TIME SCALES IN A NON-AUTONOMOUS LINEAR DISCRETE MODEL

Abstract

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists of describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few "global" variables. We present a time varying discrete model in which we distinguish two processes with different time scales. By defining the global variables as appropriate linear combinations of the state variables, we transform the system into a reduced one. The variables corresponding to the original and reduced systems can be related, therefore allowing one the study of the former in terms of the latter. The property of weak ergodicity, which has to do with the capacity of a system to become asymptotically independent of initial conditions, is explored for the original and reduced systems. The general method is also applied to aggregate a time-dependent multiregional model which appears in the field of population dynamics in two different cases: Fast migration with respect to demography and fast demography with respect to migration.