Reduction of Nonautonomous Population Dynamics Models with Two Time Scales
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Reduction of Nonautonomous Population Dynamics Models with Two Time Scales Marcos Marva´ • Rafael Bravo de la Parra
Received: 15 December 2013 / Accepted: 7 May 2014 Ó Springer Science+Business Media Dordrecht 2014
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 so-called approximate aggregation methods. The existence of different time scales in a system, together with some long-term 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 terms. Keywords Slow–fast dynamics Singular perturbations Periodic systems Asymptotically autonomous systems Epidemic models
1 Introduction The mathematical models used in population dynamics necessarily show the complexity found in natural systems. They are often governed by a large number of variables corresponding to different interacting organization levels. Some methods of reduction should be used in order to transform such models into mathematically
M. Marva´ R. Bravo de la Parra (&) Department of Physics and Mathematics, University of Alcala´ Campus Universitario, 28871 Alcala´ de Henares, Spain e-mail: [email protected] M. Marva´ e-mail: [email protected]
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M. Marva´, R. Bravo de la Parra
tractable ones. The so-called aggregation of variables methods can be included in this latter category. The term aggregation of variables appeared first in economy and later introduced in ecology (Luckyanov et al. 1983; Iwasa et al. 1987, 1989). The aggregation of a system consists of finding a certain number of global variables, functions of its state variables, and a system describing their dynamics. Aggregation is called perfect if the dynamics of the global variables is identical both in the initial system and in the aggregated one (Iwasa et al. 1989). On the other hand, approximate aggregation (Iwasa et al. 1987) deals with methods of reduction where the consistency between the dynamics of the global variables in the initial and the aggregated systems is only approximate. In Auger (1989) it is suggested a whole program of study of aggregation methods linked to the existence of different time scales in the frame of autonomous ordinary differential equations. This study was motivated in the broad sense by the hierarchy theory in ecology. The method was rigourously justified in Auger and Roussarie (1994) in terms of an adequate version of the Fenichel centre manifold theorem (Fenichel 1971, 1979). It allows to study the asymptotic behavior of the complete initial system with the help of a reduced sys
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