Variables aggregation in time varying discrete systems

Abstract

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists in 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 number of variables. We present a time discrete time varying model in which we distinguish two time scales. By using perturbation methods 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 environment and a changing environment in process of stabilization. We show that under quite general conditions the knowledge of the behavior of the aggregated system characterizes that of the general system. The general method is also applied to aggregate a multiregional time dependent Leslie model showing that the aggregated model has demographic rates depending on the equilibrium proportions of individuals in the different patches.