Many large-scale random graphs (e.g., the Internet) exhibit complex topology, nonhomogeneous spatial node distribution, and preferential attachment of new nodes. Current topology models for ad-hoc networks mostly consider a uniform spatial distribution of nodes and do not capture the dynamics of evolving, real-world graphs, in which nodes "gravitate" toward popular locations and self-organize into non-uniform clusters. In this thesis, we first investigate two constraints on scalability of ad-hoc networks – network reliability and node capacity. Unlike other studies, we analyze network resilience to node and link failure with an emphasis on the growth (i.e., evolution) dynamics of the entire system. Along the way, we also study important graph-theoretic properties of ad-hoc networks (including the clustering coefficient and the expected path length) and strengthen our generic understanding of these systems. Finally, recognizing that under existing uniform models future ad-hoc networks cannot scale beyond trivial sizes, we argue that ad-hoc networks should be modeled from an evolution standpoint, which takes into account the well-known "clustering" phenomena observed in all real-world graphs. This model is likely to describe how future ad-hoc networks will self-organize since it is well documented that information content distribution among end-users (as well as among spatial locations) is non-uniform (often heavy-tailed). Results show that node capacity in the proposed evolution model scales to larger network sizes than in traditional approaches, which suggest that non-uniformly clustered, self-organizing, very large-scale ad-hoc networks may become feasible in the future.

Many large-scale random graphs (e.g., the Internet) exhibit complex topology, nonhomogeneous spatial node distribution, and preferential attachment of new nodes. Current topology models for ad-hoc networks mostly consider a uniform spatial distribution of nodes and do not capture the dynamics of evolving, real-world graphs, in which nodes "gravitate" toward popular locations and self-organize into non-uniform clusters. In this thesis, we first investigate two constraints on scalability of ad-hoc networks – network reliability and node capacity. Unlike other studies, we analyze network resilience to node and link failure with an emphasis on the growth (i.e., evolution) dynamics of the entire system. Along the way, we also study important graph-theoretic properties of ad-hoc networks (including the clustering coefficient and the expected path length) and strengthen our generic understanding of these systems. Finally, recognizing that under existing uniform models future ad-hoc networks cannot scale beyond trivial sizes, we argue that ad-hoc networks should be modeled from an evolution standpoint, which takes into account the well-known "clustering" phenomena observed in all real-world graphs. This model is likely to describe how future ad-hoc networks will self-organize since it is well documented that information content distribution among end-users (as well as among spatial locations) is non-uniform (often heavy-tailed). Results show that node capacity in the proposed evolution model scales to larger network sizes than in traditional approaches, which suggest that non-uniformly clustered, self-organizing, very large-scale ad-hoc networks may become feasible in the future.

This thesis presents several nonlinear mathematical models applied to ecoepidemiology and evolution. A detailed study involving predator-prey type models considering an alternative resource for the predator was carried out, investigating the situation of infection in the prey and in the predator on separate models. Such study served as a theoretical contribution to the investigation of problems such as bovine tuberculosis in wild animal species presented in a specific model. We also developed models to explain the evolution of alarm calls in species of birds and mammals. The theoretical framework adopted for those evolution models is that of Population Ecology. The models were developed using Ordinary Diferential Equations (ODEs) to describe the population dynamics. The biological assumptions of the systems that we wanted to analyse were enumerated and explained / Esta tese apresenta vários modelos matemáticos não-lineares aplicados à ecopidemiologia e à evolução. Foi realizado um estudo detalhado envolvendo modelos do tipo predador-presa considerando um recurso alternativo para o predador, investigando situações de infecção na presa e no predador em modelos separados. Tal estudo, serviu de aporte teórico para a investigação de problemas como a tuberculose bovina em espécies de animais selvagens apresentado em um modelo específico. Também desenvolvemos modelos para explicar a evolução dos chamados de alarme em espécies de aves e mamíferos. O quadro teórico adotado para esses modelos de evolução é o da Ecologia de População. Nos modelos desenvolvidos usamos as Equações Diferenciais Ordinárias (EDOs) para descrever a dinâmica populacional. Consideramos pressupostos biológicos dos sistemas biológicos analisados

