Awareness of biological invasions is becoming widespread and several mathematical tools have been used to study this problem. Interacting particle systems, specifically the contact process, have been used to study systems with invasion/infection type dynamics. The Propp-Wilson algorithm is a method for exact sampling from the stationary distribution of an ergodic monotone Markov chain using a method called coupling from the past. The contact process is monotone so we can sample exactly from the stationary distribution of a modified finite grid version using the Propp-Wilson algorithm. In order to study an invasion, we would like to include at least 2 species; however, monotonicity is not well defined for contact processes with more than 2 particle types. Here we develop a general theory of monotonicity for interaction map particle systems, which are interacting particle systems with contact process type dynamics. This allows us to create monotone models with any number of particles and to use the Propp-Wilson algorithm for not only sampling from the stationary distribution, but analyzing the path of invasion leading to equilibrium. Virtual particle invasion models that fall into this new theoretical framework, which we develop here, present a wide range of biological dynamics. Computer simulation of the stochastic system and mean field analysis are two powerful tools that we use for analyzing these types of models. Statistics gathered along the path to invasion help us understand the spatial dynamics of this ecological process and what the stationary behavior looks like. This allows us to understand when the invasion is successful or if coexistence occurs and how these depend on the transition rates and interactions within the process.

Anaerobic pretreatment and aerobic post-treatment of municipal wastewater is being used more frequently. Recent investigations in this field using an AFBR/aeration chamber combination demonstrated the technical feasibility of this process. The investigation presented herein describes the use of a combined UASB/aeration chamber system for the treatment of municipal wastewater and attempts to demonstrate the technical feasibility of using the UASB process as both a pretreatment unit and a waste activated sludge digestion system. The results indicate that the UASB reactor has a TSS removal efficiency of about 37%. Of the solids removed by the unit, 33% were degraded by the action of microorganisms, and 4.6% were accumulated in the reactor. The results also show that accumulation of solids in the UASB reactor took place in the upper zone of the sludge bed.

Part I: A remarkable feature of life on Earth is that despite the apparent observed diversity, the underlying chemistry that powers it is highly conserved. From the level of the nucleobases, through the amino acids and proteins they encode, to the metabolic pathways of chemical reactions catalyzed by these proteins, biology often utilizes identical solutions in vastly disparate organisms. This universality is intriguing as it raises the question of whether these recurring features exist because they represent some truly optimal solution to a given problem in biology, or whether they simply exist by chance, having arisen very early in life's history. In this project we consider the universality of metabolism { the set of chemical reactions providing the energy and building blocks for cells to grow and divide. We develop an algorithm to construct the complete network of all possible biochemically feasible compounds and reactions, including many that could have been utilized by life but never were. Using this network we investigate the most highly conserved piece of metabolism in all of biology, the trunk pathway of glycolysis. We design a method which allows a comparison between the large number of alternatives to this pathway and which takes into account both thermodynamic and biophysical constraints, finding evidence that the existing version of this pathway produces optimal metabolic fluxes under physiologically relevant intracellular conditions. We then extend our method to include an evolutionary simulation so as to more fully explore the biochemical space. Part II: Studies of population dynamics have a long history and have been used to understand the properties of complex networks of ecological interactions, extinction events, biological diversity and the transmission of infectious disease. One aspect of these models that is known to be of great importance, but one which nonetheless is often neglected, is spatial structure. Various classes of models have been proposed with each allowing different insights into the role space plays. Here we use a lattice-based approach. Motivated by gene transfer and parasite dynamics, we extend the well-studied contact process of statistical physics to include multiple levels. Doing so generates a simple model which captures in a general way the most important features of such biological systems: spatial structure and the inclusion of both vertical as well as horizontal transmission. We show that spatial structure can produce a qualitatively new effect: a coupling between the dynamics of the infection and of the underlying host population, even when the infection does not affect the fitness of the host. Extending the model to an arbitrary number of levels, we find a transition between regimes where both a finite and infinite number of parasite levels are sustainable, and conjecture that this transition is related to the roughening transition of related surface growth models.

