Y. Geometric stability switch criteria in delay di

GEOMETRICSTABILITYSWITCHCRITERIAINDELAYDIFFERENTIALSYSTEMSWITHDELAYDEPENDENTPARAMETERS∗EDOARDOBERETTA†ANDYANGKUANG‡SIAMJ.MATH.ANAL.c2002SocietyforIndustrialandAppliedMathematicsVol.33,No.5,pp.1144–1165Abstract.Inmostapplicationsofdelaydifferentialequationsinpopulationdynamics,theneedofincorporationoftimedelaysisoftentheresultoftheexistenceofsomestagestructure.Sincethethrough-stagesurvivalrateisoftenafunctionoftimedelays,itiseasytoconceivethatthesemodelsmayinvolvesomedelaydependentparameters.Thepresenceofsuchparametersoftengreatlycomplicatesthetaskofananalyticalstudyofsuchmodels.Themainobjectiveofthispaperistoprovidepracticalguidelinesthatcombinegraphicalinformationwithanalyticalworktoeffectivelystudythelocalstabilityofsomemodelsinvolvingdelaydependentparameters.Specifically,weshallshowthatthestabilityofagivensteadystateissimplydeterminedbythegraphsofsomefunctionsofτwhichcanbeexpressedexplicitlyandthuscanbeeasilydepictedbyMapleandotherpopularsoftware.Infact,formostapplicationproblems,weneedonlylookatonesuchfunctionandlocateitszeros.Thisfunctionoftenhasonlytwozeros,providingthresholdsforstabilityswitches.Thecommonscenarioisthatastimedelayincreases,stabilitychangesfromstabletounstabletostable,implyingthatalargedelaycanbestabilizing.Thisscenariooftencontradictstheoneprovidedbysimilarmodelswithonlydelayindependentparameters.Keywords.delaydifferentialequations,stabilityswitch,characteristicequations,stagestruc-ture,populationmodelsAMSsubjectclassifications.34K18,34K20,92D25PII.S00361410003760861.Introduction.Duetothefactthatactionsandreactionstaketimetotakeeffectinreal-lifeproblems,oneoftenintroducestimedelaysinthevariablesbeingmodeled.Thisoftenyieldsdelaydifferentialanddelaydifferencemodels[11],[21],[19].Someofthesemodelshavedelaydependentparameters(forexample,[1],[2],[3],[4],[9],[10],[22]),whilemostofthemcontainonlyparametersthatareindependentoftimedelays.Inmostapplicationsofdelaydifferentialequationsinpopulationdynamics,theneedofincorporationofatimedelayisoftentheresultoftheexistenceofsomestagestructure[1],[3],[10],[11],[12],[15].Indeed,justabouteverypopulationgoesthroughsomedistinctlifestages[23],[18].Sincethethrough-stagesurvivalrateisoftenafunctionofatimedelay,itisthuseasytoconceivethatthesemodelswillinevitablyinvolvesomedelaydependentparameters.Inviewofthefactthatitisoftendifficulttoanalyticallystudymodelswithdelaydependentparametersevenifonlyasinglediscretedelayispresent,itisnaturaltoresorttothehelpofcomputerprograms.Themainobjectiveofthispaperistoprovidepracticalguidelinesthatcombinegraphicalinformationwithanalyticalworktoeffectivelystudythelocalstabilityofmodelsinvolvingdelaydependentparameters.Toapplyourresults,oneneedonlyperformsomeroutinecomputation(usingouranalyticalcriteria)andgeneratesomesimplegraphswhichcanbeeasilyproducedbypopularsoftwaresuchasMaple.Theresultsalsocanbereadilyconfirmedbysome∗ReceivedbytheeditorsJuly28,2000;acceptedforpublication(inrevisedform)September4,2001;publishedelectronicallyFebruary14,2002.†IstitutodiBiomatematica,UniversitadiUrbino,I-61029Urbino,Italy([e-mail]).‡DepartmentofMathematics,ArizonaStateUniversity,Tempe,AZ85287([e-mail]).Thisauthor’sworkwaspartiallysupportedbyNSFgrantDMS-0077790.1144GEOMETRICSTABILITYSWITCHCRITERIA1145selectivesimulationsusingthefreelyavailableanduserfriendlysoftwareXPP.Nootherprogrammingskillisrequired.Specifically,weshallshowthatthestabilityofagivensteadystateissimplydeterminedbythegraphsofsomefunctionsofτwhichcanbeexpressedexplicitlyandthuscanbeeasilydepictedbyMapleandotherpopularsoftware.Infact,formostapplicationproblems,weneedonlylookatonesuchfunctionandlocateitszeros.Thisfunctionoftenhasonlytwozeros,providingthresholdsforstabilityswitches.Thecommonscenarioisthatastimedelayincreases,stabilitychangesfromstabletounstabletostable.Wehopethisworkwillshowthatitisimportantandpossibletosystematicallystudylocalstabilityaspectsofsomemodelswithdelaydependentparameters.Inthenextsection,wepresentageneralgeometriccriterionthat,theoreticallyspeaking,canbeappliedtomodelswithmanydelays,orevendistributeddelays[5],[7].Thisisfollowedbyasectiondealingwiththesimplecaseofafirstordercharacteristicequation,providingmoreuserfriendlygeometricandanalyticcriteriaforstabilityswitches.Insection4,weaccomplishthesameforthesecondordercase.Theanalyt-icalcriteriaprovidedforthefirstandsecondordercasescanbeusedtoobtainsomeinsightfulanalyticalstatementsandcanbehelpfulforconductingsimulations.Ex-amplesareprovidedforbothfirstandsecondordercasestoillustratetheapplicationsofourcriteria.Adiscussionsectionconcludesthepaper.2.Ageneralgeometriccriterion.InthissectionwestudytheoccurrenceofanypossiblestabilityswitchingresultingfromtheincreaseofvalueofthetimedelayτforthegeneralcharacteristicequationD(λ,τ)=0.(2.1)HereD(λ,τ)=Pn(λ,τ)+Qm(λ,τ)e−λτ(2.2)andPn(λ,τ)=n
k=0pk(τ)λk,Qm(λ,τ)=m
k=0qk(τ)λk.(2.3)In(2.3),n,m∈N0,nm,andpk(·),qk(·):R+0→Rarecontinuousanddifferen-tiablefunctionsofτsuchthatPn(0,τ)+Qm(0,τ)=p0(τ)+q0(τ)=0∀τ∈R+0,(2.4)i.e.,λ=0isno