Publication: DIFUSIÓN ESPACIAL EN UN MODELO DEPREDADOR-PRESA CON EFECTO ALLEE FUERTE EN LA PRESA Y RESPUESTA FUNCIONAL RAZÓN-DEPENDIENTE
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Date
2019-11
Authors
VILLAR SEPÚLVEDA, EDGARDO ENRIQUE
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Abstract
La relación dinámica entre los depredadores y sus presas es uno de los temas principales dentro de
la ecología. Solo observar la base de los modelos depredador-presa presentada por Lotka y Volterra
puede llevar a pensar que la dinámica de este tipo de modelos es simple. Sin embargo, los modelos en
la actualidad han mostrado tener distintas complicaciones que les dan riqueza y realismo, mientras
se nutren de una complejidad muy desafiante.
El objetivo de este trabajo es analizar la ecuación de reacción-difusión que surge al agregarle
términos de difusión espacial a un modelo de tipo depredador-presa previamente estudiado. Para
el análisis del modelo, se estudian en particular dos tipos de soluciones: ondas viajeras y soluciones
estacionarias. Para estudiar las soluciones de tipo onda viajera, se realiza un análisis de bifurcación
a puntos de equilibrio del sistema asociado, y se realiza un estudio y aproximación de las variedades
estable e inestable asociadas a dichos equilibrios. Por otro lado, para el análisis de las soluciones
estacionarias se utilizan dos enfoques: se estudian los cambios de una solución estacionaria al variar
distintos parámetros del sistema, y se buscan patrones de inestabilidad de Turing, al analizar las
condiciones que conllevan su existencia. Además, se realiza un análisis de la región de parámetros en
la que se dan estos patrones.
Este trabajo cuenta con una gran cantidad de material obtenido numéricamente, con el fin de
visualizar los objetos de interés. Para realizar los análisis de tipo numérico se hace un alto uso del
software Auto, y los resultados obtenidos son justificados con la teoría de sistemas dinámicos y ecua-
ciones diferenciales. En particular, se realiza un análisis teórico, numérico y visual de la interacción
de las variedades invariantes de un punto de equilibrio en vecindades de una bifurcación homoclínica
foco-foco, el cual nunca se ha hecho previamente.
El análisis realizado al sistema de ondas viajeras nos indica que existen, en particular, tres tipos
de ondas viajeras en el modelo: frentes de onda, que conectan dos estados estacionarios distintos de
especies en los que puede o no existir coexistencia de ellas; pulsos de onda, que conectan un estado
estacionario consigo mismo en el largo plazo, en que se tiene coexistencia de especies; y trenes de
onda, que oscilan infinitamente en el espacio a medida que avanza el tiempo, sin que se tenga la
extinción de especies con el paso del tiempo.
Por otro lado, el análisis de soluciones estacionarias nos indica que existen patrones heterogéneos
de poblaciones en el espacio, que no varían con el tiempo. Esto nos permite, en particular, conocer
las distribuciones espaciales que es posible obtener al variar los distintos parámetros del sistema.
Además, el estudio de los patrones de inestabilidad de Turing permite concluir que, si ambas pobla-
ciones poseen bajas tasas de difusión, es posible que estas se distribuyan heterogéneamente, formando
patrones periódicos en el espacio.