This thesis is concerned with certain properties of stochastic growth models. A stochastic growth model is a model of infection spread, through a population of individuals, that incorporates an element of randomness. The models we consider are variations on the contact process, the simplest stochastic growth model with a recurrent infection. Three main examples are considered. The first example is a version of the contact process on the complete graph that incorporates dynamic monogamous partnerships. To our knowledge, this is the first rigorous study of a stochastic spatial model of infection spread that incorporates some form of social dynamics. The second example is a non-monotonic variation on the contact process, taking place on the one-dimensional lattice, in which there is a random incubation time for the infection. Some techniques exist for studying non-monotonic particle systems, specifically models of competing populations [38] [12]. However, ours is the first rigorous study of a non-monotonic stochastic spatial model of infection spread. The third example is an additive two-stage contact process, together with a general duality theory for multi-type additive growth models. The two-stage contact process is first introduced in \cite{krone}, and several open questions are posed, most of which we have answered. There are many examples of additive growth models in the literature [26] [16] [29] [49], and most include a proof of existence of a dual process, although up to this point no general duality theory existed. In each case there are three main goals. The first is to identify a phase transition with a sharp threshold or ``critical value'' of the transmission rate, or a critical surface if there are multiple parameters. The second is to characterize either the invariant measures if the population is infinite, or to characterize the metastable behaviour and the time to extinction of the disease, if the population is finite. The final goal is to determine the asymptotic behaviour of the model, in terms of the invariant measures or the metastable states. In every model considered, we identify the phase transition. In the first and third examples we show the threshold is sharp, and in the first example we calculate the critical value as a rational function of the parameters. In the second example we cannot establish sharpness due to the lack of monotonicity. However, we show there is a phase transition within a range of transmission rates that is uniformly bounded away from zero and infinity, with respect to the incubation time. For the partnership model, we show that below the critical value, the disease dies out within C log N time for some C>0, where N is the population size. Moreover we show that above the critical value, there is a unique metastable proportion of infectious individuals that persists for at least e^{\gamma N}$ time for some $\gamma>0$. For the incubation time model, we use a block construction, with a carefully chosen good event to circumvent the lack of monotonicity, in order to show the existence of a phase transition. This technique also guarantees the existence of a non-trivial invariant measure. Due to the lack of additivity, the identification of all the invariant measures is not feasible. However, we are able to show the following is true. By rescaling time so that the average incubation period is constant, we obtain a limiting process as the incubation time tends to infinity, with a sharp phase transition and a well-defined critical value. We can then show that as the incubation time approaches infinity (or zero), the location of the phase transition in the original model converges to the critical value of the limiting process (respectively, the contact process). For the two-stage contact process, we can show that there are at most two extremal invariant measures: the trivial one, and a non-trivial upper invariant measure that appears above the critical value. This is achieved using known techniques for the contact process. We can show complete convergence, from any initial configuration, to a combination of these measures that is given by the survival probability. This, and some additional results, are in response to the questions posed by Krone in his original paper \cite{krone} on the model. We then generalize these ideas to develop a theory of additive growth models. In particular, we show that any additive growth model, having any number of types and interactions, will always have a dual process that is also an additive growth model. Under the additional technical condition that the model preserves positive correlations, we can then harness existing techniques to conclude existence of at most two extremal invariant measures, as well as complete convergence. / Graduate

O estudo da dinâmica de populações vem adquirindo grande importância atualmente, por suas aplicações nas mais diversas áreas do conhecimento, como a biologia evolutiva, ecologia, economia e computação, entre outras. O uso de redes, ou autômatos celulares, para modelar dinâmicas populacionais é um recurso frequentemente utilizado por sua simplicidade no tratamento de problemas com alto grau de complexidade. Neste trabalho utilizamos autômatos celulares para simular dinâmicas populacionais onde analisamos transições de fases longe do equilíbrio em modelos de replicação em uma e duas dimensões, classificando-as de acordo com suas classes de universalidade. Também utilizamos redes para estudar as possíveis origens dos ciclos primos presentes nas cigarras do gênero Magicicada que habitam a América do Norte, mostrando que a predação não é necessária para o surgimento deste comportamento. / The study of population dynamics becomes even more important nowadays because of its applications in a wide range of subjects, such as evolutive biology, ecology, economics and computational sciences, among many others. The use of networks, as well as cellular automata, to simulate populational dynamics is an ordinary tool because of its simplicity in the treatement of very complicated problems. In this work we use cellular automata to simulate populational dynamics where non equilibrium phase transitions in replicator models in one and two dimensions are analyzed and characterized by their universality classes. We also use cellular automata to study the possible origins of prime number cycling present in northern american Magicicada, showing that it is possible to generate prime number year life cycles whithout any predation effects.