Darwinian Evolutionary system - a system satisfying the abstract conditions: reproduction with heritable variation, in a finite world, giving rise to Natural Selection encompasses a complex and subtle system of interrelated theories, whose substantive transplantation to any artificial medium let it be mathematical model or computational model - will be very far from easy. There are two motives in bringing Darwinian evolution into computational frameworks: one to understand the Darwinian evolution, and the other is to view Darwinian evolution - that carries out controlled adaptive-stochastic search in the space of all possible DNA-sequences for emergence and improvement of the living beings on our planet - as an optimization process, which can be simulated in appropriate frameworks to solve some intractable problems. The first motive led to emerging field of study commonly referred to as Artificial Life, and other gave way to emergence of Evolutionary Computation, which is speculated to be the only practical path to the development of ontogenetic machine intelligence. In this thesis we touch upon all the above aspects. Natural selection is the central concept of Darwinian evolution and hence capturing natural selection in computational frameworks which maintains the spirit of Darwinian evolution in the sense of conventional, terrestrial and biological perspectives is essential. Naive models of evolution define natural selection as a process which brings in differential reproductive capabilities in organisms of a population, and hence, most of the evolutionary simulations in Artificial Life and Evolutionary Computation implement selection by differential reproduction: the Attest members of the population are reproduced preferentially at the expense of the less fit members of the population. Formal models in evolutionary biology often subdivide selection into components called 'episodes of selection' to capture the different complex mechanisms of nature by which Darwinian evolution can occur. In this thesis we introduce the concept of 'episodes of selection' into computational frameworks of Darwinian evolution by means of A Conceptual Evolutionary model (ACE-model). ACE-model is proposed to be simple and yet it captures the essential features of modern evolutionary perspectives in evolutionary computation framework. ACE-model is rich enough to offer abstract and structural framework for evolutionary computation and can serve as a basic model for evolutionary algorithms. It captures selection in two episodes in two phases of evolutionary cycle and it offers various parameters by which evolutionary algorithms can control selection mechanisms. In this thesis we propose two evolutionary algorithms namely Malthus evolutionary algorithms and Malthus Spencer evolutionary algorithms based on the ACE-model and we discuss the relevance of parameters offered by ACE-model by simulation studies. As an application of ACE-model to artificial life we study misconceptions involved in defining fitness in evolutionary biology, and we also discuss the importance of introducing fitness landscape in the theories of Darwinian evolution. Another important and independent contribution of this thesis is: A Mathematical Abstraction of Evolutionary process. Evolutionary process is characterized by Evolutionary Criteria and Evolutionary Mechanism which are formalized by classical mathematical tools. Even though the model is in its premature stage to develop any theory based on it, we develop convergence criteria of evolutionary process based on this model.

Les reliefs résultent d'une interaction entre de nombreux processus qui mettent en jeu des couplages complexes : climat, tectonique, érosion, impact anthropique, etc. Dans les paysages dominés par l'érosion fluviatile, la morphologie du relief est étroitement liée à la structure du réseau hydrographique. Ce dernier joue un rôle important dans les couplages entre les processus de surface (processus fluviaux et processus de versants). De nombreuses évidences de captures entre rivières provoquant des réorganisations du réseau hydrographique suggèrent que le réseau de drainage évolue, et il est loin d'être fixe comme le suggèrent certains modèles numériques d'évolution du paysage. Néanmoins, les travaux s'intéressant aux causes et aux mécanismes de ces réorganisations sont peu nombreux, et essentiellement qualitatifs. L'objectif de cette thèse a été de quantifier et de modéliser la réorganisation des réseaux hydrographiques par mécanisme de capture par rivières pirates dans le cadre d'un enfoncement relatif entre deux bassins, où l'avancement (par érosion régressive) de la tête de vallée de la rivière pirate provoque le détournement d'une rivière qui coule à proximité. Le comportement des Modèles numériques d'Evolution des Paysages (MEP) et plus particulièrement l'influence des paramètres de contrôle, ont été étudiés. L'expérimentation a été réalisée avec le logiciel GOLEM qui a été adapté aux besoins. L'étude s'est principalement portée sur le cas de l'évolution du réseau hydrographique de la partie française de la Meuse, qui a connue de nombreuses captures par le passé (capture de la Haute-Moselle par la Meurthe, capture de l'Aire par l'Aisne) et dont les caractéristiques actuelles sont favorables à de nouvelles captures (le lit de la Meuse se trouve perché à plus de 50m au dessus des vallées qui l'entourent (le lit de la Meuse se trouve perché à plus de 50m au dessus des vallées qui l'entourent : la Marne à l'ouest et la Moselle à l'est). / Landforms result from interaction between several processes, involving complex coupling: climate, tectonics, erosion, human impact, etc. In landscapes dominated by fluvial erosion, terrain morphology is closely related to the hydrographic network system. Because the latter plays an important role in the surface processes coupling (fluvial processes and hillslope processes). Many evidences of stream piracies causing reorganization of hydrographic network suggest that the drainage system is changing, and it is far from being fixe as propose some numerical landscape evolution models. Nevertheless, works focusing on causes and mechanisms of these reorganizations are very few and they are largely qualitative. The aim of this thesis is to quantify and describe the reorganization of hydrographic networks provoked by headward piracy mechanism in the case of relative lowering between two basins, where the riverhead progress by headward erosion causing the diversion of a river flowing nearby. The problem is tackled with a numerical modeling approach. We emphasis on the evolution of the hydrographic networks of the French part of the Meuse basin, where several piracies were recorded in the past (the piracy of the Upper-Moselle by the Meurthe and the piracy of the Aire by the Aisne), and which current characteristics are favorable to new piracies (the Meuse river bed is perched at more than 50m comparing to its surrounding valleys (the Meuse river bed is perched at more than 50m comparing to its surrounding valleys: Marne at the west and the Moselle at the east).

