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Visual Models of Plants Interacting with Their Environment Radomir Mech and Przemyslaw Visual Models of Plants Interacting with Their Environment Radomir Mech and Przemyslaw

Visual Models of Plants Interacting with Their Environment Radomir Mech and Przemyslaw - PDF document

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Visual Models of Plants Interacting with Their Environment Radomir Mech and Przemyslaw - PPT Presentation

ucalgaryca Abstract Interaction with the environment is a key factor affecting the development of plants and plant ecosystems In this paper we introduce a modeling framework that makes it possible to simulate and visualize a wide range of interaction ID: 30057

ucalgaryca Abstract Interaction with the

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competitionforbetweenthree-dimensionalshootsofherbaceousplants[25]andbranchesoftrees[9,10,11,15,33,35,52].Modelsofexogenousphenomenarequireacomprehensiverepre-sentationofboththedevelopingplantandtheenvironment.Con-sequently,theyarethemostdifÞculttoformulate,implement,anddocument.Programsaddressedtothebiologicalaudienceareoftenlimitedtonarrowgroupsofplants(forexample,poplars[9]ortreesinthepinefamily[21]),andpresenttheresultsinarudimentarygraphicalform.Ontheotherhand,modelsaddressedtothecom-putergraphicsaudienceusemoreadvancedtechniquesforrealisticimagesynthesis,butputlittleemphasisonthefaithfulreproductionofphysiologicalmechanismscharacteristictospeciÞcplants.Inthispaperweproposeageneralframework(deÞnedasamod-elingmethodologysupportedbyappropriatesoftware)formod-eling,simulating,andvisualizingthedevelopmentofplantsthatbi-directionallyinteractwiththeirenvironment.Theusefulnessofmodelingframeworksforsimulationstudiesofmodelswithcom-plex(emergent)behaviorismanifestedbypreviousworkinthe-oreticalbiology,artiÞciallife,andcomputergraphics.Examplesincludecellularautomata[51],systemsforsimulatingbehaviorofcellularstructuresindiscrete[1]andcontinuous[20]spaces,andL-system-basedframeworksformodelingplants[36,46].Frame-worksmayhavetheformofageneral-purposesimulationprogramthatacceptsmodelsdescribedinasuitablemini-languageasin-e.g.[36,46],orasetoflibraryprograms[27].Comparedtospecial-purposeprograms,theyofferthefollowingbeneÞts:Attheconceptuallevel,theyfacilitatethedesign,speciÞcation,documentation,andcomparisonofmodels.Atthelevelofmodelimplementation,theymakeitpossibletode-velopsoftwarethatcanbereusedinvariousmodels.SpeciÞcally,graphicalcapabilitiesneededtovisualizethemodelsbecomeapartofthemodelingframework,anddonothavetobereimple-Finally,ßexibleconceptualandsoftwareframeworksfacilitateinteractiveexperimentationwiththemodels[46,AppendixA].Ourframeworkisintendedbothforpurposeofimagesynthesisandasaresearchandvisualizationtoolformodelstudiesinplantmor-phogenesisandecology.Thesegoalsareaddressedatthelevelsofthesimulationsystemandthemodelinglanguagedesign.Theun-derlyingparadigmofplant-environmentinteractionisdescribedinSection2.TheresultingdesignofthesimulationsoftwareisoutlinedinSection3.ThelanguageforspecifyingplantmodelsispresentedinSection4.Itextendstheconceptofenvironmentally-sensitiveL-systems[45]withconstructsforbi-directionalcommunicationwiththeenvironment.Thefollowingsectionsillustratetheproposedframeworkwithconcretemodelsofplantsinteractingwiththeirenvironment.Theexamplesinclude:thedevelopmentofplanarbranchingsystemscontrolledbythecrowdingofapices(Section5),thedevelopmentofclonalplantscontrolledbyboththecrowdingoframetsandthequalityofterrain(Section6),thedevelopmentofrootscontrolledbytheconcentrationofwatertransportedinthesoil(Section7),andthedevelopmentoftreecrownsaffectedbythelocaldistributionoflight(Section8)Thepaperconcludeswithanevaluationoftheresultsandalistofopenproblems(Section9). PlantEnvironment