O estudo da dinâmica de populações vem adquirindo grande importância atualmente, por suas aplicações nas mais diversas áreas do conhecimento, como a biologia evolutiva, ecologia, economia e computação, entre outras. O uso de redes, ou autômatos celulares, para modelar dinâmicas populacionais é um recurso frequentemente utilizado por sua simplicidade no tratamento de problemas com alto grau de complexidade. Neste trabalho utilizamos autômatos celulares para simular dinâmicas populacionais onde analisamos transições de fases longe do equilíbrio em modelos de replicação em uma e duas dimensões, classificando-as de acordo com suas classes de universalidade. Também utilizamos redes para estudar as possíveis origens dos ciclos primos presentes nas cigarras do gênero Magicicada que habitam a América do Norte, mostrando que a predação não é necessária para o surgimento deste comportamento. / The study of population dynamics becomes even more important nowadays because of its applications in a wide range of subjects, such as evolutive biology, ecology, economics and computational sciences, among many others. The use of networks, as well as cellular automata, to simulate populational dynamics is an ordinary tool because of its simplicity in the treatement of very complicated problems. In this work we use cellular automata to simulate populational dynamics where non equilibrium phase transitions in replicator models in one and two dimensions are analyzed and characterized by their universality classes. We also use cellular automata to study the possible origins of prime number cycling present in northern american Magicicada, showing that it is possible to generate prime number year life cycles whithout any predation effects.

Neste trabalho estudamos alguns modelos estocásticos reversíveis e irreversíveis por meio de varias técnicas que incluem expansões em serie, simulações numéricas e métodos analíticos. Primeiramente, construímos uma expansão supercrítica para a densidade do processo de contato em uma dimensão, que fornece a taxa crítica e o expoente crítico pelo método de aproximantes de Pad´e. Depois, examinamos um modelo de pilha de areia com restrição de altura assimétrico que apresenta fluxo de partículas não-nulo no estado estacionário e suas propriedades criticas são determinadas em função do parâmetro de assimetria p. Finalmente, estudamos de forma analítica o modelo de Glauber linear, que é idêntico, em uma dimensão, ao modelo de Glauber. Em qualquer numero de dimensões, e possível obter uma expressão para a susceptibilidade do modelo de Glauber linear a partir da expansao em s´erie perturbativa, cujos coeficientes são determinados em todas as ordens. Também discutimos como generalizar esse método para obter expansões em série para o modelo de Glauber em duas dimensões. / In this work we study some reversible and irreversible stochastic models using various techniques that include series expansions, numerical simulations and analytical methods. Firstly we write a supercritical series expansion of the particle density of the one-dimensional contact process, which gives us the critical annihilation rate and critical exponent ¯ after using the Pad´e approximants method. Secondly we examine the assimetric height restricted sandpile model, which presents a non-zero particle flux at the stationary active state and its critical properties are determined as a function of the assimetry parameter p. Finally we study analitically the linear Glauber model, which is identical in one dimension to the Glauber model. It is possible in any dimension to obtain an expression of the susceptibility of the linear Glauber model from a perturbative series expansion in which the coefficients can be determined at all orders. We also discuss how to generalize the method in order to obtain a series expansion for the Glauber model in two dimensions.

