stability of periodic solutions for a model of genetic repression with delays

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1、,JournalofJ.Math.Biology(1985)22:137-144mathematicalWologyOSpringer-Verlag1985StabilityofperiodicsolutionsforamodelofgeneticrepressionwithdelaysJ.M.Mahaffy***DepartmentofMathematics,HarveyMuddCollege,Claremont,CA91711,USAAbstract.Atechniqueisdiscussedforlo

2、catingtheHopfbifurcationofann-dimensionalsystemofdelaydifferentialequationswhicharisesfromamodelforcontrolofproteinbiosynthesis.CertainparametervaluesareshowntoallowaHopfbifurcationtoperiodicorbits.AttheHopfbifurcationtheperiodicorbitsareshowntobestableeit

3、heranalyticallyornumericallydependingontheparametervalues.Keywords:Hopfbifurcation--Delaydifferentialequations--Stabilityoforbits--Proteinbiosynthesis--Geneticrepression1.IntroductionTherehavebeennumerousstudiesofperiodicenzymesynthesesforpopulationsofprok

4、aryoticandeukaryoticcells.A,summaryofsomeoftheexperimentalstudiescanbefoundinTyson[13,14].Theoscillationsareobservedinbothsynchronousandasynchronouscellcultureswhichsuggeststhattheautogenousoscillationsinenzymeactivitymaybecontrolledbyanegativefeedbacksyst

5、emforthesynthesisoftheenzyme.Avarietyofoscillatoryphenomenainbiologyarethoughttoarisebecauseofnegativefeedback,fromhighfrequencyneuralactivitytolongerperiodcircadianrhythmsandendocrineoscillations[15].Therehasbeenconsiderableinterestastowhetherornottheclas

6、sicalmodelofrepressionproposedbyJacobandMonod[6]couldaccountforoscillations.Inthispaperwedeterminewhenoscillationscanariseinaclassofmodelsforgeneticrepressionwithtimedelaysandshowthatthereisastableperiodicorbit.Astableperiodicsolutionisonewhichcouldbeobser

7、vedexperimentally.Goodwin[4,5]proposedamathematicalmodelforgeneticrepressionwhichwasdevelopedfromthetheoryofJacobandMonodusingbiochemicalkinetics.Thismodelhasbeenextendedandstudiedextensively[seee.g.,2,9,10,13,15].Previousworkhasbeenmainlyconcernedwiththee

8、xistenceofperiodicsolutionstothesystemsofdifferentialequationsforthismodel.Inthispaperweareinterestedinstudyingthestabilityofsmallamplitudeperiodicsolutions.Mahaffy[8]showedtheexistenceofperiodicsolutionsfora