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Reduction of Discrete Dynamical Systems with Applications toDynamics Population Models

Published online by Cambridge University Press:  28 November 2013

R. Bravo de la Parra ,
M. Marvá ,
E. Sánchez  and
L. Sanz
R. Bravo de la Parra*
Affiliation:
Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
M. Marvá
Affiliation:
Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
E. Sánchez
Affiliation:
Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain
L. Sanz
Affiliation:
Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain
*
⋆Corresponding author. E-mail: rafael.bravo@uah.es
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Abstract

In this work we review the aggregation of variables method for discrete dynamicalsystems. These methods consist of describing the asymptotic behaviour of a complex systeminvolving many coupled variables through the asymptotic behaviour of a reduced systemformulated in terms of a few global variables. We consider population dynamics modelsincluding two processes acting at different time scales. Each process has associated a mapdescribing its effect along its specific time unit. The discrete system encompassing bothprocesses is expressed in the slow time scale composing the map associated to the slow oneand the k-th iterate of the map associated to the fast one. In the linear case a result isstated showing the relationship between the corresponding asymptotic elements of bothsystems, initial and reduced. In the nonlinear case, the reduction result establishes theexistence, stability and basins of attraction of steady states and periodic solutions ofthe original system with the help of the same elements of the corresponding reducedsystem. Several models looking over the main applications of the method to populationsdynamics are collected to illustrate the general results.

Keywords

discrete modelsaggregation of variablestime scalespopulation dynamic.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 107 - 129
Copyright
© EDP Sciences, 2013

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