Approximate aggregation of linear discrete models with two time scales: re-scaling slow processes to the fast scale

Abstract

Mathematical models used in ecology often inherit the complexity found in nature and thus are governed by a large number of variables. Aggregation of variables methods is used to make such models mathematically tractable by building an approximate system governing fewer variables. We extend here aggregation methods for linear discrete models with processes occurring at different time scales. In practical cases, some processes that occur at a fast time scale are often only measured at a slow time scale, like mortality. We present a general class of models with two time scales involving such kind of processes. We show how they should be re-scaled in order to be taken into account at the fast time scale in a more realistic approach. The approximate aggregation of these models is undertaken and justified in mathematical terms. We also provide an application to a model of a structured population in a two-patch environment.