Cette thèse s'inscrit dans l'étude des systèmes de particules en interaction et plus précisément dans celle des modèles de croissance aléatoire qui représentent une quantité qui grandit au cours du temps et s'étend sur un réseau. Ce type de processus apparaît naturellement lorsqu'on s'intéresse à l'évolution d'une population ou à la propagation d'une épidémie. L'un de ces modèles est celui du processus de contact introduit par T.E. Harris en 1974. Il compte parmi les plus simples à représenter une transition de phase ce qui a rendu son étude passionnante.Le processus de contact standard sur le réseau Zd est maintenant relativement bien connu sous toutes ses phases et on étudie maintenant des variantes naturelles de ce processus comme celle à laquelle nous nous intéressons ici : le processus de contact standard sur le graphe Booléen qui est un graphe aléatoire dans Rd. Notre travail a été motivé notamment par le résultat suivant de L. Ménard er A. Singh : sur ce réseau aléatoire, le processus de contact admet ube transition de phase non-triviale. C'est le premier exemple de graphe à degré non-borné sur lequel la transition de phase du processus de contact n'est pas triviale. Nous commençons notre travail par une étude du modèle Booléen surcritique pour dégager des propriétés de régularité à grande échelle. Ces propriétés nous permettent ensuite d'adapter les démarches usuelles de l'étude du processus de contact sur les réseaux déterministes au cadre aléatoire Booléen. Dans notre résultat principal, nous montrons un théorème de forme asymptotique déterministe pour notre modèle. En fait, il apparaît que les propriétés de régularité à grandes échelles mentionnées ci-dessus sont suffisantes pour montrer un théorème de forme asymptotique sur d'autres graphes aléatoires / This thesis is a contribution to the mathematical study of interacting particle systems, and more precisely of random growth models representing a spreading shape over time in a lattice. These processes occur when one is interested in the evolution of a population or the spread of an epidemic. One of those models is the contact process introduced by Harris in 1974 with the goal of representing this specific spread. It is one of the simplest interacting particle systems that exhibits a critical phenomenon and today, in the cubic lattice, its behavior is well-known on each phase. Here, we study the standard contact process the Boolean graph which is a random graph in Rd. Our work in particular was motivated by the following result of L. M´enard and A. Singh: on this random network, the contact process admits a non-trivial phase transition. This is the first example of a non-bounded degree graph on which the phase transition of the contact process is non-trivial. We begin our work with a study of the supercritical Boolean model to find large scale regularity properties that allow us to adapt the usual approaches of the study of the contact process on the deterministic networks to the Boolean random framework. In our main result, we show that our model satisfies a deterministic asymptotic shape theorem. In fact, it appears that the large scale regularity properties mentioned above are sufficient to obtain an asymptotic shape theorem on other random graphs

Neste trabalho estudamos alguns modelos estocásticos reversíveis e irreversíveis por meio de varias técnicas que incluem expansões em serie, simulações numéricas e métodos analíticos. Primeiramente, construímos uma expansão supercrítica para a densidade do processo de contato em uma dimensão, que fornece a taxa crítica e o expoente crítico pelo método de aproximantes de Pad´e. Depois, examinamos um modelo de pilha de areia com restrição de altura assimétrico que apresenta fluxo de partículas não-nulo no estado estacionário e suas propriedades criticas são determinadas em função do parâmetro de assimetria p. Finalmente, estudamos de forma analítica o modelo de Glauber linear, que é idêntico, em uma dimensão, ao modelo de Glauber. Em qualquer numero de dimensões, e possível obter uma expressão para a susceptibilidade do modelo de Glauber linear a partir da expansao em s´erie perturbativa, cujos coeficientes são determinados em todas as ordens. Também discutimos como generalizar esse método para obter expansões em série para o modelo de Glauber em duas dimensões. / In this work we study some reversible and irreversible stochastic models using various techniques that include series expansions, numerical simulations and analytical methods. Firstly we write a supercritical series expansion of the particle density of the one-dimensional contact process, which gives us the critical annihilation rate and critical exponent ¯ after using the Pad´e approximants method. Secondly we examine the assimetric height restricted sandpile model, which presents a non-zero particle flux at the stationary active state and its critical properties are determined as a function of the assimetry parameter p. Finally we study analitically the linear Glauber model, which is identical in one dimension to the Glauber model. It is possible in any dimension to obtain an expression of the susceptibility of the linear Glauber model from a perturbative series expansion in which the coefficients can be determined at all orders. We also discuss how to generalize the method in order to obtain a series expansion for the Glauber model in two dimensions.

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