations aux dérivées partielles paraboliques issus de modèles de cinétique chimique, de dynamique des populations et de la théorie de l'électromigration. On s'intéresse à des questions d'existence de solutions globales en temps, à l'unicité de solutions faibles, ainsi qu'à la limite de réaction rapide dans un système de réaction-diffusion. Dans un premier chapitre, on étudie deux systèmes aux diffusions croisées. On commence par s'intéresser à un modèle de dynamique des populations, où les effets croisés dans les interactions entre les différentes espèces sont modélisés par des opérateurs non locaux. Pour toute dimension d'espace, on prouve l'existence et l'unicité de solutions globales régulières. On s'intéresse ensuite à un système aux diffusions croisées qui apparait comme la limite de réaction rapide d'un système classique associé à la réaction chimique C1+C2=C3. On prouve alors la convergence lorsque k tend vers l'infini de la solution du système avec une vitesse de réaction finie k vers une solution globale du système limite. Le second chapitre contient de nouveaux résultats d'existence globale pour des systèmes de réaction-diffusion. Pour des réseaux de réactions chimiques élémentaires du type Ci+Cj=Ck qui suivent la loi d'Action de Masse, on montre l'existence et l'unicité de solutions globales fortes, pour des dimensions en espace N<6 dans le cas semi-linéaire et N<4 dans le cas quasi-linéaire. On montre aussi l'existence de solutions globales faibles pour une classe de systèmes paraboliques quasi-linéaires dont les non-linéarités sont au plus quadratiques et dont les données initiales sont seulement supposées positives et intégrables. Dans le dernier chapitre, on généralise un résultat d'existence globale de solutions fortes pour des systèmes de réaction-diffusion dont les non-linéarités ont une structure "triangulaire", pour lesquels on prend désormais en compte des termes d'advection et des coefficients de diffusion dépendant du temps et de la variable d'espace. Ce résultat est ensuite utilisé dans un argument de point fixe de Leray-Schauder pour prouver l'existence en toute dimension de solutions globales à un problème d'électromigration-diffusion. / This thesis is devoted to the study of parabolic systems of partial differential equations arising in mass action kinetics chemistry, population dynamics and electromigration theory. We are interested in the existence of global solutions, uniqueness of weak solutions, and in the fast-reaction limit in a reaction-diffusion system. In the first chapter, we study two cross-diffusion systems. We are first interested in a population dynamics model, where cross effects in the interactions between the different species are modeled by non-local operators. We prove the well-posedness of the corresponding system for any space dimension. We are then interested in a cross-diffusion system which arises as the fast-reaction limit system in a classical system for the chemical reaction C1+C2=C3. We prove the convergence when k goes to infinity of the solution of the system with finite reaction speed k to a global solution of the limit system. The second chapter contains new global existence results for some reaction-diffusion systems. For networks of elementary chemical reactions of the type Ci+Cj=Ck and under Mass Action Kinetics assumption, we prove the existence and uniqueness of global strong solutions, for space dimensions N<6 in the semi-linear case, and N<4 in the quasi-linear case. We also prove the existence of global weak solutions for a class of parabolic quasi-linear systems with at most quadratic non-linearities and with initial data that are only assumed to be nonnegative and integrable. In the last chapter, we generalize a global well-posedness result for reaction-diffusion systems whose nonlinearities have a "triangular" structure, for which we now take into account advection terms and time and space dependent diffusion coefficients. The latter result is then used in a Leray-Schauder fixed point argument to prove the existence of global solutions in a diffusion-electromigration system.

Les systèmes de réaction-diffusion interviennent pour décrire les transitions de phase dans de nombreux champs d'application. Cette thèse porte sur l'analyse mathématique de modèles de propagation dans des milieux diffusifs, non bornés et hétérogènes, et s'inscrit ainsi dans la lignée d'une recherche particulièrement active. La première partie concerne l'équation simple: on s'y intéressera à la structure interne des fronts, mais on exhibera aussi de nouvelles dynamiques où la vitesse d'un profil de propagation n'est pas unique. Dans la seconde partie, on s'intéresse aux systèmes à deux équations, pour lesquels l'absence de principe du maximum pose de nombreuses difficultés. Ces travaux, en portant sur un vaste éventail de situations, offrent une meilleure compréhension des phénomènes de propagation, et mettent en avant de nouvelles propriétés des problèmes de réaction-diffusion, aidant ainsi à améliorer l'analyse théorique comme alternative à l'approche empirique. / Reaction-diffusion systems arise in the description of phase transitions in various fields of natural sciences. This thesis is concerned with the mathematical analysis of propagation models in some diffusive, unbounded and heterogeneous media, which comes within the scope of an active research subject. The first part deals with the single equation, by looking at the inside structure of fronts, or by exhibiting new dynamics where the profile of propagation may not have a unique speed. In a second part, we take interest in some systems of two equations, where the lack of maximum principles raises many theoretical issues. Those works aim to provide a better understanding of the underlying processes of propagation phenomena. They highlight new features for reaction-diffusion problems, some of them not known before, and hence help to improve the theoretical approach as an alternative to empirical analysis.

