Thesis VARIEDADES INVARIANTES, BIFURCACIONES Y CUENCAS DE ATRACCIÓN EN MODELOS DEPREDADOR-PRESA
Date
2018
Authors
CONTRERAS JULIO, DANA
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Abstract
En este trabajo analizamos dos modelos de dinámica poblacional. Uno de ellos es un modelo depredador-presa con doble efecto Allee en la presa, con un comportamiento competitivo para los depredadores, dado por un sistema bidimensional de ecuaciones diferenciales ordinarias. Para este modelo investigamos la naturaleza de los umbrales Allee y las cuencas de atracción. Desde una perspectiva matemática, esto implica encontrar y caracterizar las fronteras de las cuenca de atracción en el espacio de fase. Este suele ser un desafío importante, ya que los objetos que actúan como fronterasentre dos cuencas de atracción diferentes son variedades invariantes del sistema, los cuales pueden sufrir cambios topológicos en las bifurcaciones. Las bifurcaciones locales incluyen bifurcaciones sillanodo, transcrítica y de Hopf; mientras que los fenómenos globales incluyen bifurcaciones homoclínicas,conexiones heteroclínicas y ciclos heteroclínicos. Identi camos que el umbral Allee es un ciclo límite, una órbita homoclínica o la variedad estable de un equilibrio. Ofrecemos una explicación geométrica completa de cómo el umbral Allee y las cuencas de atracción experimentan transiciones críticas.Este enfoque se complementa con un estudio de la dinámica cercana al in nito. De esta manera, determinamos las condiciones de modo que las cuencas de atracción sean conjuntos acotados o no acotados en el espacio de fase. Finalmente, estos resultados nos permiten mostrar una descripcióncompleta de los retratos de fase, los umbrales de extinción y las cuencas de atracción de nuestro modelo bajo la variación de los parámetros. Este trabajo fue publicado en [1].El otro modelo de dinámica poblacional estudiado es una extensión de un modelo de competencia de dos dimensiones de depredador-presa de mosquitos Aedes aegypti infectados con la bacteria Wolbachia y mosquitos no infectados, agregándole difusión espacial y luego estudiamos el sistemarelacionado de cuatro dimensiones de ondas viajeras. Para este objetivo investigamos las condiciones que nos aseguran la prevalencia de mosquitos infectados con Wolbachia en el modelo de cuatro dimensiones, utilizando herramientas analíticas de sistemas dinámicos en conjunto con herramientasnuméricas para describir las variedades invariantes del sistema a medida que sufren cambios topológicos al pasar por bifurcaciones. Particularmente nos centramos en el estudio del equilibrio con la totalidad de la población infectada con Wolbachia. Además, presentamos un análisis de bifurcación que incluye las condiciones para cambios de estabilidad de los puntos de equilibrio y la creación y destrucción de algunas órbitas heteroclínicas que corresponden a frentes de onda en el modelo. Las bifurcaciones locales incluyen bifurcaciones de silla-nodo y transcríticas; mientras que los fenómenos globales incluyen bifurcaciones heteroclínicas. Más aún, identi camos las condiciones para la existencia de una cantidad no numerable de órbita heteroclínicas que convergen al equilibriocon la población de mosquitos infectados con la bacteria, que es el resultado principal de este capítulo debido a que estas soluciones son frentes de onda que convergen a la totalidad de la población de mosquitos infectados con Wolbachia.Para este objetivo, hacemos un uso extensivo de las herramientas analíticas de la teoría de sistemas dinámicos y el análisis de bifurcación numérica y determinamos el diagrama de bifurcación completo.Las estrategias basadas en bifurcaciones y análisis de variedades invariantes nos permite identi car los mecanismos matemáticos que producen los reordenamientos de separatrices en el espacio de fase. En general, estos resultados nos permiten mostrar una descripción completa de los retratos de fase, los umbrales de extinción y las cuencas de atracción de nuestros modelos bajo la variación de parámetros.Finalmente hemos realizado un aporte importante para el cálculo de variedades invariantes, ideando un método para el cálculo de variedades invariantes de dimensión tres implementado en el software Auto, en conjunto con un método para poder visualizar cuencas de atracción de modelos de cuatrodimensiones.
