The Paradox of the Plankton Oscillations and Chaos in Multispecies Evolution

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TheParadoxofthePlankton:OscillationsandChaosinMultispeciesEvolution

JeffreyHornandJamesCattronDepartmentofMathematicsandComputerScienceNorthernMichiganUniversity1401PresqueIsleAvenueMarquette,Michigan,49855,USAjhorn@nmu.edu,jcattron@alumni.nmu.eduhttp://cs.nmu.edu/{˜jeffhorn,˜jcattron}

Abstract.Twotheoreticalecologistshaverecentlydiscoveredthatevenunderthesimplestmodelsofcompetition,threespeciesaresufficienttogeneratepermanentoscillations,andfivespeciescangeneratechaos(Huisman&Weissing,2001).Wecanshowthattheseresultscarryoverintogeneticalgorithm(GA)resourcesharingaftermakingoneminorchangeinthe“usual”sharingmethods.Wealsobringtogetherprevious,scatteredresultsshowingoscillatoryandchaoticbehaviorinthe“usual”GAsharingmethodsthemselves.Thusonecouldarguethatoscillationsandchaosarefairlyeasytogenerateonceindividualsareallowedtoinfluenceeachother,evenifsuchinteractionsareextremelysimple,nat-ural,andindirect,astheyareunderresourcesharing.Wesuggestthatgreatcarebetakenbeforeassumingthatanyparticularimplementationofresourcesharingleadstoauniqueandstableequilibrium.

1IntroductionandBackgroundPopulationbiologistshavelongknownaboutoscillationsandchaoticbehaviorinmultispeciescompetitionmodels.Butinthemuchsimplerandmoreabstractmodelsofevolutionemployedingeneticalgorithms,weusuallyassumesmoothconvergencetostableequilibria(especiallyunderselectionalone).Inparticular,thesimpleandnaturalmethodofnichingviathesharingofcommonresourceshasbeenshowntoinducestable,long-termequilibriawithmultiplespeciesco-existing(e.g.,Horn,1997).Yetevenunderthewidely-usedtechniqueofsimplydividingupfiniteresourcesamongcompetingindividuals,wecanfindevidenceofoscillatorybehaviorandnon-monotonicapproachestoequilibrium.

1.1ResourceSharingAnaturalnichingeffectisimplicitlyinducedbycompetitionforlimitedresources(i.e.,finiterewards).ThesharingprocedurewediscusshereiscommontothemodelsoftheoreticalecologistsaswellastothoseofGApractitioners.Thefundamentalstepsofresourcesharingareintuitive:

E.Cant´u-Pazetal.(Eds.):GECCO2003,LNCS2723,pp.298–309,2003.c󰀁Springer-VerlagBerlinHeidelberg2003TheParadoxofthePlankton:OscillationsandChaos2991.Foreachofthefiniteresourcesri,divideitupamongallindividualscontend-ingforit,inproportiontothestrengthsoftheirclaims.(Thustwoequallydeservingindividualsshouldbeallocatedequalamountsoftheresource.)

2.Foreachindividual,addallrewards/creditsearnedinthefirststep,andusethisamount(perhapsscaled)asthefitnessforGAselection.

3.Afteranewgenerationisproduced,replenish/renewtheresourcesandstartoveratthefirststepabove.

Thenotionsofcompetitionandnicheoverlapareeasytovisualizeinthecaseofresourcesharing.InFigure1,thecirclerepresentstheresourcescoveredbythecorrespondingspecies.Inalearningclassifiersystem,forexample,thecircleswouldrepresentthesubsetofexamplesthatarecorrectlyclassifiedbyindividualsofthatspecies1.Theresourcesintheoverlappednichesarecoveredbymultiple

species,andmustbesharedamongtheindividualsofallsuchspecies.(Theresourcescoveredbyonlyonespeciesmustalsobeshared,butonlyamongmembersofthatonespecies.)

AfBfABf

B

A

ABfBfAfACfBCf

Cf

ABCf

AB

C

CfAfBfABf

ACfBC

f

BA

Ck = 3niches

k = 2 niches

k = 3niches

Fig.1.Differentsituationsofoverlappingresourcecoverage.1Forthepurposesofthispaper,wewillusetheterm“niche”torefertosuchasubset

ofresourcescoveredbyaspecies.300J.HornandJ.CattronTobeexplicitabouttheactualsharingmechanism,wecalculatethesharedfitnessforthemembersofaspeciesAinthesituationshowninFigure1,bottom(i.e.,onlypairwisenicheoverlaps,withfABC=0).LetfA,fB,andfCbethe

objective(i.e.,unshared)fitnessesforrulesA,B,andCrespectively2.LetfAB

betheamountofresourcesintheoverlappingcoverageofspeciesAandB.That

is,fABistheamountofresourcessharedbyAandB.LetnA,nB,nCbethenumberofmembersofeachofthethreespecies,inourpopulationofsizeN(thusN=nA+nB+nC).WecalculatethesharedfitnessofA:

fsh,A=fA−fAB−fACnA+fABnA+nB+fACnA+nC.(1)Similarlyforfsh,Bandfsh,C.Tosimulateanactualexperiment(i.e.,arunofaGA),weusethewell-knownmethodofexpectedproportionequations.WeassumeagenerationalGAwithproportionateselectionandnocrossoverormutation:

PA,t+1=

nA,tfsh,A,t󰀃∀speciesX(nX,tfsh,X,t)=PA,tfsh,A,t󰀃

∀speciesX(PX,tfsh,X,t),(2)

wherePA,tmeanstheproportionofthepopulationtakenupbycopiesofspeciesAattime(generation)t(i.e.,nA,t+1/N),andfsh,A,tisthesharedfitness(e.g.,Equation1)ofAattimet.Resourcesharingisoftenincorporatedinadaptive,orsimulated,systems,including:learningclassifiersystems(LCS)(Booker,1982;Wilson,1994;Horn,Goldberg,&Deb,1994),immunesystemmodels(Smith,Forrest,&Perelson,1993),evolvingcellularautomata(Werfel,Mitchell,&Crutchfield,2000;Juill´e&Pollack,1998),andecologicalsimulations(Huberman,1988).Itisknownbyothernames,suchasexamplesharing(McCallum&Spackman,1990)andsharedsampling(Rosin&Belew,1997).