Model reduction by extended quasi-steady-state

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1、J.Math.Biol.40,443–450(2000)MathematicalBiologyDigitalObjectIdentifier(DOI):10.1007/s002850000026cSpringer-Verlag2000KlausR.Schneider
ThomasWilhelmModelreductionbyextendedquasi-steady-stateapproximationReceived:30March1999/Publishedonline:5May2000Abstract.Weextendthequasi-steady

2、-stateapproximation(QSSA)withrespecttotheclassofdifferentialsystemsaswellaswithrespecttotheorderofapproximation.Weillustratethefirstextensionbyanexamplewhichcannotbetreatedintheframeoftheclassicalapproach.Asanapplicationofthesecondextensionweprovethatthetrimolecularautocatalator

3、canbeapproximatedbyafastbimolecularreactionsystem.FinallywedescribeaclassofsingularlyperturbedsystemsforwhichahigherorderQSSAcaneasilybeobtained.1.IntroductionMathematicalmodelingofprocesseswithdifferenttimescalesleadsingeneraltosingularlyperturbedsystems(SPS)oftheformxPDf.x;y;

4、t;"/;(1)"yPDg.x;y/C"g.x;y;t;"/;Qwherex2Rm;y2Rn;0<"1;f;gQareboundedas"tendstozero.Thefirstequationiscalledtheslowsubsystemandthesecondrepresentsthefastone.Avarietyofperturbationmethodshavebeendevelopedtoinvestigatesingularlyper-turbedsystems:matchedasymptoticexpansions[15],WKB-m

5、ethods[16],multiplescalemethods[11],boundarylayerfunctions[24],averaging[2].Renormalizationgrouptheoryisanewunifyingmethodforglobalasymptoticanalysis[4].Geometricsingularperturbationtheoryisanotherapproachforthequalitativeanalysisofsingularlyperturbedsystems[7],especiallyitprov

6、idesamathemati-caljustificationforthereductionofsystem(1).ItisbasedontheexistenceofaninvariantmanifoldMoftheformyD.x;t;"/D2/(2)0.x/C"1.x;t/CO."K.R.Schneider:WeierstrassInstituteforAppliedAnalysisandStochastics,Mohrenstrasse39,10117Berlin,Germany.e-mail:schneider@wias-berlin.deT.

7、Wilhelm:InstituteofFreshwaterandFishEcology,Muggelseedamm310,12587Berlin,¨Germany.e-mail:wilhelm@igb-berlin.deKeywords:Quasi-steady-stateapproximation–Singularlyperturbedsystems–Trimole-cularautocatalatorMathematicsSubjectClassification(2000):92E20,34E15,34E05444K.R.Schneider,T.

8、Wilhelmforsystemsoftype(1)andreduces(1)totheregularlyp