Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2008. / Submitted by Jaqueline Oliveira (jaqueoliveiram@gmail.com) on 2008-12-15T15:59:27Z No. of bitstreams: 1 DISSERTACAO_2008_KelemGomesLourenco.pdf: 1422115 bytes, checksum: 20a32fbf536b3c9518ef0a8601e1d847 (MD5) / Approved for entry into archive by Georgia Fernandes(georgia@bce.unb.br) on 2009-02-18T17:36:21Z (GMT) No. of bitstreams: 1 DISSERTACAO_2008_KelemGomesLourenco.pdf: 1422115 bytes, checksum: 20a32fbf536b3c9518ef0a8601e1d847 (MD5) / Made available in DSpace on 2009-02-18T17:36:21Z (GMT). No. of bitstreams: 1 DISSERTACAO_2008_KelemGomesLourenco.pdf: 1422115 bytes, checksum: 20a32fbf536b3c9518ef0a8601e1d847 (MD5) / Neste trabalho analisamos o sistema de equações differenciais com duas presas e um predador do tipo Lotka-Volterra, com e sem colheita. Inicialmente estudamos a estabilidade local e global dos pontos de equilíbrio no primeiro modelo. Posteriormente, no segundo modelo, estudamos o coeficiente de estabilidade das órbitas periódicas, através da forma normal e estimativas numéricas. Através do recurso Maple 11, verificamos o comportamento das soluções e o surgimento das órbitas periódicas. ________________________________________________________________________________________ ABSTRACT / In this work we analyzed the Lotka-Volterra system of diferential equations with two preys and a predator, with and without harvesting. Initially we studied the local and global stability of the points of equilibrium in the first model. Later, in the second model, we studied the coefficient of stability of the periodic orbits, by using normal form and numerical estimatives. By using Maple 11, we verified the behavior of the solutions and the appearance of the periodic orbits.

Esta tese é composta por três ensaios. No primeiro, mostramos em nosso modelo básico que economias formadas exclusivamente por produtores e parasitas podem cair em armadilhas de pobreza, desde que ambos grupos se comportem de acordo com a dinâmica de Lotka-Volterra. Contudo, a introdução de um limite para o crescimento do produto e de expectativas por parte dos agentes exclui o resultado de armadilha em seus múltiplos equilíbrios. Nossa conclusão, entretanto, é similar para ambos modelos estudados: melhora na proteção aos direitos de propriedade por parte do Estado pode fazer com que a armadilha de pobreza seja superada, no modelo básico; e afetar a estabilidade dos equilíbrios, no modificado, fazendo com que resultados econômicos com maior produto tornem-se estáveis. No segundo ensaio, obtemos condições sob as quais a função perda do banco central é estritamente convexa em quatro estados distintos da economia: economia aquecida, em recessão, inação alta e produto alto. Encontramos, ainda, que quando in- ação e produto são funções lineares do instrumento de política monetária, a convexidade é garantida para qualquer um dos quatro estados citados. Ao estendermos nossa análise a vários instrumentos, encontramos que apenas linearidade já não é mais suficiente para a garantia do formato da função perda. Nossos resultados fornecem, ainda, condições sob as quais existirá dependência entre os instrumentos de política monetária. Por fim, o terceiro ensaio estuda regimes de metas de inação, nos quais agentes podem influenciar a política monetária através das expectativas de mercado reportadas ao banco central. Este, por sua vez, deve formular a política monetária considerando que tal influência pode ser usada em benefício dos próprios agentes. Modelamos essa relação estratégica como um jogo sequencial entre uma instituição financeira representativa e o banco central. Mostramos que quando a autoridade monetária escolhe apenas o nível da taxa de juros, existe um potencial viés inflacionário na economia. Esse viés é superado quando a oferta de moeda torna-se um segundo instrumento de política. Ainda mostramos que penalização de instituições más previsoras também pode ser um mecanismo eficiente de ancoragem de expectativas. / This thesis consists of three essays. In the first one, we show in our basic model that economies consisted exclusively by producers and parasites may fall into poverty traps, assuming that both groups behave according to the dynamics of Lotka-Volterra. However, the introduction of an upper bound on the output growth and expectations for the agents excludes the result of trap in its multiple equilibria. Our conclusion, nevertheless, is similiar for both studied models: improved protection of property rights by the state can mitigate the poverty trap possibility in the basic model, and affect the stability of equilibria in the modified one, making that economic outcomes with higher output become stable. In the second essay, we obtain conditions under which the central bank's loss function is strictly convex in four different states of the economy: booming economy, recession, high inflation and high output. Moreover, we found that when inaction and output are linear functions of monetary policy instruments, convexity is guaranteed for any of the four states mentioned. When we extend our analysis to the case of many instruments, we found that only linearity is not sufficient to guarantee the shape of loss function. Our results also provide conditions under which there exists dependence between instruments of monetary policy. Finally, the third essay studies the ination targeting regimes, in which agents can influence the monetary policy through market expectations reported to the central bank. Monetary authority, in its turn, should formulate the monetary policy considering that this influence may be used for the benefit of agents themselves. We model this strategic relationship as a sequential game between a representative financial institution and the central bank. We show that when the monetary authority chooses only the level of interest rates, there is a potential inflationary bias in the economy. This bias is solved when the money supply becomes a second instrument of policy. In addition, we show that to impose penalty on the worse predictor institutions may also be an efficient anchoring expectations mechanism.

