Lotka-Volterra模型和Bass产品销售模型的分支分析及应用

Lotka-VolterraBassBifurcationAnalysisofLotka-VolterraandBassProductDiffusionModelswithApplicationsAlgeWattageWijeratne200806O241.8519.09Lotka-VolterraBassAlgeWattageWijeratne200806DomesticClassifiedIndex:O241.8U.D.C.:519.09DissertationfortheDoctoralDegreeinScienceBifurcationAnalysisofLotka-VolterraandBassProductDiffusionModelswithApplicationsCandidate:AlgeWattageWijeratneSupervisor:ProfessorJunjieWeiAcademicDegreeAppliedfor:DoctorofScienceSpecialty:FundamentalMathematicsAffiliation:Dept.ofMathematics,SchoolofScienceDateofDefence:June,2008Degree-Conferring-Institution:HarbinInstituteofTechnologyLotka-VolterraBassLotka-VolterraBassLotka-VolterraLogistic90%Lotka-VolterraLotka-VolterrakHopfµ-BassHopfLotka-Volterra;Bass;Hopf;-;–I–HarbinInstituteofTechnologyThesisforDoctoralDegreeinScienceAbstractInthisresearch,theLotka-Volterra(LV)formulationsandthepopularBassmodelareconsideredinordertoascertaindynamicsinmarketcompetitionandprod-uctdiffusionprocessrespectively.Themainobjectiveoftheinvestigationistwofold;dealingwithapplicationsforcontemporaryeconomicissuesandperformingsometheoreticalanalysessuchasbifurcationsandtravelingwavesolutionsthatwouldbeaplausiblesubstitutefortraditionalapproachesineconomicinquiry.First,theLVsystemwasqualifiedtoexplainmarketcompetitionsbyusingarealsetofdata.Thenonlinearlogisticregressionmodelfitsthedataover90%ofaccuracy.Wecarriedoutaninverseapproachofmodelbuildingprocedurebyusingclassicalin-terpretationofmathematicalbiology.Basedontheresultsofthestabilityanalysis,weobtainedconditionsformonopoly,oligopolyandperfectcompetition.Theinves-tigationconfirmsthatthegeneralizedLVmodelconsistsofcomponentsthatmeasuremarketpower(dominance)andthecompetition,whicharecoreissuesinantitrustanalysis.AdiffusiveLotka-VolterrasystemisformulatedinChapter4thatrepresentsthedynamicsofmarketshareatduopoly.ThesystemundergoesaHopfbifurcationwhenkischosenasthebifurcationparameter.Thesteadystatesofthesystemunder-goesatranscriticalandpitchforkbifurcationwhenthediffusioncoefficientµischosenasthebifurcationparameter.ThemodifiedBassmodelwasusedtodescribedifferentsituationsofproductgrowthprocesswithdelaysandtheneighborhooddependentgrowth.WeprovidedconditionsfortheoccurrenceofHopfbifurcationandderivedalgorithmsfordeter-miningdirectionandstabilityofbifurcatingperiodicsolutions.Furthermore,theex-istence,nonexistenceanduniquenessoftravelingwavesolutionsundervariationofdelayparametershavebeenobtained.KeywordsLotka-Volterraformulations;Bassmodel;Hopfbifurcation;Transcriti-calandpitchforkbifurcation;Travelingwavesolutions.–II–ContentsContentsAbstract(inChinese).................................................................IAbstract(inEnglish)..................................................................IIListofFigures..........................................................................VIIChapter1Introduction.............................................................11.1Backgroundandsignificance...................................................11.2Lotka-Volterrasystem...........................................................31.3Bassproductdiffusionmodel...................................................41.4Bifurcationindynamicalsystems..............................................51.4.1Bifurcationingeneralterms................................................51.4.2Techniquesusedinthisresearch...........................................51.4.3Somerecentapplications...................................................51.5Objectivesoftheresearch.......................................................8Chapter2ApplicationofLotka-VolterraEquationtoRealWorldSituations92.1Introduction......................................................................92.2Literaturesurvey.................................................................92.3Experimentalprocedure.........................................................112.3.1Populationofthestudy.....................................................112.3.2SituationofChinatelecomsector..........................................132.3.3Dataanddesign.............................................................142.3.4Logisticpopulationgrowthmodel.........................................152.3.5Thecompetitionsystem....................................................172.3.6Theempiricalcompetitionsystem.........................................212.4Resultsanddiscussion..........................................................242.5Concludingremarks.............................................................25Chapter3GeneralizedLotka-VolterraSystemforMarketCompetition.....273.1Introduction......................................................................273.2TheGLVsystemandmarketcompetition......................................273.3Analysisofstability.............................................................30–III–HarbinInstituteofTechnologyThesisforDoctoralDegreeinScience3.4RepresentationofduopolybyGLVsystem:empiricalanalysistakingn=2323.5Concludingremarks..................................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