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1、13CoexistenceandchaosincomplexecologiesJ.C.Sprott,J.A.Vano,J.C.Wildenberg,M.B.Anderson&J.K.NoelDepartmentsofPhysicsandMathematics,UniversityofWisconsin,Madison,Wisconsin53706,USAAnongoingdebate1,2amongecologistsinvolvestheobservationthatcomplexanddiverseeco
2、logiesarerelativelystable3-5,whereasmostmodelspredictinstabilityandextinctionofmostspecies6-8.Furthermore,erraticfluctuationsarecommonandareusuallyattributedtorandomexternalinfluences,buttheymaybeevidenceofchaos.Whilelaboratoryexperimentswithflourbeetlesdos
3、uggestchaos9,10,ecologistshaveyettodocumentanunequivocalexampleofchaosinnature11,perhapsbecauseofdynamicalcomplexityandmeasurementlimitations.Onedifficultywithmostsimplemodelsisthatchaostendstooccuroverarelativelynarrowrangeofparameters,boundedononesidebyst
4、ablebehaviourandontheotherbyextinction.Hereweshowthatasimplemodelwithrealisticandpurelydeterministicadaptationproducessystemsinwhichmostspeciescoexistwithweaklychaoticfluctuationsindependentoftheinitialconditions.Theproposedmechanismoffersanexplanationforth
5、eobservedbiodiversityandatleastsomeofthefluctuationsandunpredictability,anditsuggestswhyitmaybedifficulttostabilizesuchsystemsbyhumanintervention.Manydifferentmathematicalmodels12havebeenusedtostudythedynamicsofinteractingspeciesoragentsinavarietyofdifferen
6、tcontextsandsystems.WechoosehereavariantofthegeneralizedLotka-Volterraequations13,14becauseouraimistoshowthattheobservedresultsareverygeneralandbecausethechosenmodelcanbeviewedasthefirstapproximationinaTaylorseriesexpansionforawideclassofmodels15.13Weconsid
7、erasystemconsistingofNcompetingspecieswithpopulationxifori=1toNsatisfying(1)whereallthebiology(economics,sociology,etc.)iscontainedinthevectorofgrowthratesriandthematrixofinteractionsaij,allelementsofwhicharepositivetoindicatecompetition.Theelementaijdescri
8、bestheaverageextenttowhich(membersof)speciesicompeteswith(membersof)speciesj.Notethatasanygivenspeciesapproachesextinction,theaveragingoftheaijelementsforthisspecieswilloccuroversmallerandsmallerpopula
