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Modelos de colonização e colapso / Colonization and collapse modelsRezende, Bruna Luiza de Faria 31 August 2017 (has links)
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Previous issue date: 2017-08-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work a basic immigration process was investigated which starts with a
single colony with a single individual at the origin of a homogeneous tree with the
other empty vertices. The process colonies are established at the vertices of the graph
and each one grows during a random time, according to a process of general counting
until a disaster that annihilates part of the population occurs. After the collapse a
random amount of individuals survives and attempts to establish, in a independent
manner, new colonies in a neighboring vertices. After a time these formed colonies
also suffer catastrophes and the process is repeated. It is important to emphasize
that the time until the disaster of each colony is independent of the others. Here
this general process was studied under two methods, Poisson growth with geometric
catastrophe and Yule growth with binomial catastrophe. That is, in each colony
the population grows following a Poisson (or Yule), process during a random time,
considered here exponential, and soon after that time its size is reduced according to
the geometric (or binomial) law. Conditions were analyzed in the set of parameters
so that these processes survived and limits were established that were relevant for
the probability of survival, the number of colonies generated during the process and
the range of the colonies in relation to the initial point. / Neste trabalho foi investigado um processo básico de imigração o qual é iniciado
com uma única colônia com um único indivíduo na origem de uma árvore homogênea
com os demais vértices vazios. As colônias do processo se estabelecem nos vértices
do grafo e cada uma cresce durante um tempo aleatório, de acordo com um processo
de contagem geral até ocorrer um desastre que aniquila parte da população. Após
o colapso uma quantidade aleatória de indivíduos sobrevive e tenta estabelecer, de
forma independente, novas colônias em vértices vizinhos. Depois de um tempo essas
colônias formadas também sofrem catástrofes e o processo se repete. É importante
enfatizar que o tempo até o desastre de cada colônia independe do das demais. Aqui
esse processo geral foi estudado sujeito a dois métodos, crescimento de Poisson com
catástrofe geométrica e crescimento de Yule com catástrofe binomial. Ou seja, em
cada colônia a população cresce seguindo um processo de Poisson (ou Yule), durante
um tempo aleatório, considerado aqui exponencial, e logo após esse tempo seu
tamanho é reduzido de acordo com a lei geométrica (ou binomial). Foram analisadas
condições no conjunto de parâmetros para que esses processos sobrevivam e foram
estabelecidos limites relevantes para a probabilidade de sobrevivência, o número de
colônias geradas durante o processo e o alcance das colônias em relação ao ponto
inicial.
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Processus de branchements et graphe d'Erdős-Rényi / Branching processes and Erdős-Rényi graphCorre, Pierre-Antoine 29 November 2017 (has links)
Le fil conducteur de cette thèse, composée de trois parties, est la notion de branchement.Le premier chapitre est consacré à l'arbre de Yule et à l'arbre binaire de recherche. Nous obtenons des résultats d'oscillations asymptotiques de l'espérance, de la variance et de la distribution de la hauteur de ces arbres, confirmant ainsi une conjecture de Drmota. Par ailleurs, l'arbre de Yule pouvant être vu comme une marche aléatoire branchante évoluant sur un réseau, nos résultats permettent de mieux comprendre ce genre de processus.Dans le second chapitre, nous étudions le nombre de particules tuées en 0 d'un mouvement brownien branchant avec dérive surcritique conditionné à s'éteindre. Nous ferons enfin apparaître une nouvelle phase de transition pour la queue de distribution de ces variables.L'objet du dernier chapitre est le graphe d'Erdős–Rényi dans le cas critique : $G(n,1/n)$. En introduisant un couplage et un changement d'échelle, nous montrerons que, lorsque $n$ augmente les composantes de ce graphe évoluent asymptotiquement selon un processus de coalescence-fragmentation qui agit sur des graphes réels. La partie coalescence sera de type multiplicatif et les fragmentations se produiront selon un processus ponctuel de Poisson sur ces objets. / This thesis is composed by three chapters and its main theme is branching processes.The first chapter is devoted to the study of the Yule tree and the binary search tree. We obtain oscillation results on the expectation, the variance and the distribution of the height of these trees and confirm a Drmota's conjecture. Moreover, the Yule tree can be seen as a particular instance of lattice branching random walk, our results thus allow a better understanding of these processes.In the second chapter, we study the number of particles killed at 0 for a Brownian motion with supercritical drift conditioned to extinction. We finally highlight a new phase transition in terms of the drift for the tail of the distributions of these variables.The main object of the last chapter is the Erdős–Rényi graph in the critical case: $G(n,1/n)$. By using coupling and scaling, we show that, when $n$ grows, the scaling process is asymptotically a coalescence-fragmentation process which acts on real graphs. The coalescent part is of multiplicative type and the fragmentations happen according a certain Poisson point process.
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