Made available in DSpace on 2016-12-12T20:15:49Z (GMT). No. of bitstreams: 1 Cristiane Stegemann.pdf: 4817823 bytes, checksum: f859f3883a8981d6a95d5baba3847fc3 (MD5) Previous issue date: 2012-07-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / One of the tumor growth model is formed by a three-dimensional continuous-time dynamical system, modeled by a set of three autonomous, first-order ordinary differential equations. Mathematical models for tumor growth are used as mechanisms to better understand this disease, find patterns for identification through simulations of the spatial distribution of tumors, or even analysis of interactions of cell populations in order to predict their future behavior. In this work, we introduce some systems that model population growth, which substantiate the choice of the equations of growth of tumors that will later be used in computer simulations. From the analytical point of view, one can determine all equilibrium points of the system and for one of them to study its stability. For to the latter task, we will use the eigenvalues of the Jacobian matrix. The numerical results were obtained by the study of parameter spaces and bifurcation diagrams. The parameter spaces were constructed from the change in a couple of parameters and by calculating a third magnitude, which in this work will be the period and the Lyapunov exponent. These results indicate the existence of specific regions in the parameter space where periodic structures were arranged in a period-adding bifurcation cascade. It is shown that, in the innermost region of the periodic structures, it is possible to visualize the superestable line. Finally, for certain parameter values, the periodic structures are presented spirally arranged, although no law of formation has been found. / Um dos modelos de crescimento de tumores é formado por um sistema dinâmico tridimensional a tempo contínuo, modelado por um conjunto de três equações diferenciais ordinárias de primeira ordem autônomas. Modelos matemáticos para crescimento de tumores são utilizados como mecanismos para entender melhor esta doença, encontrar padrões para sua identificação através de simulações da distribuição espacial de tumores, ou mesmo análises de interações de populações celulares com o intuito de predizer seu comportamento futuro. Neste trabalho, serão apresentados alguns sistemas que modelam crescimento populacional, o que fundamentará a escolha das equações de crescimento de tumores que, posteriormente, serão utilizadas nas simulações computacionais. Do ponto de vista analítico, pode-se determinar todos os pontos de equilíbrio do sistema e, para um deles, estudar sua estabilidade. Para esta última tarefa, serão utilizados os autovalores da matriz Jacobiana. Os resultados numéricos foram obtidos via estudo de espaços de parâmetros e diagramas de bifurcação. Os espaços de parâmetros são construídos a partir da variação de um par de parâmetros e do cálculo de uma terceira grandeza, que neste trabalho, serão o período e o expoente de Lyapunov. Tais resultados indicam a existência de regiões específicas no espaço de parâmetros em que a estruturas periódicas são arranjadas em uma cascata de bifurcação por adição de período. Será mostrado que, na região mais interna das estruturas periódicas, é possível visualizar a linha de superestabilidade. Por fim, para determinados valores dos parâmetros, as estruturas periódicas se apresentam dispostas em espiral, embora nenhuma lei de formação tenha sido encontrada.