In this work we analyze two models of population dynamics. One of them is a predator-prey model with double Allee effect in the prey, with a competitive behavior for predators, given by a two-dimensional system of ordinary differential equations. For this model we investigate the nature of Allee thresholds and basins of attraction. From a mathematical perspective, this implies to fi nd andcharacterise the corresponding basin boundaries in phase space. This is typically a major challengesince the objects that act as boundaries between two different basins are invariant manifolds of thesystem which may also undergo topological changes at bifurcations. Local bifurcations include saddlenode,transcritical and Hopf bifurcations; while global phenomena include homoclinic bifurcations,heteroclinic connections and heteroclinic cycles. We identify the Allee threshold to be either alimit cycle, a homoclinic orbit or the stable manifold of an equilibrium. We give a full geometricalexplanation of how the Allee threshold and basins of attraction undergo critical transitions. Thisapproach is complemented with a study of the dynamics near in nity. In this way, we determine theconditions such that the basins of attraction are bounded or unbounded sets in phase space. All inall, these results allow us to show a complete description of phase portraits, extinction thresholds,and basins of attraction of our model under variation of parameters. This work was published in [1].The other model of population dynamics studied is an extension of a predator-prey model ofcompetition of interaction between wild Aedes aegypti female mosquitoes and those infected withbacteria Wolbachia, given by a two-dimensional system of ordinary differential equations. We addspatial diffusion and study the traveling waves solutions of the four-dimensional system of ordinarydifferential equations assosiated. For this purpose we investigate the conditions that assure the prevalenceof mosquitoes infected with Wolbachia in the four-dimensional model, using analytical tools ofdynamic systems combined with numerical tools to describe the invariant manifolds of the system asit undergo topological changes at bifurcations. Particularly we focus on the study of the equilibriumwith the whole populationinfected with Wolbachia. In addition, we present a bifurcation analysis thatincludes the conditions of stability changes of the equilibrium points and the creation and destructionof some heteroclinic orbits that correspond to the wave fronts in the model. Local bifurcations includesaddle-node and transcritical bifurcations; While global phenomena include heteroclinic bifurcations.Moreover, we identify the conditions for the existence of an uncountable quantity of heteroclinicorbits that converge to the equilibrium with the population of mosquitoes infected with the bacteria,which is the main result of this chapter because these solutions are wavefronts that converge to theentire population of mosquitoes infected with Wolbachia.This strategies based on bifurcation and invariant manifold analysis allows us to identify themathematical mechanisms that produce rearrangements of separatrices in phase space. In general,these results allow us to show a complete description of phase portraits, extinction thresholds, andbasins of attraction of our model under variation of parameters. Finally, we have made an importantcontribution to the calculation of invariant manifolds, devising a method for the calculation of threedimensionalinvariant manifolds implemented in software Auto, combined with a method to visualizebasins of attraction of four-dimensional models.
In this work we analyze two models of population dynamics. One of them is a predator-prey model with double Allee effect in the prey, with a competitive behavior for predators, given by a two-dimensional system of ordinary differential equations. For this model we investigate the nature of Allee thresholds and basins of attraction. From a mathematical perspective, this implies to fi nd andcharacterise the corresponding basin boundaries in phase space. This is typically a major challengesince the objects that act as boundaries between two different basins are invariant manifolds of thesystem which may also undergo topological changes at bifurcations. Local bifurcations include saddlenode,transcritical and Hopf bifurcations; while global phenomena include homoclinic bifurcations,heteroclinic connections and heteroclinic cycles. We identify the Allee threshold to be either alimit cycle, a homoclinic orbit or the stable manifold of an equilibrium. We give a full geometricalexplanation of how the Allee threshold and basins of attraction undergo critical transitions. Thisapproach is complemented with a study of the dynamics near in nity. In this way, we determine theconditions such that the basins of attraction are bounded or unbounded sets in phase space. All inall, these results allow us to show a complete description of phase portraits, extinction thresholds,and basins of attraction of our model under variation of parameters. This work was published in [1].The other model of population dynamics studied is an extension of a predator-prey model ofcompetition of interaction between wild Aedes aegypti female mosquitoes and those infected withbacteria Wolbachia, given by a two-dimensional system of ordinary differential equations. We addspatial diffusion and study the traveling waves solutions of the four-dimensional system of ordinarydifferential equations assosiated. For this purpose we investigate the conditions that assure the prevalenceof mosquitoes infected with Wolbachia in the four-dimensional model, using analytical tools ofdynamic systems combined with numerical tools to describe the invariant manifolds of the system asit undergo topological changes at bifurcations. Particularly we focus on the study of the equilibriumwith the whole populationinfected with Wolbachia. In addition, we present a bifurcation analysis thatincludes the conditions of stability changes of the equilibrium points and the creation and destructionof some heteroclinic orbits that correspond to the wave fronts in the model. Local bifurcations includesaddle-node and transcritical bifurcations; While global phenomena include heteroclinic bifurcations.Moreover, we identify the conditions for the existence of an uncountable quantity of heteroclinicorbits that converge to the equilibrium with the population of mosquitoes infected with the bacteria,which is the main result of this chapter because these solutions are wavefronts that converge to theentire population of mosquitoes infected with Wolbachia.This strategies based on bifurcation and invariant manifold analysis allows us to identify themathematical mechanisms that produce rearrangements of separatrices in phase space. In general,these results allow us to show a complete description of phase portraits, extinction thresholds, andbasins of attraction of our model under variation of parameters. Finally, we have made an importantcontribution to the calculation of invariant manifolds, devising a method for the calculation of threedimensionalinvariant manifolds implemented in software Auto, combined with a method to visualizebasins of attraction of four-dimensional models.
Description
Catalogado desde la version PDF de la tesis.
Keywords
CUENCAS DE ATRACCION , DENGUE , EFECTO ALLEE , MODELOS POBLACIONALES , SISTEMAS DINAMICOS , VARIEDADES INVARIANTES