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1、Lotka-VolterraEquationAnalysisAim²TohaveabasicunderstandingoftheLotka-Volterra(LV)Equationonhowitresultedinaunstableoscillation²Tostudytothekeyparametersfortheireffectsontheamplitudeandthefrequencyofoutputoscillation²FurthermodifytheLVequationtomodelourrealsystem²Proposeasetofrealisticpa
2、rameterswhichwillachievethesystemdesignaimLVEquationThisisanequationwhichmodelstheprey-predatorbehaviours.Itconsistsoftwodifferentialequations:dX/dt=aX-bXY--------------(1)dY/dt=cXY-dY--------------(2)Xisthenumber(concentration)ofthepreys,whereasYrepresentsthenumberofthepredators.Thefour
3、keyparameters:“a”–theexponentialgrowthrateofX;“b”–therateofXiskillingbyY;“c”–thegrowthrateofYbychancesofkillingX“d”–theexponentialdeathrateofY;Thekeyfuturesofthisequationarethepreyshaveanexponentialgrowth,anditsdeathisonlycausedbypredators.Predators’growthisdependingonitsprey,andithasits
4、uniquehalf-lifetodie.Thissystemistooideal,andoften,thedeathrateofpreywillbeincludedbyadding“-eX2”intotheequation(1).However,dependsontheaimingaccuracyofourmodelling,ifthedeathrateofXistoosmall,wecouldignoreitinourequation.Insomemodelling,theparameters“b”and“c”aretreatedthesame,i.e.asingl
5、edeathofXwillresultinonebirthofY,and“b”or“c”isjusttheencounteringrateofXandY.BasicAnalysisBymanipulatingtheequations(1)and(2),wecouldgainsomeunderstandingofthesedifferentialequations:Seemanipulationontheright.Wecouldseethatequation(3)equalstoaconstant(thisdependsontheinitialconditionofXa
6、ndY),thismeansthattheplotofYagainstXwillbeaclosedencirclement;thisexplainswhythereisanoscillation.StationaryPoint(1)Whenthesystemreachesequilibrium,i.e.therateofchangeiszero.Fromequations(1)and(2)bysettingdX/dt=0anddY/dt=0,wecouldgetaX–bXY=0andcXY–dY=0.Thesolutionstothesesetofequationswi
7、llbe(X,Y)=(d/c,a/b)or(0,0)(trivial).Combinedwiththeresultfromthebasicanalysis,thesystemwillhaveatendencytogotostationarypoint;anyencirclementwillbearoundthestationarypoint.ThesizeoftheencirclementdependsonhowfartheinitialconditionsXandYisawayfromthestationarypoint,andthes
