L’objectif de cette thèse est d’étudier le comportement en temps long de certains processus de Markov déterministes par morceaux (PDMP) dont le flot suivi par la composante spatiale commute aléatoirement entre plusieurs flots possédant un unique équilibre attractif (éventuellement le même pour chaque flot). Nous donnerons dans un premier temps un exemple d’étude d’un tel processus construit dans le plan à partir de flots associés à des équations différentielles linéaires stables où il est déjà possible d’observer des comportements contre-intuitifs. La deuxième partie de ce manuscrit est dédiée à l’étude et la comparaison de deux modèles de compétition pour une ressource dans un environnement hétérogène. Le premier modèle est un modèle alétoire simulant l’hétérogénéité temporelle d’un environnement sur les espèces en compétition à l’aide d’un PDMP. Son étude utilise des outils maintenant classiques sur l’étude des PDMP. Le deuxième modèle est un modèle déterministe (présentant sous forme d’un système d’équations différentielles) modélisant l’impact de l’hétérogénéité spatiale d’un environnement sur ces mêmes espèces. Nous verrons que malgré leur nature très différente, le comportement en temps long de ces deux systèmes est relativement similaire et est essentiellement déterminé par le signe des taux d’invasion de chacune des espèces qui sont des quantités dépendant exclusivement des paramètres du système et modélisant la vitesse de croissance (ou de décroissance) de ces espèces lorsqu’elles sont au bord de l’extinction. / The objective of this thesis is to study the long time behaviour of some piecewise deterministic Markov processes (PDMP). The flow followed by the spatial component of these processes switches randomly between several flow converging towards an equilibrium point (not necessarily the same for each flow). We will first give an example of such a process built in the plan from two linear stable differential equations and we will see that its stability depends strongly on the switching times. The second part of this thesis is dedicated to the study and comparison of two competition models in a heterogeneous environment. The first model is a probabilistic model where we build a PDMP simulating the effect of the temporal heterogeneity of an environment over the species in competition. Its study uses classical tools in this field. The second model is a deterministic model simulating the effect of the spatial heterogeneity of an environment over the same species. Despite the fact that the nature of the two models is very different, we will see that their long time behavior is very similar. We define for both model several quantities called invasion rates modelizing the growth (or decreasing) rate speed of a species when it is near to extinction and we will see that the signs of these invasion rates fully describes the long time behavior for both systems.

L'attraction d'ondes dans des systèmes contra-propagatifs est un phénomène général, établi initialement en Physique dans le contexte de l'attraction de polarisation entre deux ondes contra-propagatives se propageant dans des fibres optiques. Ce phénomène a été observé expérimentalement, et ses propriétés étudiées via des simulations numériques. Les modèles qui s'y rattachent sont des systèmes hyperboliques d'équations aux dérivées partielles, avec des conditions aux bords dépendant du temps sur un intervalle fini. Le mécanisme sous-jacent peut être expliqué par l'existence de tores singuliers dans les équations stationnaires correspondantes. Le but de cette thèse est d'analyser en détail l'exemple le plus simple dans cette famille de modèles. Nous montrons que la plupart des phénomènes de processus d'attraction d'ondes sont en fait existants dans un modèle linéaire avec intéraction résonnante. Nous établissons l'existence et la régularité des solutions et analysons la relaxation vers la solution stationnaire qui caractérise les propriétés de l'attraction d'ondes. / Wave attraction in counter-propagating waves systems is a general phenomenon that was first established in Physics in the context of the attraction of the polarization between two counter-propagating waves in optical fibers. This phenomenon has been observed experimentally, and its properties were studied through numerical simulations. The models are Hamiltonian hyperbolic systems of partial differential equations, with time-dependent boundary conditions on a finite interval. The underlying mechanism can be traced back to the existence of singular tori in the corresponding stationary equations. In this work we analyze in detail the simplest example in this family of models. We show that most of the phenomena of the wave attraction process are already present in a linear model with resonant interaction. We establish the existence and regularity of the solutions and analyze the relaxation towards a stationary solution that features the wave attraction properties.

