Reduction of slow-fast asymptotically autonomous systems with applications to gradostat models

Abstract

Two distinguishing features characterize the population dynamic models considered in the present work. On the one hand, we consider several interacting organization levels associated to different time scales. On the other hand, the environment tends to be constant in the long term. The mathematical representation of these properties leads to slow-fast asymptotically autonomous systems. These characteristics add some realism in the models. However, the analytical study of this class of systems is generally hard to perform.Here we present a reduction technique that can be included among the so-called approximate aggregation methods. The existence of different time scales, together with the long term features, are used to build up a simpler system, which can be described by means of a lower number of state variables. The asymptotic behavior of the simplified model helps to study the original one.The reduction procedure is formulated in a general way. Following, two illustrations of asymptotically autonomous models with two time scales, in a gradostat, are given: a consumer–resource model and a competition model. Finally, a wider range of applications is suggested.