En la Tesis titulada "Análisis y mejoras en la evaluación de diversos términos de los modelos de evolución de playas en escalas de medio plazo" se desarrolla un modelo de evolución de playas de medio-largo plazo y se aplica a zonas de refracción-difracción. La Tesis se divide en tres secciones principales:SECCIÓN 2. Fundamentos y desarrollo del modelo de evolución de medio-largo plazoSe desarrolla y valida con datos de campo el modelo de evolución de playas de medio-largo plazo.SECCIÓN 3. Oleaje en rotura en zonas de refracción-difracción en los modelos de evolución de playasSe propone y valida un método para la determinación de las características del oleaje en rotura en zonas de refracción-difracción.SECCIÓN 4. Perfil de equilibrio en zonas de refracción-difracciónSe deriva una expresión analítica para definir el perfil de equilibrio en zonas de refracción-difracción y se calibra con datos de campo. / In this Thesis entitled "Analysis and improvements in the evaluation of several terms of medium-term beach evolution models" a medium-long term beach evolution model is developed and applied in refraction-diffraction areas. The Thesis is composed of the following main sections:SECTION 2. Fundaments and development of a medium-long term beach evolution modelIn this section a medium-long term beach evolution model is developed and validated based on field data.SECTION 3. Breaking wave characteristics in refraction-diffraction areas for beach evolution modelsIn this section a method to determine the breaking wave characteristics in refraction-diffraction areas is proposed and validated.SECTION 4. Equilibrium beach profile for refraction-diffraction areasIn this section an analytical expression to calculate the equilibrium beach profile for refraction-diffraction areas is derived, being calibrated based on field data.

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional (that derivative is called the 􀀀derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold behavior. The results obtained here may be signi cant for the ght and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still a ecting many people in West-Africa and other parts of the world. The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [129, 142], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. Moreover, the explicit form of its solution is, in general, impossible to obtain. We explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with the conventional time derivative with a new parameter. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and prove that the system describing the polymer chain degradation contains complex and simple harmonic poles whose e ects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, di usion or relaxation equations. The question is to say whether or not this is always true! Whence, three generalized evolution equations with an additional fractional parameter are solved analytically with conventional techniques. These are processes related to stationary state system, relaxation and di usion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However, unlike other models, the generalized di usion and relaxation models are proven not to be governed by Mittag-Le er functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called ; can be used to control the stationarity of such generalized models. / Mathematical Sciences / Ph. D. (Applied Mathematics)

This study focusses on modelling important aspects of the evolution of pulsar wind nebulae using two different approaches. The first uses a hydrodynamic model to simulate the morphological evolution of a spherically-symmetric composite supernova remnant that is expanding into a homogeneous interstellar medium. In order to extend this model, a magnetic field is included in a kinematic fashion, implying that the reaction of the fluid on the magnetic field is taken into account, while neglecting any counter-reaction of the field on the fluid. This approach is valid provided that the ratio of electromagnetic to particle energy in the nebula is small, or equivalently, for a large plasma β environment. This model therefore allows one to not only calculate the evolution of the convection velocity but also, for example, the evolution of the average magnetic field. The second part of this study focusses on calculating the evolution of the energy spectra of the particles in the nebula using a number of particle evolution models. The first of these is a spatially independent temporal evolution model, similar to the models that can be found in the literature. While spatially independent models are useful, a large part of this study is devoted to developing spatially dependent models based on the Fokker-Planck transport equation. Two such models are developed, the first being a spherically-symmetric model that includes the processes of convection, diffusion, adiabatic losses, as well as the non-thermal energy loss processes of synchrotron radiation and inverse Compton scattering. As the magnetic field geometry can lead to the additional transport process of drift, the previous model is extended to an axisymmetric geometry, thereby allowing one to also include this process. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2014