We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

L'objet de cette thèse est d'étudier le comportement en temps long de solutions d'équations d'évolution non locales ainsi que la limite singulière d'équations et de systèmes d'équations aux dérivées partielles, où intervient un petit paramètre epsilon. Au Chapitre 1, nous considérons une équation de réaction-diffusion non locale avec conservation au cours du temps de l'intégrale en espace de la solution; cette équation a été initialement proposée par Rubinstein et Sternberg pour modéliser la séparation de phase dans un mélange binaire. Le problème de Neumann associé possède une fonctionnelle de Lyapunov, c'est-à-dire une fonctionnelle qui décroit selon les orbites. Après avoir prouvé que la solution est confinée dans une région invariante, nous étudions son comportement en temps long. Nous nous appuyons sur une inégalité de Lojasiewicz pour montrer qu'elle converge vers une solution stationnaire quand t tend vers l'infini. Nous évaluons également le taux de la convergence et calculons précisément la solution stationnaire limite en dimension un d'espace. Le Chapitre 2 est consacré à l'étude de l'équation différentielle non locale que l'on obtient en négligeant le terme de diffusion dans l'équation d'Allen-Cahn non locale étudiée au Chapitre 1. Sans le terme de diffusion, la solution ne peut pas être plus régulière que la fonction initiale. C'est la raison pour laquelle on ne peut pas appliquer la méthode du Chapitre 1 pour l'étude du comportement en temps long de la solution. Nous présentons une nouvelle méthode basée sur la théorie des réarrangements et sur l'étude du profil de la solution. Nous montrons que la solution est stable pour les temps grands et présentons une caractérisation détaillée de sa limite asymptotique quand t tend vers l'infini. Plus précisément, la fonction limite est une fonction en escalier, qui prend au plus deux valeurs, qui coïncident avec les points stables d'une équation différentielle associée. Nous montrons aussi par un contre-exemple non trivial que, quand une hypothèse sur la fonction initiale n'est pas satisfaite, la fonction limite peut prendre trois valeurs, qui correspondent aux points instable et stables de l'équation différentielle associée. Nous étudions au Chapitre 3 une équation différentielle ordinaire non locale qui a éte proposée par M. Nagayama. Une difficulté essentielle est que le dénominateur dans le terme de réaction non local peut s'annuler. Nous appliquons un théorème de point fixe lié a une application contractante pour démontrer que le problème à valeur initiale correspondant possède une solution unique qui reste connée dans un ensemble invariant. Ce problème possède une fonctionnelle de Lyapunov, qui est un ingrédient essentiel pour démontrer que la solution converge vers une solution stationnaire constante par morceaux quand t tend vers l'infini. Au Chapitre 4, nous considérons un modèle d'interface diffuse pour la croissance de tumeurs, où intervient une équation d'ordre quatre de type Cahn Hilliard. Après avoir introduit un modèle de champ de phase associé, on étudie formellement la limite singulière de la solution quand le coefficient du terme de réaction tend vers l'infini. Plus précisément, nous montrons que la solution converge vers la solution d'un problème à frontière libre. AMS subject classifications. 35K57, 35K50, 35K20, 35R35, 35R37, 35B40, 35B25. / The aim of this thesis is to study the large time behavior of solutions of nonlocal evolution equations and to also study the singular limit of equations and systems of parabolic partial differential equations involving a small parameter epsilon. In Chapter 1, we consider a nonlocal reaction-diffusion equation with mass conservation, which was originally proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. The corresponding Neumann problem possesses a Lyapunov functional, namely a functional which decreases in time along solution orbits. After having proved that the solution is conned in an invariant region, we study its large time behavior and apply a Lojasiewicz inequality to show that it converges to a stationary solution as t tends to infinity. We also evaluate the rate of convergence and precisely compute the limiting stationary solution in one space dimension. Chapter 2 is devoted to the study of a nonlocal evolution equation which one obtains by neglecting the diffusion term in the nonlocal Allen-Cahn equation studied in Chapter 1. Without the diffusion term, the solution can not be expected to be more regular than the initial function. Moreover, because of the absence of the diusion term, the method of Chapter 1 can not be applied to study the large time behavior of the solution. We present a new method based up on rearrangement theory and the study of the solution profile. We show that the solution stabilizes for large times and give a detailed characterization of its asymptotic limit as t tends to infinity. More precisely, it turns out that the limiting function is a step function, which takes at most two values, which are stable points of a corresponding ordinary dierential equation. We also show by means of a nontrivial counterexample that, when a certain hypothesis on the initial function does not hold, the limiting function may take three values. One of them is the unstable point and the two others are the stable points of the ordinary dierential equation. We study in Chapter 3 a nonlocal ordinary dierential equation which has been proposed by M. Nagayama. The nonlocal term involves a denominator which may vanish. We apply a contraction fixed point theorem to prove the existence of a unique solution which stays confined in an invariant region. We also show that the corresponding initial value problem possesses a Lyapunov functional and prove that the solution stabilizes for large times to a step function, which takes at most two values. In Chapter 4, we consider a diffuse-interface tumor-growth model which involves a fourth order Cahn-Hilliard type equation. Introducing a related phase-field model, we formally study the singular limit of the solution as the reaction coecient tends to infinity. More precisely, we show that the solution converges to the solution of a moving boundary problem. AMS subject classifications. 35K57, 35K50, 35K20, 35R35, 35R37, 35B40, 35B25.