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Abstract(s)
Nesta dissertação, vamos considerar o modelo Lotka-Volterra. Este foi obtido na década 1920’s
independentemente por Lotka e Volterra. O modelo é dado por um par de equações diferenciais
não lineares de primeira ordem e considera a interação entre as duas populações. Existem
três grandes tipos de interação: competição, cooperação e predador -presa. Neste trabalho,
estudamos o modelo Lotka-Volterra com interação do tipo predador-presa. Para modelar a
dinâmica entre as duas populações podemos adicionar termos ao modelo original de forma a
torná-lo mais realista e sempre que possível estimar a sua estabilidade. No primeiro modelo a
ser analisado, será introduzido um termo nas presas e será estudada sua estabilidade. Um dos
termos a ser adicionado pode ser um controle, numa ou nas duas populações e pode ser visto
como introdução ou remoção de elementos nas populações. No segundo e terceiro modelo,
iremos introduzir um termo que deverá ser visto como um controle. Este será introduzido nos
predadores e será do tipo ON-OFF. Em ambos os modelos iremos mostrar graficamente que
os modelos aparentam convergir para um ponto numa zona específica. Todos serão modelados
usando equações às diferenças mas para isso é necessário escolher um esquema numérico. Entre
os mais comuns estão os métodos de Euler, Runge-Kutta e Mickens. Iremos usar o método de
Mickens.
In this dissertation, we consider the Lotka-Volterra model. It was obtained in 1920’s independently by Lotka and Volterra. The model is given by two first-order nonlinear differential equations and consider the interaction between two populations. The three main types of interaction are competition, cooperation, and predator-prey. In this work, we study the Lotka-Volterra model of the predator-prey type. To model the dynamics between these two populations there can be added terms in an attempt to make it more realistic and if it is possible, to estimate its stability. In the first model, we add a term in the preys and its stability will be studied. One of the terms that can be added may be a control, in one or two populations and it can be seen as an introduction or removal of elements of the population or populations. In the second and third model, we will add a term that should be seen as a control. It will be added in the predators and will be an ON-OFF control. In these two last models, it will be shown graphically that the trajectories tend to converge to a point in a specific zone. All of them will be modeled by difference equations but, to do that, we need to choose some numerical scheme. The most common ones are Euler, Runge-Kutta and more recently Mickens method. We will use the Mickens Method.
In this dissertation, we consider the Lotka-Volterra model. It was obtained in 1920’s independently by Lotka and Volterra. The model is given by two first-order nonlinear differential equations and consider the interaction between two populations. The three main types of interaction are competition, cooperation, and predator-prey. In this work, we study the Lotka-Volterra model of the predator-prey type. To model the dynamics between these two populations there can be added terms in an attempt to make it more realistic and if it is possible, to estimate its stability. In the first model, we add a term in the preys and its stability will be studied. One of the terms that can be added may be a control, in one or two populations and it can be seen as an introduction or removal of elements of the population or populations. In the second and third model, we will add a term that should be seen as a control. It will be added in the predators and will be an ON-OFF control. In these two last models, it will be shown graphically that the trajectories tend to converge to a point in a specific zone. All of them will be modeled by difference equations but, to do that, we need to choose some numerical scheme. The most common ones are Euler, Runge-Kutta and more recently Mickens method. We will use the Mickens Method.
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Keywords
Abordagens Ao Modelo de Lotka-Volterra