The growing flow of people and goods around the globe has allowed new, non-native species to establish and spread in already fragile ecosystems. The introduction of invasive species can have a detrimental impact on the already established species. Thus, it is important that we understand the mechanisms that facilitate or prevent invasion. Since reaction-diffusion invasion models produce travelling waves we can study invasion by looking at the mechanisms that allow for wave propagation failure, or wave-blocking. In this thesis we consider a perturbed reaction-diffusion model in which the perturbation resides in either the reaction or diffusion term. In doing so we exploit the underlying symmetry of our problem to define a region in the appropriate parameter space that leads to wave blocking. As a demonstrative example we apply our theory to the bistable equation and consider the effects of various perturbations.

Cette thèse porte sur l’étude expérimentale et théorique de quatre instabilités associées à l’émergence de motifs réguliers sur des substrats érodables ou fortement déformables,instabilités liées à l’hydrodynamique sur un relief modulé.La première partie porte sur l’étude de l’instabilité d’une plaque élastique fixée aux deux bouts et soumise à un écoulement fluide permanent. La solution plane est instable vis-à-vis d’ondes propagatives, lorsque l’écoulement est suffisamment fort. La sélection de fréquence et de longueur d’onde est caractérisée expérimentalement en fonction de la vitesse de l’écoulement. Ces quantités suivent remarquablement les lois d’échelle obtenues par l’analyse de stabilité linéaire du problème. Le principe de l’expérience pourrait être appliqué à la récupération d’énergie.La deuxième partie porte sur une analyse théorique de la formation de rides géantes sur la comète 67P, récemment observées par la sonde Rosetta. Nous montrons comment le dégazage de vapeur se produit au travers d’une couche poreuse granulaire superficielle et comment l’alternance jour/nuit conduit à des gradients de pression gigantesques qui engendrent des vents thermiques de surface. Ces motifs apparaissent comme étant les analogues de rides qui se forment à la surface de lit sableux dans un écoulement visqueux.L’analyse de stabilité linéaire du problème permet de prédire quantitativement l’émergence de ces rides à la longueur d’onde et à la vitesse de propagation observées. Cette description fournit un outil robuste et fiable pour décrire les processus d’érosion et d’accrétion dans l’évolution des petits corps.Dans la troisième partie, nous proposons un modèle pour l’apparition de motifs de sublimation sur Pluton, tels que ceux observés sur Sputnik Planum. La formation et l’évolution de ces motifs proviennent de la sublimation/condensation différentielle de la glace d’azote.Nous montrons que l’atmosphère de Pluton possède des propriétés (température et pression)peu variables en espace et en temps. Nous analysons les différents mécanismes d’instabilité en compétition et concluons à un mécanisme original, basé sur le mélange et e transport de chaleur dans l’atmosphère, plutôt qu’au mécanisme des pénitents, basé sur l’auto-éclairement de la surface de glace.Enfin, nous avons étudié théoriquement l’instabilité de formation des rides éoliennes en considérant les trajectoires des grains résonantes avec le relief. Cette modélisation prend en compte de manière simple et effective les effets collectifs du transport de sédiments. Le modèle est validé à partir de simulations numériques existantes, elles mêmes calées sur des expériences contrôlées. / This thesis is devoted to the experimental and theoretical investigations of four instabilitiesassociated with the emergence of regular patterns over erodible/flexible substrates, andrelated to hydrodynamics over a modulated relief.First, the instability of a flexible sheet clamped at both ends and submitted to a permanentwind is investigated. The flat sheet solution is unstable towards propagative waves, forstrong enough wind. We experimentally study the selection of frequency and wavenumberas a function of the wind velocity. These quantities obey simple scaling laws derived froma linear stability analysis of the problem. This phenomenon may be applied for energyharvesting.Second, an explanation is proposed for the giant ripples observed by spacecraft Rosettaat the surface of the comet 67P. We show that the outgassing flow across a porous surfacegranular layer and the strong pressure gradient associated with the day-night alternanceare responsible for thermal superficial winds. We show that these unexpected patterns areanalogous to ripples emerging on granular beds submitted to viscous shear flows. Linearstability analysis of the problem quantitatively predicts the emergence of bedforms at theobserved wavelength and their propagation. This description provides a reliable tool topredict the erosion and accretion processes controlling the evolution of small solar systembodies.Third, we propose a model for rhythmic, dune-like patterns observed on SputnikPlanum of Pluto. Their emergence and evolution are related to the differential condensation/sublimation of nitrogen ice. We show that the temperature and pressure in Pluto’satmosphere are almost homogeneous and steady, and that heat flux from the atmospheredue to convection and turbulent mixing is responsible for the emergence of these sublimationpatterns, in contrast to the penitentes instability due to solar radiation.Last, we report an analytical model for the aeolian ripple instability by considering theresonant grain trajectories over a modulated sand bed, taking the collective effect in thetransport layer into account. The model is tested against existing numerical simulationsthat match experimental observations.