The provision of clean drinking water typically involves treatment processes to remove contaminants. The conventional process involves coagulation with hydrolysing metal salts, typically of aluminium (‘alum’) or trivalent iron (‘ferric’). Along with the product water this also produces a waste by-product, or sludge. The fact of increasing sludge production — due to higher levels of treatment and greater volume of water supply — conflicts with modern demands for environmental best practice, leading to higher financial costs. A further issue is the significant quantity of water that is held up in the sludge, and wasted. / One means of dealing with these problems is to dewater the sludge further. This reduces the volume of waste to be disposed of. The consistency is also improved (e.g. for the purpose of landfilling). And a significant amount of water can be recovered. The efficiency, and efficacy, of this process depends on the dewaterability of the sludge.In fact, good dewaterability is vital to the operation of conventional drinking water treatment plants (WTP’s). The usual process of separating the particulates, formed from a blend of contaminants and coagulated precipitate, relies on ‘clarification’ and ‘thickening’, which are essentially settling operations of solid–liquid separation.WTP operators — and researchers — do attempt to measure sludge dewaterability, but usually rely on empirical characterisation techniques that do not tell the full story and can even mislead. Understanding of the physical and chemical nature of the sludge is also surprisingly rudimentary, considering the long history of these processes. / The present work begins by reviewing the current state of knowledge on raw water and sludge composition, with special focus on solid aluminium and iron phases and on fractal aggregate structure. Next the theory of dewatering is examined, with the adopted phenomenological theory contrasted with empirical techniques and other theories.The foundation for subsequent analyses is laid by experimental work which establishes the solid phase density of WTP sludges. Additionally, alum sludges are found to contain pseudoböhmite, while 2-line ferrihydrite and goethite are identified in ferric sludges. / A key hypothesis is that dewaterability is partly determined by the treatment conditions. To investigate this, numerous WTP sludges were studied that had been generated under diverse conditions: some plant samples were obtained, and the remainder were generated in the laboratory (results were consistent). Dewaterability was characterised for each sludge in concentration ranges relevant to settling, centrifugation and filtration using models developed by LANDMAN and WHITE inter alia; it is expressed in terms of both equilibrium and kinetic parameters, py(φ) and R(φ) respectively.This work confirmed that dewaterability is significantly influenced by treatment conditions.The strongest correlations were observed when varying coagulation pH and coagulant dose. At high doses precipitated coagulant controls the sludge behaviour, and dewaterability is poor. Dewaterability deteriorates as pH is increased for high-dose alum sludges; other sludges are less sensitive to pH. These findings can be linked to the faster coagulation dynamics prevailing at high coagulant and alkali dose.Alum and ferric sludges in general had comparable dewaterabilities, and the characteristics of a magnesium sludge were similar too.Small effects on dewaterability were observed in response to variations in raw water organic content and shearing. Polymer flocculation and conditioning appeared mainly to affect dewaterability at low sludge concentrations. Ageing did not produce clear changes in dewaterability.Dense, compact particles are known to dewater better than ‘fluffy’ aggregates or flocs usually encountered in drinking water treatment. This explains the superior dewaterability of a sludge containing powdered activated carbon (PAC). Even greater improvements were observed following a cycle of sludge freezing and thawing for a wide range of WTP sludges. / Further aspects considered in the present work include deviations from simplifying assumptions that are usually made. Specifically: investigation of long-time dewatering behaviour, wall effects, non-isotropic stresses, and reversibility of dewatering (or ‘elasticity’).Several other results and conclusions, of both theoretical and experimental nature, are presented on topics of subsidiary or peripheral interest that are nonetheless important for establishing a reliable basis for research in this area. / This work has proposed links between industrial drinking water coagulation conditions, sludge dewaterability from settling to filtration, and the microstructure of the aggregates making up that sludge. This information can be used when considering the operation or design of a WTP in order to optimise sludge dewaterability, within the constraints of producing drinking water of acceptable quality.