Extremal Coupled Map Lattices

We propose a model for co-evolving ecosystems that takes into account two levels of description of an organism, for instance genotype and phenotype. Performance at the macroscopic level forces mutations at the microscopic level. These, in turn, affect the

ExtremalCoupledMapLattices

GuillermoAbramson andJos´eLuisVega

Max-Planck-Insitutf¨urPhysikkomplexerSysteme,N¨othnitzerStr.38,01187Dresden,Germany

(toappearinTheEuropeanPhysicalJournalB,inpress,1999)

(February5,2008)

Weproposeamodelforco-evolvingecosystemsthattakesintoaccounttwolevelsofdescription

ofanorganism,forinstancegenotypeandphenotype.Performanceatthemacroscopiclevelforcesmutationsatthemicroscopiclevel.These,inturn,a ectthedynamicsofthemacroscopicvariables.Insomeregionsofparameterspace,thesystemself-organisesintoastatewithlocalisedactivityandpowerlawdistributions.

arXiv:adap-org/9905001v1 30 Apr 1999

PACS:05.45.+b,64.60.Lx,87.10.+e

Complexextendedsystemsshowingalackofscaleintheirfeaturesappeartobewidespreadinnature,beingasperseasearthquakes[1,2],creepphenomena[3],ma-terialfracturing uiddisplacementinporousme-diainterfacegrowthrivernetworksandbiologicalevolutionAtvariancewithequi-libriumstatisticalmechanics,thesesystemsdonotneedany netuningofaparametertobeinacriticalstate.Toexplainthisbehaviour,Bak,TangandWiesenfeldin-troducedtheconceptofself-organised(SOC)throughthesimplesand-pilemodel[18,19].Inrecentyears,severalmodelswithextremaldynamicshavebeenshowntoexhibitSOCinthepresenceofasu cientlydecorrelatedsignalSeveralyearsago,BakandSneppen(BS)proposedaSOCmodel[21]fortheco-evolutionofspecies.IntheBSmodeleachspeciesoccupiesasiteonalattice.Eachsitejisassigneda tness,namelyarandomnumberbetween0and1.Ateachtimestepinthesimulationthesmallest tnessisfound.Thenthe tnessesoftheminimumandofitsnearestneighboursareupdatedaccordingtotherule

fn+1=F(fn)

(1)

NfollowsapowerlawdistributionN(s)～s τwheresisthesizeoftheavalancheandτ～1.07Otherquantitiesalsoexhibitapowerlawbehaviourwiththeirowncriticalexponents.Prominentamongthemarethe rstandallreturntimedistributionsofactivity(asiteisde nedasactivewhenits tnessistheminimumone),

Pf(t)～t τf,Pa(t)～t τa,

(2)

thatassignsanew tnessfn+1attimen+1tothecho-senlatticesite.Thiscorrespondstotheextinctionoftheless tanditsimpactontheecosystem.Indeed,intheoriginalBSmodel,thefunctionFisjustarandomfunc-tionwithauniformdistributionbetween0and1.Thesystemreachesastationarycriticalstateinwhichthedistributionof tnessesiszerobelowacertainthresholdanduniformaboveit(theactualvalueofthethresholddependsontheupdatingrule).Ithasbeenshownthattheexactnatureoftheupdatingruleisnotrelevant.Indeed,theuseofachaoticmapinsteadofarandomup-date,preservestheuniversalityclass(evenifthe naldistributionmaybealtered).

Asaresultofthedynamicalrules,theBSsystemexhibitssequencesofcausallyconnectedevolutionaryeventscalledavalanchesThenumberofavalanches

1

whereτf～1.58andτa～0.42IntheBSmodel,twobasicingredientsareneededforSOCtooccuri)Orderfromextremaldynamics(minimumrule).

ii)Disorderfromtheupdatingrule(stochasticorother-wise).

Anoversimpli cationintheBSmodelisapparent:eachspeciesisdescribedbyasinglevariable.Themini-mumruleisappliedtothisvariable,andsoisthee ectofmutations.Innaturalevolvingsystems,however,atleasttwo—interacting—levelsoforganisationarepresent,andbothplayaroleintheevolution.Mutationsoccuratamicroscopiclevel,namelyatthemolecularlevelofthegenotype.Thisa ectsamacroscopiclevel,de ningthephenotype.Naturalselectionactsonthephenotype,al-lowingthesurvivalofsomeandtheextinctionofothers,accordingtotheobservedpowerlawdistributions.

Inthispaperweproposeanewmodel,namelyanEx-tremalCoupledMapLattice(ECML),thattakesintoaccountthistwolevelstructure.Theecosystemisrep-resentedasanensembleofNspeciesarrangedonaone-dimensionallattice.Eachspeciesisdescribedbymeansofmacroscopicvariablexisubjecttoanonlineardynam-icsandcoupledtoitsnearestneighbours.Ingeneral,xicanbeidenti edwithapopulation,orsomeotherfunctionofthephenotype.Thecontrolparameterλiofthenonlineardynamicsisidenti edwiththemicroscopiclevel(genotype).To ndtheexactfunctionconnectingthesetwolevelsofdescriptionisbeyondthescopeofthissimpli edmodel.Forthisreason,wechosealogisticmapfortheindependentevolutionofxiwithλiactingasthenonlinearparameterinthemapWiththisinmind,

We propose a model for co-evolving ecosystems that takes into account two levels of description of an organism, for instance genotype and phenotype. Performance at the macroscopic level forces mutations at the microscopic level. These, in turn, affect the

theevolutionofeachspeciesisgivenby:

i

xin+1=(1 )f(xn)+

We propose a model for co-evolving ecosystems that takes into account two levels of description of an organism, for instance genotype and phenotype. Performance at the macroscopic level forces mutations at the microscopic level. These, in turn, affect the

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FIG.2.Distributionof rstreturntime.Theparame-terscorrespondtothoseinFig.1:N=500, λ=0.1,(a)λ0=3.7, =0.5.Thedashedlinecorrespondstoapowerlawwithexponentα=1.80(3).(b)λ0=3.9, =0.5.Thedashedline,shownasareference,correspondstoanexponen-tialdecay.(c)λ0=3.8, =0.1.Thedashedlinecorrespondstoapowerlawwithexponentα=1.57(2).(d)λ0=3.9, =0.1.Thedashedlinecorrespondstoapowerlawwithexponentα=1.53(1), ttedtotheasymptoticregion.

Thispictureholds,qualitatively,evenintheabsenceoftheevolutionarydynamics.Indeed,ifineverytimestepwetrackthepositionoftheminimum,butdonotchangethevalueofitsλ(thiscorresponds,e ectively,tosett …… 此处隐藏：6877字，全部文档内容请下载后查看。喜欢就下载吧 ……