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Revista Integración

Print version ISSN 0120-419X

Integración - UIS vol.35 no.1 Bucaramanga Jan./June 2017

https://doi.org/10.18273/revint.v35n1-2017008 

Original article

A model of competing species that exhibits zip bifurcation

Un modelo de especies en competencia que exhibe bifurcación zip

Luis F. Echeverri1* 

Óscar I. Giraldo2 

Edwin Zarrazola1 

1Universidad de Antioquia, Instituto de Matemáticas, Medellín, Colombia

2Universidad Nacional de Colombia, Escuela de Matemáticas, Medellín, Colombia


Abstract:

The purpose of this paper is to present a concrete model of competing population species that exhibits a phenomenon called zip bifurcation. The Zip Bifurcation was introduced by Farkas in 1984 for a three dimensional ODE prey-predator system describing a chemostat. We will study a three dimensional system of ordinary differential equations that model the competition of two predators species for one single prey species. The system is based on concrete trigonometric functions modeling the growth rate of the prey and the functional response of the predator. The model exhibits different kinds of behavior and shows examples of the so called “competitive exclusión principle,” and the competition of one “r-strategist” and one “K-strategist.” Additionally, in order to illustrate the zip bifurcation, we will present some numerical simulations for our model.

Keywords: Predator prey model; Zip bifurcation; r-strategist; K-strategist

Resumen:

El objetivo de este trabajo es presentar un modelo concreto de poblaciones de especies en competición que exhibe la bifurcación Zip. La bifurcación zip fue introducida por Farkas en 1984 para un sistema tridimensional de ecuaciones diferenciales ordinarias que describe un quimiostato. Estudiaremos un sistema tridimensional de ecuaciones diferenciales ordinarias que modela la competición de dos poblaciones distintas de predadores por una única población presa. El sistema usa funciones trigonométricas concretas para representar la tasa de crecimiento de la presa y la respuesta funcional del predador. El modelo exhibe diferentes clases de comportamientos y muestra ejemplos de los llamados principio de exclusión competitiva y la competición

Palabras clave: Modelo depredador-presa; Bifurcación Zip; r-estratega; K-estratega

Texto Completo disponible en PDF

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0Para citar este artículo: L.F. Echeverri, Ó.I. Giraldo, E. Zarrazola, A model of competing species that exhibits zip bifurcation, Rev. Integr. Temas Mat. 35 (2017), No. 1, 127-141.

Received: June 2016; Accepted: June 2017

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