This study focusses on modelling important aspects of the evolution of pulsar wind nebulae using two different approaches. The first uses a hydrodynamic model to simulate the morphological evolution of a spherically-symmetric composite supernova remnant that is expanding into a homogeneous interstellar medium. In order to extend this model, a magnetic field is included in a kinematic fashion, implying that the reaction of the fluid on the magnetic field is taken into account, while neglecting any counter-reaction of the field on the fluid. This approach is valid provided that the ratio of electromagnetic to particle energy in the nebula is small, or equivalently, for a large plasma β environment. This model therefore allows one to not only calculate the evolution of the convection velocity but also, for example, the evolution of the average magnetic field. The second part of this study focusses on calculating the evolution of the energy spectra of the particles in the nebula using a number of particle evolution models. The first of these is a spatially independent temporal evolution model, similar to the models that can be found in the literature. While spatially independent models are useful, a large part of this study is devoted to developing spatially dependent models based on the Fokker-Planck transport equation. Two such models are developed, the first being a spherically-symmetric model that includes the processes of convection, diffusion, adiabatic losses, as well as the non-thermal energy loss processes of synchrotron radiation and inverse Compton scattering. As the magnetic field geometry can lead to the additional transport process of drift, the previous model is extended to an axisymmetric geometry, thereby allowing one to also include this process. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2014

La thèse qui suit est organisée en deux volets: un premier volet portant sur les modèles de masse de galaxies et un second volet sur la conception de revêtements optiques et le contrôle de leurs propriétés mécaniques. Les modèles de masse présentés dans cette thèse ont été réalisés sur un sous-échantillon de dix galaxies de l'étude SINGS comprenant neuf galaxies normales et une galaxie naine. Ce travail visait à fixer le rapport masse-luminosité du disque à tout rayon en utilisant les résultats de modèles d'évolution galactique chimio-spectrophotométriques ajustés spécifiquement à chaque galaxie grâce à son profil de photométrie multi-bandes. Les résultats montrent que les disques stellaires tels que normalisés par les rapports masse-luminosité issus des modèles ont des masses cohérentes dans toutes les bandes étudiées de l'ultra-violet, du visible ainsi que du proche infrarouge (bandes FUV à IRAC2). Ces disques peuvent être considérés comme maximaux par rapport aux données cinématiques des galaxies étudiées. Ceci est dû au fait que le rapport M/L est plus élevé au centre que sur les bords. Les disques étant maximaux et physiquement justifiés, on ne peut dès lors ignorer les effets de composants tels que les bulbes ou les barres et les corrections nécessaires doivent être apportées aux profils de luminosité et de vitesses de rotation de la galaxie. Dans les travaux de la seconde partie, le logiciel en développement libre OpenFilters a été modifié afin de tenir compte des contraintes mécaniques dans la conception numérique de revêtements optiques. Les contraintes mécaniques dans les couches minces ont un effet délétère sur leurs performances optiques. Un revêtement destiné à rendre réflectives les lames d'un étalon Fabry-Perot utilisé en astronomie a été conçu et fabriqué afin d'évaluer les performances réelles de la méthode. Ce cas a été choisi à cause de la diminution de la finesse d'un étalon Fabry-Perot apporté par la courbure des lames sous l'effet des contraintes. Les résultats montrent que les mesures concordent avec les modèles numériques et qu'il est donc possible à l'aide de ce logiciel d'optimiser les revêtements pour leur comportement mécanique autant que pour leurs propriétés optiques. / The following thesis includes two parts: the first concentrates on mass models of galaxies while the second concerns the design of broadband optical coatings and the control of their mechanical properties. Mass models presented here were performed on a subsample of ten galaxies from the SINGS survey. It includes nine late-type regular galaxies as well as a dwarf galaxy. The work consisted in the determination at all radii of M/L ratio of galactic discs by choosing, amongst a grid of chemospectrophotometric galactic evolution models, the one that fits better the galaxy's multi-band photometric profile. Results from this work show that stellar discs weighted by the model's mass-to-luminosity ratio display consistent masses in all studied bands from the ultraviolet to the near-infrared (FUV to IRAC2 bands). The resulting discs are physically motivated and maximal, due to their mass-to-luminosity ratio being higher in the center of the galaxies than at their outskirts. In this maximal-disc situation, bars and bulges of galaxies can no longer be ignored and the appropriate corrections need to be made on the luminosity or circular velocity profiles of the galaxy. In the second part, an additional module of the open source software Open Filters was written in order to take into account the mechanical properties of thin films in the design of optical coatings. The mechanical stresses arising in optical coatings can have a deleterious effect on the optical performances of coatings, and this tool was meant to provide design capacities to minimize this effect. Broadband reflective coatings for Fabry-Perot plates were designed and fabricated as a proof of concept of the method. This specific case was interesting because the curvature induced by mechanical stresses induces decrease of the finesse of Fabry-Perot etalons. Actual curvature of the deposited samples are coherent with the numerical calculations and undergo a substantial reduction for the two strategies devised here with this new design too. We therefore conclude that it is possible with this new software module to optimize coatings for their mechanical behaviour as well as for their optical properties.