EcoLab Documentation

EcoLabDocumentationRussellK.StandishOctober8,1999AbstractEcoLabisasystemthatimplementsanabstractecologymodel.[3]ItiswrittenasasetofTcl/Tk[2]commandssothatthemodelparameterscaneasilybechangedontheybymeansofeditingascript.Thisdocumentdescribestheclasslibrariesthatimplementthedynamicarraystructures,andtheinterfacetotheTclinterpreter,aswellalltheTclcommandsandvariablesthataredenedintheEcoLabsystem.YoushouldalsoexaminethesampleEcoLabscript(ecolab.tcl)providedwiththepacakge,andthesampleproblemscripts,e.gpred-prey.tclandlifetime.tcl.Contents1TheModel31.1Lotka-VolterraDynamics......................31.2Mutation...............................42StructureoftheEcolabSimulationSystem63MakingEcolab84Constructinganexperiment84.1InputParameters...........................94.2Themainbuttonbar.........................94.3Instrumentation............................104.3.1Plot..............................104.3.2Histogram...........................104.3.3Display............................104.3.4Connectplot.........................114.4AuxilliaryCommands........................114.4.1getglobalvars/dataserver.................114.4.2checkpoint/restart......................114.5EcolabModelcommands.......................124.5.1generate............................1214.5.2condense...........................124.5.3mutate.............................124.5.4migrate............................134.5.5maxeig.............................134.5.6lifetimes............................134.5.7get...............................135CreatingNewInstruments146CreatingaNewModel157SpatialVariation167.1ParallelExecution..........................168tcl++188.1GlobalVariables...........................188.2CreatingaTclcommand.......................188.3tclvar.................................198.4tclcmd.................................208.5tclreturn................................218.6tclindex................................229arrays239.1arrayandiarray...........................239.1.1IndexingandGather/Scatter................269.1.2Rounding...........................269.2sparsemat..............................269.3Globalfunctions...........................279.3.1Reductionfunctions.....................289.3.2Arrayfunctions........................289.3.3Randomnumberfunctions..................289.3.4StreamFunctions.......................2810initarraysmodule2911Globalsclass3112Analysis3412.1newwin................................3412.2ReductionFunctions.........................3412.3PaletteVariable............................3412.4displaystub..............................3412.5connectstub.............................3521TheModelWestartwithageneralisedformoftheLotka-Volterraequation_n=rn+nn:(1)Herenisthepopulationdensity,thecomponentnibeingthenumberofindi-vidualsofspeciesi,risthedierencebetweenreproductionanddeath,istheinteractionmatrix,withijbeingtheinteractionbetweenspeciesiandj,*referringtoelementwisemultiplicationandmutateisthemutationoperator.1.1Lotka-VolterraDynamicsThemostobviousthingaboutequation(1)isitsxedpoint^n=1r;(2)where_n=0.Forthispointtobebiologicallymeaningful,allcomponentsof^nmustbepositive,givingrisetothefollowinginequalities:^ni=1ri0;8i(3)Thestabilityofthispointisrelatedtothenegativedenitenessofderivativeof_nat^n.Thecomponentsofthederivativearegivenby@_ni@nj=ijri+Xkiknk!+ijni(4)Substitutingeq(2)gives@_ni@nj^n=ij1ri(5)Stabilityofthexedpointrequiresthatthismatrixshouldbenegativede-nite.Sincethe1riareallnegativebyvirtueof(3),eachminordeterminantofthismatrixisequaltoaminordeterminantofmultipliedbyapositivenum-ber,stabilityoftheequilibriumisequivalenttobeingnegativedenite.Aweakerconditionistorequirethatthesystemremainboundedwithtime:Xi_ni=rn+nn0;8n:XiniN9N(6)Asnbecomeslargeinanydirection,thisfunctionalisdominatedbythequadraticterm,sothisimpliesthatnn08n:ni0.Negativede-nitenessofissucient,butnotnecessaryforthiscondition.Forexample,thepredator-preyrelations(heavilynormalised)havethefollowingmatrixas:3=1220whichhaseigenvalues3=2;5=2.Ifweletn=(x;y);x;y0,thennn=2x2,whichisclearlynon-positiveforallnonpositivex.Consideraddinganewrowandcolumnto.Whatisconditionisthenewrowandcolumnrequiredtosatisfysuchthatequation(6)issatised.Breakupinthefollowingway:0BBBB@...A......B...CD1CCCCA0BBBB@...n1...n21CCCCACondition(6)becomes:n1An1+n1(B+C)n2+Dn220(7)Leta=maxn=1nAn;andb=maxiBi+Ci:Thenasucientbutnotnecessaryconditionforcondition(7)isan21+bn1n2+Dn220Themaximumvaluewithrespectton2isan21(bn1)2=4D,sothisrequiresthatb2pad(8)1.2MutationWithmutation,equation(1)reads_n=rn+nn+mutate(;r;n):(9)Thedicultywithaddingmutationtothismodelishowtodenethemap-pingbetweengenotypespaceandphenotypespace,orinotherwords,whatdenestheembryology.Afewstudies,includingRay’sTierraworld,dothiswithanexplicitmappingfromthegenotypetotosomeparticularorganismproperty(e.g.interpretedasmachinelanguageinstructions,orasweightinaneuralnet).Theseorganismstheninteractwithoneanothertodeterminethepopulationdynamics.Inthismodel,howe