Reduction of Nonautonomous Population Dynamics Mod

Authors: Marcos Marvá Rafael Bravo de la Parra

Publish Date: 2014/05/18

Volume: 62, Issue: 3, Pages: 285-303

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Abstract

The purpose of this work is reviewing some reduction results to deal with systems of nonautonomous ordinary differential equations with two time scales They could be included among the socalled approximate aggregation methods The existence of different time scales in a system together with some longterm features are used to build up a simpler system governed by a lesser number of state variables The asymptotic behavior of the latter system is then used to describe the asymptotic behaviour of the former one The reduction results are stated in two particular but important cases periodic systems and asymptotically autonomous systems The reduction results are illustrated with the help of simple spatial SIS epidemic models including either periodic or asymptotically autonomous termsWe summarize in the next theorem the results on singular perturbations methods for slowfast dynamics on the infinite interval as presented for the first time in the work of Hoppensteadt 1966 that the author subsequently included in a more readable way in reviews of differential equations with small parameters and quasistatic state analysis of differential equations Hoppensteadt 1971 1993 2010 In Verhulst 2007 it is found a review of singular perturbation methods for slowfast systems where they are mentioned some other works notably by Tikhonov et al 1985 that preceded those of Hoppensteadt though for bounded intervals of time It is also found in Verhulst 2007 the peculiarities of this theory applied to autonomous equations following the works by Fenichel 1971

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