The reliability of approximate reduction techniques in population models with two time scales

As a result of the complexity inherent in some natural systems, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often find multiregional models for structured populations in which individuals are classified regarding their age and their spatial location. Dealing with such structured populations leads to high dimensional models. Moreover, in many instances the dynamics of the system is controlled by processes whose time scales are very different from each other. For example, in multiregional models migration is often a fast process in comparison to the growth of the population. Approximate reduction techniques take advantage of the presence of different time scales in a system to introduce approximations that allow one to transform the original system into a simpler low dimensional system. In this way, the dynamics of the original system can be approximated in terms of that of the reduced system. This work deals with the study of that approximation. In particular, we work with a non-autonomous discrete time model previously presented in the literature and obtain different bounds for the error we incur when we describe the dynamics of the original system in terms of the reduced one. The results are illustrated by some numerical simulations corresponding to the reduction of a Leslie type model for a population structured in two age classes and living in a two patch system.