Esta tese é composta por três ensaios. No primeiro, mostramos em nosso modelo básico que economias formadas exclusivamente por produtores e parasitas podem cair em armadilhas de pobreza, desde que ambos grupos se comportem de acordo com a dinâmica de Lotka-Volterra. Contudo, a introdução de um limite para o crescimento do produto e de expectativas por parte dos agentes exclui o resultado de armadilha em seus múltiplos equilíbrios. Nossa conclusão, entretanto, é similar para ambos modelos estudados: melhora na proteção aos direitos de propriedade por parte do Estado pode fazer com que a armadilha de pobreza seja superada, no modelo básico; e afetar a estabilidade dos equilíbrios, no modificado, fazendo com que resultados econômicos com maior produto tornem-se estáveis. No segundo ensaio, obtemos condições sob as quais a função perda do banco central é estritamente convexa em quatro estados distintos da economia: economia aquecida, em recessão, inação alta e produto alto. Encontramos, ainda, que quando in- ação e produto são funções lineares do instrumento de política monetária, a convexidade é garantida para qualquer um dos quatro estados citados. Ao estendermos nossa análise a vários instrumentos, encontramos que apenas linearidade já não é mais suficiente para a garantia do formato da função perda. Nossos resultados fornecem, ainda, condições sob as quais existirá dependência entre os instrumentos de política monetária. Por fim, o terceiro ensaio estuda regimes de metas de inação, nos quais agentes podem influenciar a política monetária através das expectativas de mercado reportadas ao banco central. Este, por sua vez, deve formular a política monetária considerando que tal influência pode ser usada em benefício dos próprios agentes. Modelamos essa relação estratégica como um jogo sequencial entre uma instituição financeira representativa e o banco central. Mostramos que quando a autoridade monetária escolhe apenas o nível da taxa de juros, existe um potencial viés inflacionário na economia. Esse viés é superado quando a oferta de moeda torna-se um segundo instrumento de política. Ainda mostramos que penalização de instituições más previsoras também pode ser um mecanismo eficiente de ancoragem de expectativas. / This thesis consists of three essays. In the first one, we show in our basic model that economies consisted exclusively by producers and parasites may fall into poverty traps, assuming that both groups behave according to the dynamics of Lotka-Volterra. However, the introduction of an upper bound on the output growth and expectations for the agents excludes the result of trap in its multiple equilibria. Our conclusion, nevertheless, is similiar for both studied models: improved protection of property rights by the state can mitigate the poverty trap possibility in the basic model, and affect the stability of equilibria in the modified one, making that economic outcomes with higher output become stable. In the second essay, we obtain conditions under which the central bank's loss function is strictly convex in four different states of the economy: booming economy, recession, high inflation and high output. Moreover, we found that when inaction and output are linear functions of monetary policy instruments, convexity is guaranteed for any of the four states mentioned. When we extend our analysis to the case of many instruments, we found that only linearity is not sufficient to guarantee the shape of loss function. Our results also provide conditions under which there exists dependence between instruments of monetary policy. Finally, the third essay studies the ination targeting regimes, in which agents can influence the monetary policy through market expectations reported to the central bank. Monetary authority, in its turn, should formulate the monetary policy considering that this influence may be used for the benefit of agents themselves. We model this strategic relationship as a sequential game between a representative financial institution and the central bank. We show that when the monetary authority chooses only the level of interest rates, there is a potential inflationary bias in the economy. This bias is solved when the money supply becomes a second instrument of policy. In addition, we show that to impose penalty on the worse predictor institutions may also be an efficient anchoring expectations mechanism.

