Time Scales in Density Dependent Discrete Models

Abstract

The aim of this work is to extend approximate aggregation methods for multi-time scale systems of nonlinear ordinary differential equations to time discrete models. Approximate aggregation consist on describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete time models with two different time scales, the fast one considered linear and the slow one generally nonlinear. We transform the system to make the global variables appear, and use a version of center manifold theory to build up the aggregated system. Simple forms of the aggregated system are enough for the local study of the asymptotic behaviour of the general system provided that it has certain stability under perturbations. The general method is applied to aggregate a multiregional density dependent Leslie model into a density dependent Leslie model in which the demographic rates are expressed in terms of the equilibrium proportions of individuals in the different patches.