L’ADN est reconnu depuis longtemps comme une des molécules fondamentales des organismes vivants.Support de l’information génétique, la molécule d’ADN possède aussi des propriétés qui en font unmatériel de choix pour construire à l’échelle nanométrique. Deux simples brins d’ADN complémentaireset antiparallèles (c.à.d. de directivité opposée) peuvent, par exemple, s’hybrider s’ils se rencontrent ensolution, c’est à dire s’associer l’un à l’autre. La cohésion de la molécule « double-brin » ainsi forméeest maintenue par une série de liaisons faibles entre les bases complémentaires de chaque brin. Cetteréaction d’hybridation de l’ADN est réversible : un double-brin stable à basse température retrouveral’état simple-brin à plus haute température.Notre capacité à lire (séquencer) et écrire (synthétiser) l’ADN est à l’origine de l’émergence dudomaine des nanotechnologies ADN. Cette capacité à prévoir quantitativement les interactions (cinétiqueset thermodynamiques) entre deux partenaires moléculaires quels qu’ils soient est propre à l’ADN: on peut facilement synthétiser deux molécules de même taille et nature, de manière à ce qu’elles interagissent– ou non – selon la séquence qui leur est propre. Il existe aussi toute une batterie d’enzymescapables de catalyser différentes réactions au sein d’un brin d’ADN ou entre deux brins d’ADN, parexemple : une polymérase catalyse la synthèse d’un brin d’ADN à partir de son complémentaire ; unenickase coupe un seul des deux brins d’une molécule double-brin à un emplacement spécifique ; uneexonucléase hydrolyse un brin d’ADN en fragments plus courts, tandis qu’une ligase lie deux brinscourts en un brin unique, plus long. Dans cette thèse, nous commençons par vérifier que les trois modules de l’oligator (activation,autocatalyse et inhibition) peuvent être réarrangés de manière arbitraire, afin de créer différents circuitsde réactions dynamiques. Nous appellerons cette collection de réactions catalysées par trois enzymes(polymérase, nickase et exonucléase) la boite à outils ADN. La construction et le contrôle de circuitscomplexes nécessitent de pouvoir observer les modules désirés de manière spécifique et en temps réel.A cette fin, nous mettons au point une nouvelle technique de fluorescence utilisant une interaction– souvent négligée – entre les bases d’ADN et un fluorophore qui y est attaché : celui-ci émet unefluorescence dont l’intensité dépend de l’état (simple ou double brin) et de la séquence à proximité dufluorophore. / Biological organisms process information through the use of complex reaction networks. These can bea great source of inspiration for the tailoring of dynamic chemical systems. Using basic DNA biochemistry–the DNA-toolbox– modeled after the cell regulatory processes, we explore the construction ofspatio-temporal dynamics from the bottom-up.First, we design a monitoring technique of DNA hybridization by harnessing a usually neglectedinteraction between the nucleobases and an attached fluorophore. This fluorescence technique –calledN-quenching– proves to be an essential tool to monitor and troubleshoot our dynamic reaction circuits.We then go on a journey to the roots of the DNA-toolbox, aiming at defining the best design rulesat the sequence level. With this experience behind us, we tackle the construction of reaction circuitsdisplaying bistability. We link the bistable behavior to a topology of circuit, which asks for specificDNA sequence parameters. This leads to a robust bistable circuit that we further use to explore themodularity of the DNA-toolbox. By wiring additional modules to the bistable function, we make twolarger circuits that can be flipped between states: a two-input switchable memory, and a single-inputpush-push memory. Because all the chemical parameters of the DNA-toolbox are easily accessible,these circuits can be very well described by quantitative mathematical modeling. By iterating thismodular approach, it should be possible to construct even larger, more complex reaction circuits: eachsuccess along this line will prove our good understanding of the underlying design rules, and eachfailure may hide some still unknown rules to unveil.Finally, we propose a simple method to bring DNA-toolbox made reaction circuits from zerodimensional,well-mixed conditions, to a two-dimensional environment allowing both reaction anddiffusion. We run an oscillating reaction circuit in two-dimensions and, by locally perturbing it, areable to provoke the emergence of traveling and spiral waves. This opens up the way to the building ofcomplex, tailor-made spatiotemporal patterns.

Neste trabalho estudamos problemas de reação-difusão semilineares do tipo \'u IND..t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \'PERTENCE A\' \'OMEGA\' \'PARTIAL\' U/\'PARTIAL\'V (x, t) = 0, x \'PERTENCE A\' \'PARTIAL\'\' OMEGA\'. Desenvolvemos uma teoria abstrata para a obtenção da continuidade da dinâmica assintótica de (P) sob perturbações singulares do domínio espacial W e aplicamos a uma série de exemplos dos assim chamados domínios finos / In this work we study semilinear reaction-diffusion problems of the type \'u IND.t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \' PERTENCE A\' \'OMEGA\' \'PARTIAL\'u/\'ARTIAL\' v (x, t) = 0, x \"PERTENCE A\' \'PARTIAL\' \' OMEGA\' We develop a abstract theory to obtain the continuity of the asymptotic dynamics of (P) under singular perturbations of the spatial domain W and we apply that to many examples in thin domains