Variables aggregation in time varying discrete systems

In this work we extend approximate aggregation methods in time discrete lin ear models to the case of time varying environments. Approximate aggregatio n consists in describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced syst em with a few number of variables. We present a time discrete lime varying model in which we distinguish two time scales. By using perturbation method s we transform the system to make the global variables appear and build up the aggregated system. The asymptotic relationships between the general and aggregated systems are explored in the cases of a cyclically varying envir onment and a changing environment in process of stabilization. We show that under quite general conditions the knowledge of the behavior of the aggreg ated system characterizes that of the general system. The general method is also applied to aggregate a multiregional time dependent Leslie model show ing that the aggregated model has demographic rates depending on the equili brium proportions of individuals in the different patches.