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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,inwhichagentscompeteforli
2、mitedresources,exhibitapparentlychaoticfluctuations.Thefluctuationsoccurdespitemostmodelsofthesesystemshavingtobedelicatelytunedtoproducesuchbehaviour,andthechaosoccurswhenthesystemisonthevergeofdestruction.Thispaperprovidesasimpleexplanationoftheapparentubiquityofchaosins
3、uchsystemsandtheirresilience.Theproposedmechanismevolvesrobustsystemsthatareweaklychaotic,andthatexhibitadaptation,self-organization,sporadicvolatility,andpunctuatedequilibria.Anongoingdebate(1,2)amongecologistsinvolvestheobservationthatcomplexanddiverseecologiesarerelativ
4、elystable(3–5),whereasmostmodelspredictinstabilityandextinctionofmostspecies(6–8).Furthermore,erraticfluctuationsarecommonandareusuallyattributedtorandomexternalinfluences,buttheymaybeevidenceofchaos.Whilelaboratoryexperimentswithflourbeetlesdosuggestchaos(9,10),ecologists
5、12haveyettodocumentanunequivocalexampleofchaosinnature(11),perhapsbecauseofdynamicalcomplexityandmeasurementlimitations.Onedifficultywithmostsimplemodelsisthatchaostendstooccuroverarelativelynarrowrangeofparameters,boundedononesidebystablebehaviourandontheotherbyextinction
6、.Hereweshowthatasimplemodelwithrealisticandpurelydeterministicadaptationproducessystemsinwhichmostspeciescoexistwithweaklychaoticfluctuationsindependentoftheinitialconditions.Theproposedmechanismoffersanexplanationfortheobservedbiodiversityandatleastsomeofthefluctuationsan
7、dunpredictabilityinnature,anditsuggestswhyitmaybedifficulttostabilizesuchsystemsbyhumanintervention.Manydifferentmathematicalmodels(12)havebeenusedtostudythedynamicsofinteractingspeciesoragentsinavarietyofdifferentcontextsandsystems.WechoosehereavariantofthegeneralizedLotk
8、a-Volterraequations(13,14)becauseouraimistoshowthattheobservedresultsareverygeneralandbec
