The goal of this paper is to explain the mathematics involved in modeling biochemical oscillations. We first discuss several important biochemical concepts fundamental to the construction of descriptive mathematical models. We review the basic theory of differential equations and stability analysis as it relates to two-variable models exhibiting oscillatory behavior. The importance of the Hopf Bifurcation will be discussed in detail for the central role it plays in limit cycle behavior and instability. Once we have exposed the necessary mathematical framework, we consider several specific models of biochemical oscillators in three or more variables. This will include a detailed analysis of Goodwin's equations and their modification first studied by Painter. Additionally, we consider the consequences of introducing both distributed and discrete time delay into Goodwin's model. We will show that the presence of distributed time lag modifies Goodwin's model in no significant way. The final section of the paper will discuss discrete time lag in the context of a minimal model of the circadian rhythm. In the main, this paper will address mathematical, as opposed to biochemical, issues. Nevertheless, the significance of the mathematics to the biochemistry will be considered throughout. / Master of Science

Résumé : La principale caractéristique des marchés OTC (Over-The-Counter) est l’absence d’un mécanisme de négociation centralisée (comme des ventes aux enchères, des spécialistes ou des limit-order books). Les acheteurs et les vendeurs sont donc souvent dans l'ignorance des prix actuellement disponibles auprès d'autres contreparties potentielles et ont une connaissance limitée de l’amplitude des transactions récemment négociées ailleurs sur le marché. C'est la raison pour laquelle les marchés OTC sont qualifiés de relativement opaques et nommés «Dark Markets» par Duffie (2012) dans sa récente monographie afin de refléter le fait que les investisseurs sont en quelque sorte dans le noir au sujet du meilleur prix disponible et de la personne à contacter pour faire la meilleure transaction. Dans ce travail, nous sommes particulièrement intéressés à l’évolution temporelle de la transmission de l’information au cours des séances de négociation. Plus précisément, nous cherchons à établir la stabilité asymptotique de la dynamique de partage de l'information au sein d’une large population d’investisseurs caractérisés par la fréquence/intensité des rencontres entre investisseurs. L’effort optimal déployé par un agent en recherche d’informations dépend de son niveau actuel d'information et de la distribution transversale des efforts de recherche des autres agents. Dans le cadre défini par Duffie-Malamud-Manso (2009), à l’équilibre, les agents recherchent au maximum jusqu'à ce que la qualité de leur information atteigne un certain niveau, déclenchant une nouvelle phase de recherche minimale. Dans le contexte de percolation d'information entre agents, l'information peut être transmise parfaitement ou imparfaitement. La première étude de ce problème de percolation a été faite par Duffie-Manso (2007), puis par Duffie-Giroux-Manso (2010). Dans cette deuxième étude, le cas de la percolation de l'information par des groupes de plus de deux investisseurs a été abordé et résolu. Cette dernière étude a conduit au problème de l'extension des sommes de Wild dans Bélanger-Giroux (2013). D'autre part, dans Duffie-Malamud-Manso (2009), chaque agent est doté de signaux quant à l'issue probable d'une variable aléatoire d'intérêt commun dans l’optique de transmission d’informations dans une large population d'agents. Un tel contexte conduit à des systèmes d'équations non linéaires d’évolution. Leur objectif est d'obtenir une politique d'équilibre déterminée par un ensemble de paramètres d'une politique de cible traduisant le fait que l’effort de recherche qui doit être minimal lorsqu’un agent possède suffisamment d’information. Dans ce travail, nous sommes en mesure d'obtenir l'existence de l’état d’équilibre, même lorsque la fonction d'intensité n'est pas un produit. De plus, nous sommes également en mesure de montrer la stabilité asymptotique pour toute loi initiale par un changement de noyaux. Enfin, nous élargissons les hypothèses de Bélanger-Giroux (2012) pour montrer la stabilité exponentielle par le critère de Routh-Hurwitz pour un autre exemple de système à un nombre fini d’équations. // Abstract : Over-the-counter (OTC) markets have the main characteristic that they do not use a centralized trading mechanism (such as auctions, specialist, or limit-order book) to aggregate bids and offers and to allocate trades. The buyers and sellers have often a limited knowledge of trades recently negotiated elsewhere in the market. They are also negotiating in potential ignorance of the prices currently available from other counterparties. This is the reason why OTC markets are said to be relatively opaque and are qualified as «Dark Markets» by Duffie (2012) in his recent monograph to reflect the fact that investors are somewhat in the dark about the most attractive available deals and about whom to contact. In this work, we are particularly interested in the evolution over time of the distribution across investors of information learned from private trade negotiations. Specifically, we aim to establish the asymptotic stability of equilibrium dynamics of information sharing in a large interaction set. An agent’s optimal current effort to search for information sharing opportunities depends on that agent’s current level of information and on the cross-sectional distribution of information quality and search efforts of other agents. Under the Duffie-Malamud-Manso (2009) framework, in equilibrium, agents search maximally until their information quality reaches a trigger level and then search minimally. In the context of percolation of information between agents, the information can be transmitted directly or indirectly. The first studies of such a problem were made by Duffie-Manso (2007) and then by Duffie-Giroux-Manso (2010). In that second study the case of the percolation of information by groups of more than 2 investors was addressed and solved for a perfect information transmission kernel. That last study has led Bélanger-Giroux (2013) to the problem of extending the Wild sums for a general interacting kernel (not only for the kernel which adds the information). On the other hand, in Duffie-Malamud-Manso (2009), the authors explain that, for the information sharing in a large population, each agent is endowed with signals regarding the likely outcome of a random variable of common concern, like the price of an asset of common interest. Such a setting leads to nonlinear systems of evolution equations. The agents’ goal is to obtain an equilibrium policy specified by a set of parameters of a trigger policy; more specifically the minimal search effort trigger policies. We concentrate our study on those trigger policies in order to provide more intuitive and practical results. Doing so, we are able to obtain the existence of the steady state even when the intensity function is not a product. And in our framework, we are even able to show the asymptotic stability starting with any initial law. This can be done because we are able to show that, by a change of kernels, the systems of ODE’s, which are expressed by a set of kernels (one 1-airy and one 2-airy) are equivalent to systems expressed with a single 2-airy kernel even with a constant intensity equal to one (by a change of time). We show also that starting from any distribution, the solution converges to the limit proportions. Furthermore, we are able to show the exponential stability using the Routh-Hurwitz criterion for an example of a finite system of differential equations. The solution of such a system of equations describes the cross distribution of types in the market.

