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UniformUnifor

    Uniform

App1.Math.J.ChineseUniv.

    ;2008,23(4):410412

    ;Uniformpersistenceofmultispeciesecological

    ;competitionpredator-praysystemwithHollingIIItype

    ;functionalresponse

    ;LIZhanLILing:xiaoWANGKe.

    ;Abstract.ThemainpurposeofthispaperistostudythepersistenceofthegenerMmultispecies

    ;competitionpredatorpraysystemwithHollingIIItypefunctionalresponse.Inthissystem,

    ;the

    ;competitionamongpredatorspeciesandamongpreyspeciesaresimultaneouslyconsidered.By

    ;usingthecomparisontheoryandqualitativeanalysis,thesufficientconditionsforuniformstrong

    ;persistenceareobtained.

    ;?1Introduction

    ;Recently,theecologicalpredatorpreysystemswithHollingIIItypefunctionalresponse ;havebeenstudiedbymanyauthors,theirworks[1-5]areconcernedwithtwoorthreespecies ;models.However,innature,multispeciessystemiscommonanduniversa1.Inaddition,the ;stateismorecomplicatedandrealisticthanthetwoorthreesystems.Recentpaper[6gives

    ;thesufficientconditionsforuniformpersistenceinanonautonomousmultispeciesecolo

    gical

    ;competitionpredatorpraysystemwithHollingIIItypefunctionalresponse.

    ;Themainpurposeofthispaperistostudythepersistenceofthegeneralmultispecies ;competitionpredatorpraysystemwithHollingIIItypefunctionalresponse.Inthissystem,

    ;thecompetitionamongpredatorspeciesandamongpreyspeciesaresimultaneouslyconsidered.

    ;Forrealityweimprovetheresultsofpapers68

    IAndatthesametimeweobtainconditions

    ;foruniformstrongpersistence.

    ;Received:2007-0528

    ;MRSubjectClassification:34K15,34C25,92D25

    ;Keywords:competitionpredatorpraysystem,HollingIIItypefunctionalresponse,persistence

    ;DigitalObjectIdentmer(D0I):10.1007/s117660081886-1

    ;SupportedbytheNationalNaturalScienceFoundationofChina(10701020) ;LIZhan,eta1.Uniformpersistenceofmultispeeiesecologicalcompetitionpredatorpra

ysystemwith--?411

    ;?2Mainresultsandproofs

    ;Wewillinvestigatethefollowingnonautonomousmultispeciesecologicalcompetitionpredator-

    ;praysystemofdifferentialequations.Wetakexi(t)asthedensityofpreyspeciesxiattimet,

    ;yi(t)asthedensityofpredatorspeciesYiattimet.Weproposethefollowingmodelgoverning

    ;thesystem:

    ;=)?.t

    ;n

    ;=yj[-r~(t)+?k=l

    ;(t)xk(t)

    ;djk(t)x~

    ;,j%(t)+x?eJ()()]=1

    ;withinitialdatagivenby

    ;(0)=Xio0;(0)=y3o0,i=1,2,…,n;J=1,2,…,m

    ;where6t(t),(),0(),cij(t),(t),do(),e~j(t)arecontinuous,nonnegativefunctionsand ;bi(t),aij(t),eij(),arestrictlypositive.

    ;Definition1.Wecallthesystemx(t)tobeauniformlystrongpersistence,ifx(t)satisfies ;0<1.

    ;im.

    ;infx(t)limsupx(t)<+?.to.t_+oo

    ;Lemma1.BoththepositiveandnonnegativeconesofR+areinvariantwithrespectto ;system(M).ItfollowsfromLemma1,thatanysolutionofsystem(M)whichhasanonnegative

    ;initialconditionremainsnonnegative.

    ;Welet0<liminft---,+~f(t)limsupt_?+..f(t)=,<...Fromthe,thereexists>0suchthat

    ;bi(t)6+,ai~(t)aii一?,Vt.