abstract: There has been important progress in understanding ecological dynamics through the development of the theory of ecological stoichiometry. This fast growing theory provides new constraints and mechanisms that can be formulated into mathematical models. Stoichiometric models incorporate the effects of both food quantity and food quality into a single framework that produce rich dynamics. While the effects of nutrient deficiency on consumer growth are well understood, recent discoveries in ecological stoichiometry suggest that consumer dynamics are not only affected by insufficient food nutrient content (low phosphorus (P): carbon (C) ratio) but also by excess food nutrient content (high P:C). This phenomenon, known as the stoichiometric knife edge, in which animal growth is reduced not only by food with low P content but also by food with high P content, needs to be incorporated into mathematical models. Here we present Lotka-Volterra type models to investigate the growth response of Daphnia to algae of varying P:C ratios. Using a nonsmooth system of two ordinary differential equations (ODEs), we formulate the first model to incorporate the phenomenon of the stoichiometric knife edge. We then extend this stoichiometric model by mechanistically deriving and tracking free P in the environment. This resulting full knife edge model is a nonsmooth system of three ODEs. Bifurcation analysis and numerical simulations of the full model, that explicitly tracks phosphorus, leads to quantitatively different predictions than previous models that neglect to track free nutrients. The full model shows that the grazer population is sensitive to excess nutrient concentrations as a dynamical free nutrient pool induces extreme grazer population density changes. These modeling efforts provide insight on the effects of excess nutrient content on grazer dynamics and deepen our understanding of the effects of stoichiometry on the mechanisms governing population dynamics and the interactions between trophic levels. / Dissertation/Thesis / Ph.D. Applied Mathematics 2014

Esta tese é composta por três ensaios. No primeiro, mostramos em nosso modelo básico que economias formadas exclusivamente por produtores e parasitas podem cair em armadilhas de pobreza, desde que ambos grupos se comportem de acordo com a dinâmica de Lotka-Volterra. Contudo, a introdução de um limite para o crescimento do produto e de expectativas por parte dos agentes exclui o resultado de armadilha em seus múltiplos equilíbrios. Nossa conclusão, entretanto, é similar para ambos modelos estudados: melhora na proteção aos direitos de propriedade por parte do Estado pode fazer com que a armadilha de pobreza seja superada, no modelo básico; e afetar a estabilidade dos equilíbrios, no modificado, fazendo com que resultados econômicos com maior produto tornem-se estáveis. No segundo ensaio, obtemos condições sob as quais a função perda do banco central é estritamente convexa em quatro estados distintos da economia: economia aquecida, em recessão, inação alta e produto alto. Encontramos, ainda, que quando in- ação e produto são funções lineares do instrumento de política monetária, a convexidade é garantida para qualquer um dos quatro estados citados. Ao estendermos nossa análise a vários instrumentos, encontramos que apenas linearidade já não é mais suficiente para a garantia do formato da função perda. Nossos resultados fornecem, ainda, condições sob as quais existirá dependência entre os instrumentos de política monetária. Por fim, o terceiro ensaio estuda regimes de metas de inação, nos quais agentes podem influenciar a política monetária através das expectativas de mercado reportadas ao banco central. Este, por sua vez, deve formular a política monetária considerando que tal influência pode ser usada em benefício dos próprios agentes. Modelamos essa relação estratégica como um jogo sequencial entre uma instituição financeira representativa e o banco central. Mostramos que quando a autoridade monetária escolhe apenas o nível da taxa de juros, existe um potencial viés inflacionário na economia. Esse viés é superado quando a oferta de moeda torna-se um segundo instrumento de política. Ainda mostramos que penalização de instituições más previsoras também pode ser um mecanismo eficiente de ancoragem de expectativas. / This thesis consists of three essays. In the first one, we show in our basic model that economies consisted exclusively by producers and parasites may fall into poverty traps, assuming that both groups behave according to the dynamics of Lotka-Volterra. However, the introduction of an upper bound on the output growth and expectations for the agents excludes the result of trap in its multiple equilibria. Our conclusion, nevertheless, is similiar for both studied models: improved protection of property rights by the state can mitigate the poverty trap possibility in the basic model, and affect the stability of equilibria in the modified one, making that economic outcomes with higher output become stable. In the second essay, we obtain conditions under which the central bank's loss function is strictly convex in four different states of the economy: booming economy, recession, high inflation and high output. Moreover, we found that when inaction and output are linear functions of monetary policy instruments, convexity is guaranteed for any of the four states mentioned. When we extend our analysis to the case of many instruments, we found that only linearity is not sufficient to guarantee the shape of loss function. Our results also provide conditions under which there exists dependence between instruments of monetary policy. Finally, the third essay studies the ination targeting regimes, in which agents can influence the monetary policy through market expectations reported to the central bank. Monetary authority, in its turn, should formulate the monetary policy considering that this influence may be used for the benefit of agents themselves. We model this strategic relationship as a sequential game between a representative financial institution and the central bank. We show that when the monetary authority chooses only the level of interest rates, there is a potential inflationary bias in the economy. This bias is solved when the money supply becomes a second instrument of policy. In addition, we show that to impose penalty on the worse predictor institutions may also be an efficient anchoring expectations mechanism.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