Sistemas de reação-difusão têm sido largamente estudados em diferentes contextos e através de diferentes métodos, motivados pela sua constante aparição em modelos de interação em contextos químicos, biológicos e ainda em fenômenos ecológicos. Neste trabalho nos propomos a estudar existência e unicidade - tanto do ponto de vista local como global - de soluções para uma classe de sistemas de reação-difusão acoplados, denidos em R^2, utilizando como ferramenta a teoria de semigrupos de operadores lineares. Apresentamos dois importantes exemplos: o modelo de Rosenzweig-MacArthur e um particular caso da classe de equações lambda-omega. Para o primeiro obtemos um resultado de existência e unicidade global utilizando um método de comparação envolvendo sub e super-soluções. Investigamos ainda a existência de soluções de ondas viajantes periódicas através do teorema de Bifurcação de Hopf. Já para o caso da equação lambda-omega obtemos a existência e unicidade de solucões, entretanto, a partir da aplicação da teoria de semigrupos de operadores lineares. / Reaction-diffusion systems have been widely studied in a broad variety of contexts in a large amount of disctinct approaches. It is due firstly by their constant appearance in interaction models in disciplines such as chemistry, biology and, more specific, ecology. The aim of this thesis is to provide an existence-uniqueness result - both from the local as well as from the global point of view - for solutions of a particular class of coupled reaction-diffusion systems defined over R^2. It is done applying the well established theory of semigroups of linear operators. Two remarkable examples of such systems are discussed: the Rosenzweig-MacArthur predator-prey model and a special case of lambda-omega class of equations. For the former one, an existence and uniqueness result is obtained through a comparison method - based on the notions of lower and upper solutions. Moreover, we investigate the existence of periodic travelling wave solutions via a Hopf bifurcation theorem. For the lambda-omega model another existence and uniqueness for solutions is obtained, on its turn, through the machinery obtained previously from the theory of semigroups for linear operators.

We elaborate upon the Herty-Illner macroscopic traffic models which include special non-local forces. The first chapter presents these in relation to the traffic models of Aw-Rascle and Zhang, arguing that non-local forces are necessary for a realistic description of traffic. The second chapter considers travelling wave solutions for the Herty-Illner macroscopic models. The travelling wave ansatz for the braking scenario reveals a curiously implicit nonlinear functional differential equation, the jam equation, whose unknown is, at least to conventional tools, inextricably self-argumentative! Observing that analytic solution methods fail for the jam equation yet succeed for equations with similar coefficients raises a challenging problem of pure and applied mathematical interest. An unjam equation analogous to the jam equation explored by Illner and McGregor is derived. The third chapter outlines refinements for the Herty-Illner models. Numerics allow exploration of the refined model dynamics in a variety of realistic traffic situations, leading to a discussion of the broadened applicability conferred by the refinements: ultimately the prediction of stop-and-go waves. The conclusion asserts that all of the above contribute knowledge pertinent to traffic control for reduced congestion and ameliorated vehicular flow. / Graduate

