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1、12CoexistenceandChaosinComplexEcologiesJ.C.Sprott,J.A.Vano,J.C.Wildenberg,M.B.Anderson,J.K.NoelDepartmentsofPhysicsandMathematics,UniversityofWisconsin,Madison,Wisconsin53706,USAManycomplexdynamicalsystemsinecology,economics,neurology,andelsewhere,inwhichagentscompeteforlimitedresource
2、s,exhibitapparentlychaoticfluctuations.Thefluctuationsoccurdespitemostmodelsofthesesystemshavingtobedelicatelytunedtoproducesuchbehaviour,andthechaosoccurswhenthesystemisonthevergeofdestruction.Thispaperprovidesasimpleexplanationoftheapparentubiquityofchaosinsuchsystemsandtheirresilien
3、ce.Theproposedmechanismevolvesrobustsystemsthatareweaklychaotic,andthatexhibitadaptation,self-organization,sporadicvolatility,andpunctuatedequilibria.Anongoingdebate(1,2)amongecologistsinvolvestheobservationthatcomplexanddiverseecologiesarerelativelystable(3–5),whereasmostmodelspredict
4、instabilityandextinctionofmostspecies(6–8).Furthermore,erraticfluctuationsarecommonandareusuallyattributedtorandomexternalinfluences,buttheymaybeevidenceofchaos.Whilelaboratoryexperimentswithflourbeetlesdosuggestchaos(9,10),ecologists12haveyettodocumentanunequivocalexampleofchaosinnatu
5、re(11),perhapsbecauseofdynamicalcomplexityandmeasurementlimitations.Onedifficultywithmostsimplemodelsisthatchaostendstooccuroverarelativelynarrowrangeofparameters,boundedononesidebystablebehaviourandontheotherbyextinction.Hereweshowthatasimplemodelwithrealisticandpurelydeterministicada
6、ptationproducessystemsinwhichmostspeciescoexistwithweaklychaoticfluctuationsindependentoftheinitialconditions.Theproposedmechanismoffersanexplanationfortheobservedbiodiversityandatleastsomeofthefluctuationsandunpredictabilityinnature,anditsuggestswhyitmaybedifficulttostabilizesuchsyste
7、msbyhumanintervention.Manydifferentmathematicalmodels(12)havebeenusedtostudythedynamicsofinteractingspeciesoragentsinavarietyofdifferentcontextsandsystems.WechoosehereavariantofthegeneralizedLotka-Volterraequations(13,14)becauseouraimistoshowthattheobservedresultsareverygeneralandbec