>Magister Scientiae - MSc / In principle, an enrichment of resources in predator-prey systems show prompts destabilisation of a framework, accordingly, falling trophic communication, a phenomenon known to as the \Paradox of Enrichment" [54]. After it was rst genius postured by Rosenzweig [48], various resulting examines, including recently those of Mougi-Nishimura [43] as well as that of Bohannan-Lenski [8], were completed on this problem over numerous decades. Nonetheless, there has been a universal none acceptance of the \paradox" word within an ecological eld due to diverse interpretations [51]. In this dissertation, some theoretical exploratory works are being surveyed in line with giving a concise outline proposed responses to the paradox. Consequently, a quantity of di usion-driven models in mathematical ecology are evaluated and analysed. Accordingly, piloting the way for the spatial structured pattern (we denote it by SSP) formation in nonlinear systems of partial di erential equations [36, 40]. The central point of attention is on enrichment consequences which results toward a paradoxical state. For this purpose, evaluating a number of compartmental models in ecology similar to those of [48] will be of great assistance. Such displays have greater in uence in pattern formations due to diversity in meta-population. Studying the outcomes of initiating an enrichment into [9] of Braverman's model, with a nutrient density (denoted by n) and bacteria compactness (denoted by b) respectively, suits the purpose. The main objective behind being able to transform [9]'s system (2.16) into a new model as a result of enrichment. Accordingly, n has a logistic- type growth with linear di usion, while b poses a Holling Type II and nonlinear di usion r2 nb2 [9, 40]. Five fundamental questions are imposed in order to address and guide the study in accordance with the following sequence: (a) What will be the outcomes of introducing enrichment into [9]'s model? (b) How will such a process in (i) be done in order to change the system (2.16)'s stability state [50]? (c) Whether the paradox does exist in a particular situation or not [51]? Lastly, (d) If an absurdity in (d) does exist, is it reversible [8, 16, 54]? Based on the problem statement above, the investigation will include various matlab simulations. Therefore, being able to give analysis on a local asymptotic stability state when a small perturbation has been introduced [40]. It is for this reason that a bifurcation relevance comes into e ect [58]. There are principal de nitions that are undertaken as the research evolves around them. A study of quantitative response is presented in predator-prey systems in order to establish its stability properties. Due to tradeo s, there is a great likelihood that the growth rate, attack abilities and defense capacities of species have to be examined in line with reviewing parameters which favor stability conditions. Accordingly, an investigation must also re ect chances that leads to extinction or coexistence [7]. Nature is much more complex than scienti c models and laboratories [39]. Therefore, di erent mechanisms have to be integrated in order to establish stability even when a system has been under enrichment [51]. As a result, SSP system is modeled by way of reaction-di usion di erential equations simulated both spatially and temporally. The outcomes of such a system will be best suitable for real-world life situations which control similar behaviors in the future. Comparable models are used in the main compilation phase of dissertation and truly re ected in the literature. The SSP model can be regarded as between (2018-2011), with a stability control study which is of an original.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19108号 / 情博第554号 / 新制||情||98(附属図書館) / 32059 / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 阿久津 達也, 教授 西田 豊明, 教授 山本 章博 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

The applications of differential equations are many. However, many differential equations modelling real-world scenarios are very complex and it can be of great difficulty to find an exact solution if one even exists. Thus, it is of importance to be able to approximate solutions of differential equations. Here, a method using neural networks is explored and its performance is compared to that of a numerical method. To illustrate the method, two first order, two second order and two first order systems of ordinary differential equations are explored. The systems are the Lotka-Volterra system and the SEIR (Susceptible, Exposed, Infected, Removed) epidemiological model. The first four examples have exact solutions to compare to and the observations are then used as a basis when discussing the results of the systems. The results of the thesis show that while the neural network method takes longer to deliver an approximation, it continuously gives better approximations than the implicit Euler method used for comparison. The main contribution of this thesis is the comparison done of the performances of the neural network method and the implicit Euler method. / Det finns många användningsområden för differentialekvationer. Däremot är många differentialekvationer som modellerar verkligheten komplexa och det kan vara svårt, om inte omöjligt, att hitta en exakt lösning. På grund av detta är det viktigt att ha metoder som kan approximera lösningar till differentialekvationer. Därför undersöks här en metod som använder sig av neurala nätverk. Dess resultat blir sedan jämförda med en numerisk metod. För att illustrera metoden presenteras två ekvationer av första ordningen, två ekvationer av andra ordningen och två system av differentialekvationer. Systemen som undersöks är Lotka-Volterra ekvationerna samt SEIR (Susceptible, Exposed, Infected, Removed) modellen. De första fyra exemplen som undersöks har exakta lösningar att jämföra med och dessa observationer används sedan vid diskussionerna gällande systemen. Resultaten visar att medan metoden som använder neurala nätverkar tar längre tid att exekvera, så ger metoden bättre approximationer än den implicita Euler metoden som användes som jämförelse. Det huvudsakliga bidraget med det här examensarbetet är jämförelsen av hur de två metoderna presterar.

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