Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models

¹ Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares ( Madrid), Spain
² 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

Abstract

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