O objetivo deste trabalho é estudar a ocorrência de mudanças de regime típicas de comportamentos em sistemas complexos, em particular no contexto de sistemas dinâmicos aplicados. Para isso, desenvolvemos um modelo matemático que representa a interação entre uma cultura de ostras utilizadas para consumo humano e os processos de eutrofização e biorremediação do ecossistema que as contém. As interações entre as populações de ostras e do fitoplâncton entre si e com a matéria suspensa, subproduto das relações entre os componentes do meio e seu processo de eutrofização, alteram os níveis de oxigenação e a consequente qualidade da ´agua devido `a realização de maior ou menor quantidade de fotossíntese pelas vegetações mais profundas do meio. Neste trabalho propomos um sistema dinâmico de três variáveis para modelar esse sistema e analisamos seus pontos de equilíbrio usando duas técnicas, método de Quirk-Ruppert e os critérios de Routh-Hurwitz, além de resolvê-lo numericamente para um conjunto de parâmetros realísticos (fenomenológicos) obtidos a partir da literatura especializada. Nossos resultados indicam que o limite de extração diária de ostras que pode ser realizado sem levar a cultura ao colapso gira em torno de 4.8% da população / The objective of this work is to study the occurrence of regime shifts that are typical in the behavior of complex systems, in particular in the context of applied dynamical systems. Accordingly, we have developed a mathematical model that represents the interaction between a culture of oysters used for human consumption and the eutrophication and bioremediation processes of the ecosystem containing the culture. The interactions between the oyster populations and the phytoplankton between themselves and with the suspended matter, that appears as a by-product of the relationship between the components of the medium and its eutrophication process, change the oxygenation levels and the resulting water quality due to the realization of a greater or lesser amount of photosynthesis by the vegetation of the deeper levels. In this paper we propose a dynamical system of three variables to model the system and analyze its points of equilibrium using two techniques, the Quirk-Ruppert method and the Routh-Hurwitz criteria, besides solving the equations numerically for a realistic phenomenological) set of parameters obtained from the literature. Our results indicate that the daily extraction threshold that can be achieved without collapsing the culture of oysters amounts to approximately 4.8% of the total population

O objetivo deste trabalho é estudar a ocorrência de mudanças de regime típicas de comportamentos em sistemas complexos, em particular no contexto de sistemas dinâmicos aplicados. Para isso, desenvolvemos um modelo matemático que representa a interação entre uma cultura de ostras utilizadas para consumo humano e os processos de eutrofização e biorremediação do ecossistema que as contém. As interações entre as populações de ostras e do fitoplâncton entre si e com a matéria suspensa, subproduto das relações entre os componentes do meio e seu processo de eutrofização, alteram os níveis de oxigenação e a consequente qualidade da ´agua devido `a realização de maior ou menor quantidade de fotossíntese pelas vegetações mais profundas do meio. Neste trabalho propomos um sistema dinâmico de três variáveis para modelar esse sistema e analisamos seus pontos de equilíbrio usando duas técnicas, método de Quirk-Ruppert e os critérios de Routh-Hurwitz, além de resolvê-lo numericamente para um conjunto de parâmetros realísticos (fenomenológicos) obtidos a partir da literatura especializada. Nossos resultados indicam que o limite de extração diária de ostras que pode ser realizado sem levar a cultura ao colapso gira em torno de 4.8% da população / The objective of this work is to study the occurrence of regime shifts that are typical in the behavior of complex systems, in particular in the context of applied dynamical systems. Accordingly, we have developed a mathematical model that represents the interaction between a culture of oysters used for human consumption and the eutrophication and bioremediation processes of the ecosystem containing the culture. The interactions between the oyster populations and the phytoplankton between themselves and with the suspended matter, that appears as a by-product of the relationship between the components of the medium and its eutrophication process, change the oxygenation levels and the resulting water quality due to the realization of a greater or lesser amount of photosynthesis by the vegetation of the deeper levels. In this paper we propose a dynamical system of three variables to model the system and analyze its points of equilibrium using two techniques, the Quirk-Ruppert method and the Routh-Hurwitz criteria, besides solving the equations numerically for a realistic phenomenological) set of parameters obtained from the literature. Our results indicate that the daily extraction threshold that can be achieved without collapsing the culture of oysters amounts to approximately 4.8% of the total population