Sistemas de reação-difusão têm sido largamente estudados em diferentes contextos e através de diferentes métodos, motivados pela sua constante aparição em modelos de interação em contextos químicos, biológicos e ainda em fenômenos ecológicos. Neste trabalho nos propomos a estudar existência e unicidade - tanto do ponto de vista local como global - de soluções para uma classe de sistemas de reação-difusão acoplados, denidos em R^2, utilizando como ferramenta a teoria de semigrupos de operadores lineares. Apresentamos dois importantes exemplos: o modelo de Rosenzweig-MacArthur e um particular caso da classe de equações lambda-omega. Para o primeiro obtemos um resultado de existência e unicidade global utilizando um método de comparação envolvendo sub e super-soluções. Investigamos ainda a existência de soluções de ondas viajantes periódicas através do teorema de Bifurcação de Hopf. Já para o caso da equação lambda-omega obtemos a existência e unicidade de solucões, entretanto, a partir da aplicação da teoria de semigrupos de operadores lineares. / Reaction-diffusion systems have been widely studied in a broad variety of contexts in a large amount of disctinct approaches. It is due firstly by their constant appearance in interaction models in disciplines such as chemistry, biology and, more specific, ecology. The aim of this thesis is to provide an existence-uniqueness result - both from the local as well as from the global point of view - for solutions of a particular class of coupled reaction-diffusion systems defined over R^2. It is done applying the well established theory of semigroups of linear operators. Two remarkable examples of such systems are discussed: the Rosenzweig-MacArthur predator-prey model and a special case of lambda-omega class of equations. For the former one, an existence and uniqueness result is obtained through a comparison method - based on the notions of lower and upper solutions. Moreover, we investigate the existence of periodic travelling wave solutions via a Hopf bifurcation theorem. For the lambda-omega model another existence and uniqueness for solutions is obtained, on its turn, through the machinery obtained previously from the theory of semigroups for linear operators.

<p>We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads tends to zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic travelling waves of granular monomer chains exist.</p> / Master of Science (MSc)

This thesis is concerned with mathematical modelling of oncolytic virotherapy: the use of genetically modified viruses to selectively spread, replicate and destroy cancerous cells in solid tumours. Traditional spatially-dependent modelling approaches have previously assumed that virus spread is due to viral diffusion in solid tumours, and also neglect the time delay introduced by the lytic cycle for viral replication within host cells. A deterministic, age-structured reaction-diffusion model is developed for the spatially-dependent interactions of uninfected cells, infected cells and virus particles, with the spread of virus particles facilitated by infected cell motility and delay. Evidence of travelling wave behaviour is shown, and an asymptotic approximation for the wave speed is derived as a function of key parameters. Next, the same physical assumptions as in the continuum model are used to develop an equivalent discrete, probabilistic model for that is valid in the limit of low particle concentrations. This mesoscopic, compartment-based model is then validated against known test cases, and it is shown that the localised nature of infected cell bursts leads to inconsistencies between the discrete and continuum models. The qualitative behaviour of this stochastic model is then analysed for a range of key experimentally-controllable parameters. Two-dimensional simulations of in vivo and in vitro therapies are then analysed to determine the effects of virus burst size, length of lytic cycle, infected cell motility, and initial viral distribution on the wave speed, consistency of results and overall success of therapy. Finally, the experimental difficulty of measuring the effective motility of cells is addressed by considering effective medium approximations of diffusion through heterogeneous tumours. Considering an idealised tumour consisting of periodic obstacles in free space, a two-scale homogenisation technique is used to show the effects of obstacle shape on the effective diffusivity. A novel method for calculating the effective continuum behaviour of random walks on lattices is then developed for the limiting case where microscopic interactions are discrete.

This thesis firstly presents a nonlinear extended deterministic Susceptible-Infected (SI) model for assessing the impact of public health education campaign on curtailing the spread of the HIV pandemic in a population. Rigorous qualitative analysis of the model reveals that, in contrast to the model without education, the full model with education exhibits the phenomenon of backward bifurcation (BB), where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the effective reproduction number (Reff ), is less than unity. Furthermore, an explicit threshold value is derived above which such an education campaign could lead to detrimental outcome (increase disease burden), and below which it would have positive population-level impact (reduce disease burden in the community). It is shown that the BB phenomenon is caused by imperfect efficacy of the public health education program. The model is used to assess the potential impact of some targeted public health education campaigns using data from numerous countries. The second problem considered is a Susceptible-Infected-Removed (SIR) model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For Case (i), we construct travelling front solutions whose profiles are heteroclinic orbits which connect either the disease-free state to an infected state or two endemic states with each other. For Case (ii), it is shown that the profile has the following properties: the number of susceptible individuals is monotone increasing and the number of infectives approaches zero, while their product converges to a constant. Numerical simulations are shown which confirm these analytical results. Abnormal behavior like travelling waves with non-monotone profile or oscillations are observed.