    ;Substitutingtheseexpressionsintothefirstequationofsystem(M),weget ;t?xi[bi(t)ai~(t)xit[+?一(aiiE)t,Vt.

    ;NextweusetheComparisonTheorem,hencewehavelimsupt+o.Xia口上

    ;ii.

    ;.

    ;:=Pi.Soforany

    ;>0.thereexistsT2>T1>0suchthat

    ;+,VtT2.(2)

    ;Similarly,from(1)andthesecondequationofsystem(M),weseethatforanyE>0,there ;exists>T2>0suchthat

    ;叻协[rj(t)+?d~k(t)ejj(t)y~

    ;【一(一?1++E)(ejj+e)Yj]VtT3

    ;?d~k--r,

    ;BytheComparisonTheoremagainweobtainlimsupt-?+oo:=qj?Thus,for ;m??丽

    ;412App1.Math.J.ChineseUniv.Vo1.23.No.4

;anyE>0,thereexists>?>0suchthat

    ;Yjqj+E,VtT4.(3)

    ;Nowtheexpressions(1),(2),(3)andthefirstequationofsystem(M)leadtothefollowing:for

    ;anyg>0,thereexists>T4>0suchthatforanyt?,wehave

    ;啪一((p一列.

    ;UsingtheComparisonTheoremagain,thenitfollowsthat ;l

    ;

    ;imin

    ;..

    ;f一一一,…,n

    ;Soforany>0,thereexistsT>>0suchthat

    ;t()?i.

    ;IfOli>0,thenfrom(2),(4)andthesecondequationofsystem(M),itiseasytoobtain

    ;)+?

    ;k=l

    ;

    ;?

    ;Consequently,wehave

    ;l

    ;

    ;imin

    ;..

    ;f1

    ;+

    ;糍一ej]==,,2j…,m.

    ;Moreover,itisobviousthatifgJ>0,then<Pi;andifqj>0,{>0,then< ;gJ(i=1,2,…,;J=1,2,…,m).Soweonlysupposethat>0,OLi>0,>0(i=

    ;1,2,…,;J=1,2,…,m).Thentheproofiscompleted.

    ;Rferences

    ;1LiLiming.Stationaryoscillationofhigher-dimensionalnon-autonomoussystems,Acta

    Math

    ;ApplSinica,1989,12(2):196204.

    ;2JiaJianwen.Persistenceandperiodicsolutionofnon-autonomousandpredatorpreysy

    stems

    ;withHollingIIIfunctionalresponse,JBiomath,2001,16(1):59-62. ;3SunShulin,YuanCunde.StudyofthreemixedspeciesmodelwithHollingIIIfunctionalre

    sponse

    ;andperiodiccoefficients,JournalofShanxiTeacher’SUniversity,2001,15(3):17-20.

    ;4XiongZuoliang.StudyontheparametersmodelintheIIfunctionalresponseperiodicofthr

    ee

    ;populationsmodel,JBiomath,1998,13(1):3942.

    ;5XieXiangdong,CaiSuilin.OntheuniquenessoflimitcycleforaclassofHollingmodelwit

    h

;functionalresponse,JBiomath,1994,9(1):8184.

    ;6ChenChao,ChenFengde.Conditionsforglobalattractivityofmultispeciesecologicalcompetition

    ;predatorsystemwithHollingIIItypefunctionalresponse,JBiomath,2004,19(2):136-140. ;7MukherjeeDebasis.Persistenceandglobalstabilityofapopulationinapollutedenvironment

    ;withdelay,JBiologicalSystems,2002,10:225232.

    ;8MaZ,CuiG,WangW.Persistenceandextinctionofapopulationinapollutedenvironment, ;‘MathBiosci,1990,101:75—81.

    ;1Dept.ofMath.,ZhengzhouUniv.,Zhengzhou450001,China

    ;2ScienceCollege,HenanUniv.ofSci.andTech.,Luoyang471003,China

    ;3Dept.ofMath.,HarbinInst.ofTech.,Weihai264209,China

    ;

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