Symbology-from-set-theory-applied-to-ecological-sy

arXiv:q-bio/0506040v1[q-bio.PE]28Jun2005Symbologyfromsettheoryappliedtoecologicalsystems:Gause’sexclusionprincipleandapplicationsJ.C.Floresa,baUniversidaddeTarapac´a,DepartamentodeF´isica,Casilla7-D,Arica-ChilebCentrodeEstudiosdelHombreenelDesierto,CIHDE,Casilla7-D,Arica-ChileAbstract:Weintroduceasymbolicrepresentationlikesettheorytoconsiderecologicinteractionsbetweenspecies(ECOSET).Theecologicex-clusionprinciple(Gause)isputinasymbolicwayandusedasoperationaltooltoconsidermorecomplexcaseslikeinteractionwithsterilespecies(SITtechnique),twospecieswithtwosuperposedsources(nichedifferentiation)andN+PspeciescompetingbyNresources,etc.Displacement(regionalorcharacters)isalsoconsideredbyusingthisbasictool.Oursymbolicnotationgivesusanoperativeandeasywaytoconsiderelementaryprocessinecol-ogy.Someexperimentaldata(laboratoryorfield)forecologicprocessarere-consideredundertheopticofthisset-theory.Keywords:Coexistenceandcompetition;Foodwebtheory;Ecology;Settheory.1IIntroduction.-Interactionsbetweenspeciesinecologyisaftersometimetheobjectofstudyofmathematicalbranches(Gertsevetal(2004)andreferencestherein).Forinstance,mathematicalmodelsforpredator-preyareusuallytreatedwithnonlinearcoupleddifferentialequationslikeLotka-Volterra(Begonetal(1999),Murray(1993),andreferencestherein).Itistruethat,asageneralrule,modellationofecologicalsystemsisadifficulttasksincecomplexityinbiologicalsciencesisalmostalwayspresent.Forinstance,themoreknownapplicationofLotka-Volterrasystem,thatistheHare-Lynxpredator-preydatarecorderbyHudsonBayCompanyin1953,presentssometroubles.Theexpectedperiodicsolutionbetweenpreyandpredatorisnotdeterminedfromdata(Murray(1993)).Thereasonsarenotclearbutonecanexpectacomplexdynamicsforrealpredator-preysysteminanotisolatedregion.Veryrefinedexperimentinlaboratoriesareactuallyrealized(Costantinoetal(1995)),neverthelessthemoreimpressivelaboratoryexperimentswerecarried-outsometimeagobyecologistG.F.GausewithprotozoanParame-siumspecies.Theseexperimentshavedeterminedsomegeneralprinciplesappliedtointer-competitionandcoexistenceinecology.Themorebasicforusisthestatementthat:whenwehavetwospeciescompetingby(exactly)thesamenichethenoneofthemdisappear(Gause(1934)).Thatis,finally,onespeciesfulltheniche.Thestatementisveryrestrictivesinceitrequiresquitespecialconditions(sectionIV)andinthispaperitwillbeusedasabasicprincipletoconsidermorecomplexsituations.MathematicalmodelslikeLotka-Volterrafortwospeciesincompetitionpredictcoexistenceforsomerangeofparameters(highintraspecificcompe-tition).Infact,anymathematicalsystemwithatractors(out-sideoftheaxis)inthistwodimensionalphase-spacemakesthesameprediction.So,forusthismathematicalresultisnotproofofrealviolationoftheprinciple.Gause’sexclusionprinciplewillbeassumedforinterspecificcompetitionasabasicstatementinthispaper.Nevertheless,weaccordthatthedistinctionbetweentheminimumamountofnichedifferentiation(inrealecologicalsys-tem)toproduceitsbreakisadifficultpointtoconsiderinpractice.Foranappropriatediscussionseereference(Begonetal(1999)).Inthispaperwewillconsideramathematicalmodelingforecologicalsystemsbutusingasymbology,calledECOSET,similartobasicsettheory.Thisschemahastheadvantageofacondensednotationforavarietyof2ecologicalinteractingsystems.Therearesomesimilaritieswithusualsettheorybutalsosomedifferences.Forinstance,Gause’sexclusionprinciple(16)hasnotequivalenceintheusualsettheory.Inourconstruction,Gause’sexclusionprinciplewillbeusedasabasicoperationaltooltoconsidercomplexsituationslikemorethanonespeciesandmorethanoneresource.AgreatpartoffieldandlaboratoryexamplesfoundinthisarticlewereholdfromreferenceBegonetal(1999).Inthispaper,specieswillberepresentedbycapitalletterlikeA,B,...etc.WewillusethesymbolSonlyforaprimarysourceatthebasis(consume)ofaecologicalchain.WewillgivetoSastrongersense:itwilldesignadefinedensembleofsourceswhichmakeviablethedevelopmentofspecies.Namely,itisconsideredinthesenseofanecologicalniche.Weassumethatthisprimaryresourceisauto-sustainedorexternallysustained.Theabsenceofspeciesinagivenregionwillbedenotedbyφinanalogywiththesymbolofusualsettheory.IISymbologyfordepredation()andbasicdefinitions.-ConsideraprimaryresourceSandspeciesA,B,C...φ.ThenotationAB(BconsumesA),(1)means:speciesBexploitsspeciesAasaresourceinasenseofdepredation.Inthisway,abasicchainbecomesforinstanceSABφ(abasicdepredationchain),(2)namely,speciesBexploitsAasaresourceand,Aexploitstheprimaryre-sourceS.NotethatthespeciesφattheendofthechainmeansthatBisnotpreyforothers.Fortwono-interactingspeciesAandB(alsoforecologicalprocess)inagivenecologicalregionwewriteA⊕B.Whentwospecies(AandB)consumethesameprimaryresource(S),inanotdependingway,wewrite3S(A⊕B),(independentdepredation).(3)Whentwospecies(AandB)exploitthesameresource(S)inainterde-pendentwaywewriteS(A⊗B)(interdependingdepredation).(4)ThenotationSA⊗B(withoutbraces)means:speciesAconsumesSwiththehelpofspeciesB.Notethatwhenoneoftheinterdependentspeciesifφthen,afteratime,nodepredationonSmustoccur.Inasymbolicway,{S(A⊗φ)}⇒Sφ,(5)wherethesymbol⇒hasatemporalinterpretationor,itdefinesatemp