Cette thèse s'inscrit dans le cadre de la modélisation des interactions entre hôtes et auxiliaires de lutte biologique. L'objectif principal est de faire l'analyse mathématique et la simulation numérique des modèles spatiotemporels construits. Il s'agit de déterminer la typologie et la catégorisation des structures spatiales émergentes en fonction des paramètres de contrôle. Nous considérons dans la première partie de la thèse, une chaîne alimentaire de deux espèces, c'est à dire une population de proies et une population de prédateurs modélisées par un système de réaction-diffusion. Nous étudions l'analyse qualitatives des solutions, les bifurcations globales et locales, et déterminons les conditions de variation spatiales et temporales des motifs. Nous démontrons l'existence de "Travelling waves" par les outils d'analyse fonctionnelle en généralisant la méthode développée par S. Ahmad. Une étude mathématique similaire est menée dans le cadre d'une chaîne alimentaire de trois espèces constituée d'une proie, d'un prédateur et d'un super-prédateur. Le dernier chapitre de cette thèse est consacré à la construction et l'étude d'un modèle mathématique de type réaction-diffusion de la thérapie génétique du cancer. Le modèle prend en considération à la fois la dynamique de la population des cellules cancéreuses, des virus réplicatifs et de la réponse immunitaire qui reconnait les antigènes viraux dans les cellules cancéreuses. Nous établissons les conditions de stabilité de l'état d'équilibre endémique et celui correspondant à l'élimination de la tumeur. Si la tumeur ne peut pas être complétement guérie, nous déterminons les conditions d'une thérapie optimale et estimons par simulation le temps de survie du patient.

As faltas podem ocorrer em diversos componentes de um sistema elétrico de potência, dentre os quais as linhas de transmissão se destacam como elementos susceptíveis. Devido a suas dimensões físicas, ambiente de operação e interconexão com outros sistemas, a localização exata de uma falta em uma linha de transmissão não é trivial dificultando a tarefa das equipes de manutenção para recomposição do fornecimento de energia. Neste contexto, este trabalho apresenta uma metodologia computacional desenvolvida em ambiente MATLAB® para diagnóstico de faltas em linhas de transmissão baseada na análise de transitórios de alta freqüência nas tensões do sistema. A abordagem desenvolvida neste trabalho inicialmente utiliza a equação de Park para achar a tensão zero, direta e em quadratura e posteriormente detectar a ocorrência da falta através do coeficiente de diferenças. Posteriormente, a teoria de ondas viajantes associada à transformada wavelet contínua são utilizadas objetivando determinar a frequência característica do sinal transitório, a qual é relacionada com o local da falta. Para validar o algoritmo de localização proposto foram realizadas diversas simulações de faltas com o programa ATP (Alternative Transient Program) utilizando dados de uma linha aérea real. A efetividade da abordagem proposta foi avaliada considerando a resistência de falta, o ângulo de incidência da falta e a impedância de curto-circuito do sistema. Os resultados promissores demonstraram a aplicabilidade da formulação proposta para localização faltas em linhas de transmissão aéreas utilizando medições oscilográficas digitais de apenas um terminal. / Faults can occur in several components of an electric power system, among which the transmission line stands out. Due to its physical dimensions, operating environment and interconnection systems, the exact transmission line fault location is non-trivial hindering maintenance crew system restoration. In this context, this work presents a computational methodology developed in MATLAB® environment for fault diagnosis on transmission lines based on voltage high frequency transient analysis. Thus, the approach adopted in this work use firstly the Park equations to find the zero, direct and quadrature voltage and then detect the fault occurrence through the difference coefficient. After, the traveling wave theory associated with continuous wavelet transform application is used for determining the characteristic frequency of the transient signal, which is correlated with the fault location. To validate the proposed fault location algorithm, several fault simulations were executed using ATP software using real overhead line data. The proposed fault location scheme performance was evaluated considering the fault resistance, inception fault angle and short-circuit impedance of the system. The promising results demonstrate the applicability of the proposed formulation for fault location in overhead transmission lines, using one terminal digital oscillographic measurement.