The dynamic relationship between predators and their prey is one of the main topics in ecology. Just observing the base of the predator-prey models presented by Lotka and Volterra can lead to think that the dynamics of this kind of models is simple. Nevertheless, models today have shown to have different complications that give them richness and realism, while they nourish in a very challenging complexity. The objective of this work is to analize the reaction-diffusion equation that arise when we add spatial diffusion terms to a previously studied predator-prey model. For the analysis of the model, we study two particular kind of solutions: traveling waves and stationary solutions. To study solutions of traveling wave type, we make a bifurcation analysis to equilibrium points of the associated system, and we make a study and approximation of the stable and unstable manifolds of those equilibria. On the other hand, for the analysis of stationary solutions we use two approaches: we study the changes of a stationary solution when different parameters of the model are changed, and we look for Turing instability patterns, analyzing the conditions that carry their existence. Furthermore, we make an analysis of the parameter region in which these patterns are possible. This work counts with a large amount of material obtained numerically, in order to visualize in- teresting objects. To make the numerical analysis we make a high use of the software Auto, and the obtained results are justified with dynamical systems theory and differential equations. In particular, we make a theoretical, numeric and visual analysis of the interaction of the invariant manifolds of an equilibrium point in a neighbourhood of a homoclinic focus-focus Shilnikov bifurcation, which has never been done previously. The analisis made to the system of traveling waves tells us that there exist, in particular, three kind of traveling waves in the model: wave fronts, that connect two different stationary states of species in which there may or may not exist coexistence of species; wave pulses, that connect a sta- tionary state with itself in the long time, in which there exist coexistence of species; and wave trains, that oscillate infinitely in the space, as time progresses, without having species extinction with time progression. On the other hand, the analysis of stationary solutions tells us that there exist spatial heteroge- neous population patterns, that do not change with time. This allows us, in particular, to know the spatial distributions that can be obtained by varying the different system parameters. Furthermore, the study of the Turing instability patterns allows us to conclude that, if both populations have low diffusion rates, it is possible that they distribute heterogenously, making periodic patterns in space. Keywords: Traveling waves; bifurcations; invariant manifolds; stationary solutions; Turing insta- bility patterns.
The dynamic relationship between predators and their prey is one of the main topics in ecology. Just observing the base of the predator-prey models presented by Lotka and Volterra can lead to think that the dynamics of this kind of models is simple. Nevertheless, models today have shown to have different complications that give them richness and realism, while they nourish in a very challenging complexity. The objective of this work is to analize the reaction-diffusion equation that arise when we add spatial diffusion terms to a previously studied predator-prey model. For the analysis of the model, we study two particular kind of solutions: traveling waves and stationary solutions. To study solutions of traveling wave type, we make a bifurcation analysis to equilibrium points of the associated system, and we make a study and approximation of the stable and unstable manifolds of those equilibria. On the other hand, for the analysis of stationary solutions we use two approaches: we study the changes of a stationary solution when different parameters of the model are changed, and we look for Turing instability patterns, analyzing the conditions that carry their existence. Furthermore, we make an analysis of the parameter region in which these patterns are possible. This work counts with a large amount of material obtained numerically, in order to visualize in- teresting objects. To make the numerical analysis we make a high use of the software Auto, and the obtained results are justified with dynamical systems theory and differential equations. In particular, we make a theoretical, numeric and visual analysis of the interaction of the invariant manifolds of an equilibrium point in a neighbourhood of a homoclinic focus-focus Shilnikov bifurcation, which has never been done previously. The analisis made to the system of traveling waves tells us that there exist, in particular, three kind of traveling waves in the model: wave fronts, that connect two different stationary states of species in which there may or may not exist coexistence of species; wave pulses, that connect a sta- tionary state with itself in the long time, in which there exist coexistence of species; and wave trains, that oscillate infinitely in the space, as time progresses, without having species extinction with time progression. On the other hand, the analysis of stationary solutions tells us that there exist spatial heteroge- neous population patterns, that do not change with time. This allows us, in particular, to know the spatial distributions that can be obtained by varying the different system parameters. Furthermore, the study of the Turing instability patterns allows us to conclude that, if both populations have low diffusion rates, it is possible that they distribute heterogenously, making periodic patterns in space. Keywords: Traveling waves; bifurcations; invariant manifolds; stationary solutions; Turing insta- bility patterns.
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Keywords
ONDAS VIAJERAS , BIFURCACIONES , VARIEDADES INVARIANTES , SOLUCIONES ESTACIONARIAS , PATRONES DE INESTABILIDAD DE TURING