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

《stability of periodic solutions for a model of genetic repression with delays》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。

,JournalofJ.Math.Biology(1985)22:137-144mathematicalWologyOSpringer-Verlag1985StabilityofperiodicsolutionsforamodelofgeneticrepressionwithdelaysJ.M.Mahaffy***DepartmentofMathematics,HarveyMuddCollege,Claremont,CA91711,USAAbstract.AtechniqueisdiscussedforlocatingtheHopfbifurcationofann-dimensionalsystemofdelaydifferentialequationswhicharisesfromamodelforcontrolofproteinbiosynthesis.CertainparametervaluesareshowntoallowaHopfbifurcationtoperiodicorbits.AttheHopfbifurcationtheperiodicorbitsareshowntobestableeitheranalyticallyornumericallydependingontheparametervalues.Keywords:Hopfbifurcation--Delaydifferentialequations--Stabilityoforbits--Proteinbiosynthesis--Geneticrepression1.IntroductionTherehavebeennumerousstudiesofperiodicenzymesynthesesforpopulationsofprokaryoticandeukaryoticcells.A,summaryofsomeoftheexperimentalstudiescanbefoundinTyson[13,14].Theoscillationsareobservedinbothsynchronousandasynchronouscellcultureswhichsuggeststhattheautogenousoscillationsinenzymeactivitymaybecontrolledbyanegativefeedbacksystemforthesynthesisoftheenzyme.Avarietyofoscillatoryphenomenainbiologyarethoughttoarisebecauseofnegativefeedback,fromhighfrequencyneuralactivitytolongerperiodcircadianrhythmsandendocrineoscillations[15].TherehasbeenconsiderableinterestastowhetherornottheclassicalmodelofrepressionproposedbyJacobandMonod[6]couldaccountforoscillations.Inthispaperwedeterminewhenoscillationscanariseinaclassofmodelsforgeneticrepressionwithtimedelaysandshowthatthereisastableperiodicorbit.Astableperiodicsolutionisonewhichcouldbeobservedexperimentally.Goodwin[4,5]proposedamathematicalmodelforgeneticrepressionwhichwasdevelopedfromthetheoryofJacobandMonodusingbiochemicalkinetics.Thismodelhasbeenextendedandstudiedextensively[seee.g.,2,9,10,13,15].Previousworkhasbeenmainlyconcernedwiththeexistenceofperiodicsolutionstothesystemsofdifferentialequationsforthismodel.Inthispaperweareinterestedinstudyingthestabilityofsmallamplitudeperiodicsolutions.Mahaffy[8]showedtheexistenceofperiodicsolutionsforann-dimensionalmodelofrepressionwithdelaysandalsodemonstratedatechniqueforcalculatingwhen*OnleavefromNorthCarolinaStateUniversity;**SupportedinpartbyN.S.F.Grant#MCS81-02828 138J.M.MahaffyasmallamplitudeperiodicsolutionfromaHopfbifurcationoccurs.Thistech-niqueforfindingaHopfbifurcationiscombinedwithatechniquedevelopedbyStech[1l,12]fordeterminingthestabilityofaHopfbifurcation.InSect.2wepresentthemodelandfindaregionwhereperiodicsolutionsmayexistandthenuseamethodforlocatingwhereaHopfbifurcationoccurs.AtheoremisgiventhatdeterminesaregionwhereaHopfbifurcationcanoccurasthedelayvaries.Foracollectionofexamplesthecriticaldelayiscomputednumerically.InSect.3wepresentformulaewhichallowonetocomputethestabilityoftheHopfbifurcationforthemodel.ForaparticularrangeofparametervaluesweshowthattheHopfbifurcationisalwaysstable.Wealsogivenumericalresultswhichsuggestthatforthen-dimensionalrepressionmodeltheHopfbifurcationalwaysresultsinastableperiodicorbit.2.TheHopfbifurcationinageneticrepressionmodelThemathematicalmodelforgeneticcontrolbynegativefeedbackorrepressionwasfirstderivedbyGoodwin[5].Usingnon-dimensionalvariableswepresentthen-dimensionalmodelforrepressionwithadiscretedelay,r,representingtranscriptionandtranslation.Itisgivenbythefollowingsystemofdifferentialequations:1Xl(t)-blxl--blxl(t)l+[x,(t-r)+~,]p=-f(x,(t-r))-blxl(t),(2.1)Yci(t)=xi_l(t)-bixi(t),i=2,...,n,wherebirepresentnon-dimensionaldecayrates,pistheHillcoefficientforrepression,andffiaretheconstantsusedtotranslatetheequilibriumofthemodeltotheoriginandcanbefoundfromtheuniquesolutiontothesystemofequationsgivenbyf(0)=0andXi-l=big,,i=2,...,n.Thenonlinearsystem(2.1)maybewrittenYc(t)=Ax(t)+Bx(t-r)+H(x(t-r)),(2.2)whereA=[a0]isannxnmatrixwhoseonlynon-zeroelementsarea~i=-bionthediagonalandl'sonthesubdiagonalandBisannxnmatrixwhoseonlynon-zeroelementisf'(0)inthe(1,n)position.H(O)isanonlinearn-vectorfunctionwithanexpansionoftheform3H(rE4)j=2where~(g,)aretheappropriatesymmetric,boundedj-linearformsontheBanachspaceC([-r,0];R")withtheusualsupnorm.Amoredetaileddescriptionof/-/j(4')ispresentedinSect.3. Stabilityofperiodicsolutionsforamodelofgeneticrepressionwithdelays139Thecharacteristicequationfor(2.1)isgivenbydet[A+Be-xr-hi]=0,whichuponexpansionbecomesI~(b,+A)-f'(0)e-Xr=0(2.3)i=1wheref'(0)=-[p2~-1/(1+X~)2].Ifweassumethatforr=0(theordinarydifferen-tialequationcase)allsolutionsAof(2.3)haveReA<0,thenitwasshowninMahaffy[7]thatwhenever-f'(0)>flb~=-/3(2.4)i=lthenthereexistsanto>0suchthatforr=ro,(2.3)hastwopurelyimaginarysolutions=+iz,oandallothersolutionsAhaveReA<0.IfweconsiderrasthebifurcationparameterthenasrincreaseswehaveatransversecrossingoftheimaginaryaxisbyapairofeigenvaluesA,thusaHopfbifurcationoccurs.Iff'(0)0suchthatthesystem(2.1)islocallyunstable.Wesummarizeourfindingsinthefollowingtheorem:Theorem2.1.Letflo---p-l(p-1)(p+I)/p.If0~Osuchthat(2.1)islocallyunstableforallr>ro.Iffl>tflo,thenallsolutionsAof(2.3)haveReA<0forallr~O.Proof.FromMahaffy[8]weseethatthecriticalvaluefloiswhen/30=If'(0)l.Solvingfortheequilibriumsolutionwefindthatflxn=1/(1+2~),soif(0)=-pWl~-p-kl_p~-l/(1+~)2=_pfl2~+l.HenceI=ppoXonbut1=flo~on+poXonfromtheequilibriumsolution.Combiningtheseweseethat/30=p-l(p_1)~p+l~/pandXon=(p-1).-~/p 140J.M.MahaffyNowfromaboveweseethat~rllIf'(o)l--p/3=~zg+'--p/3L~-"]=/3+/3[(P-l)-p/(l+.~P.)].If/3~o.whichimpliesx.-o>(p_1)-1FromthisweseethatIf'(0)[=/3+/3[(p-1)-p/(1+~z~.)]>/3+/3[(p-1)-p/(1+(p-1)-1)]--/3.Iffl>/30,thenasimilarargumentgivesIf'(0)[0suchthatif13e(13o-e,13o),then(2.1)hasasmallamplitudestableperiodicsolution.Proof.Theorem2.1showsthatfor13e(13o-el,flo)forsomeel>0thereexistsacriticalto>0suchthataHopfbifurcationoccurs.BythedefinitionofVoand(2.5)onecanshowthatas13~13o,ro~.AsIf'(0)1=13oat13o,thenwiththe-p+l_equilibriumsolutiononecanshowthatXo,,-1/p13oand~o,=(p-1)/p13o.Substitutingtheseinto(3.1)and(3.2)wecanshowthat132p(v-3)h2(0)-2(p-1)133p2(p--2)(p--7)h3(0)=6(p-1)2(3.5)at/3=13o. 144J.M.MahaffyAt/3=/3othereisnoHopfbifurcationwithrespecttotheparameterr;however,itiseasilyseenthatthecoefficientsh2(0)andh3(0)varycontinuouslywithrespecttotheparameter/3.For/3near/30theargumentprincipleusedintheproofofTheorem2.1givesUotobeverysmallas/3~/30;however,Uoro~7ras/3~/30fromtheargumentprincipleandourdefinitionsofu0andr0.Fromtheaboveinformationweexamine7at/3=/30.At/3=/30,u0ro=~",Uo=0,andf'(0)=-/30,sowemaysubstitutethesevalueswith(3.5)into(3.4)andobtain/33op2(p--2)(p-7)2/34p2(p-3)24/34p2(p-3)23,K-6(p-1)22/304(p-1)22/304(p-1)2/33p2(p+1)4(p-1)'whichisstrictlynegative.For/3