ReceptionResponse ReceptionResponse Figure1:Conceptualmodelofplantandenvironmenttreatedascommunicatingconcurrentprocesses2CONCEPTUALMODELAsdescribedbyHart[30],everyenvironmentallycontrolledphe-nomenoncanbeconsideredasachainofcausallylinkedevents.Afterastimulusisperceivedbytheplant,informationinsomeformistransportedthroughtheplantbody(unlessthesiteofstimulusperceptioncoincideswiththesiteofresponse),andtheplantre-acts.Thisreactionreciprocallyaffectstheenvironment,causingitsmodiÞcationthatinturnaffectstheplant.Forexample,rootsgrowinginthesoilcanabsorborextractwater(dependingonthewaterconcentrationintheirvicinity).Thisinitiatesaßowofwaterinthesoiltowardsthedepletedareas,whichinturnaffectsfurthergrowthoftheroots[12,24].Accordingtothisdescription,theinteractionofaplantwiththeenvironmentcanbeconceptualizedastwoconcurrentprocessesthatcommunicatewitheachother,thusformingafeedbackloopofinformationßow(Figure1).Theplantprocessperformsthefollowingfunctions:receptionofinformationabouttheenvironmentintheformofscalarorvectorvaluesrepresentingthestimuliperceivedbyspe-ciÞcorgans;transportandprocessingofinformationinsidetheplant;generationoftheresponseintheformofgrowthchanges(e.g.developmentofnewbranches)anddirectoutputofinformationtotheenvironment(e.g.uptakeandexcretionofsubstancesbyaroottip).Similarly,theenvironmentalprocessincludesmechanismsforperceptionoftheplantÕsactions;simulationofinternalprocessesintheenvironment(e.g.diffusionofsubstancesorpropagationoflight);presentationofthemodiÞedenvironmentinaformperceivablebytheplant.Thedesignofasimulationsystembasedonthisconceptualmodelispresentednext.3SYSTEMDESIGNThegoalistocreateaframework,inwhichawiderangeofplantstructuresandenvironmentscanbeeasilycreated,modiÞed,and usedforexperimentation.Thisrequirementledustothefollowingdesigndecisions:Theplantandtheenvironmentshouldbemodeledbyseparateprogramsandrunastwocommunicatingprocesses.Thisdesigncompatiblewiththeassumedconceptualmodelofplant-envi-ronmentinteraction(Figure1);consistentwiththeprinciplesofstructureddesign(moduleswithclearlyspeciÞedfunctionsjointlycontributetothesolu-tionofaproblembycommunicatingthroughawelldeÞnedinterface;informationlocaltoeachmoduleishiddenfromothermodules);appropriateforinteractiveexperimentationwiththemodels;inparticular,changesintheplantprogramcanbeimplementedwithoutaffectingtheenvironmentalprogram,andviceversaextensibletodistributedcomputingenvironments,wheredif-ferentcomponentsofalargeecosystemmaybesimulatedusingseparatecomputers.Theusershouldhavecontroloverthetypeandamountofinfor-mationexchangedbetweentheprocessesrepresentingtheplantandtheenvironment,sothatalltheneededbutnosuperßuousinformationistransferred.PlantmodelsshouldbespeciÞedinalanguagebasedonL-systems,equippedwithconstructsforbi-directionalcommuni-cationbetweentheplantandtheenvironment.Thisdecisionhasthefollowingrationale:Asuccinctdescriptionofthemodelsinaninterpretedlan-guagefacilitatesexperimentationinvolvingmodiÞcationstothemodels;L-systemscapturetwofundamentalmechanismsthatcontroldevelopment,namelyßowofinformationfromamothermod-uletoitsoffspring(cellulardescent)andßowofinformationbetweencoexistingmodules(endogenousinteraction)[38].Thelattermechanismplaysanessentialroleintransmittinginformationfromthesiteofstimulusperceptiontothesiteoftheresponse.Moreover,L-systemshavebeenextendedtoallowforinputofinformationfromtheenvironment(seeSection4);ModelingofplantsusingL-systemshasreachedarelativelyadvancedstate,manifestedbymodelsrangingfromalgaetoherbaceousplantsandtrees[43,46].Giventhevarietyofprocessesthatmaytakeplaceintheenviron-ment,theyshouldbemodeledusingspecial-purposeprograms.Genericaspectsofmodeling,notspeciÞctoparticularmodels,shouldbesupportedbythemodelingsystem.Thisincludes:anL-system-basedplantmodelingprogram,whichinterpretsL-systemssuppliedasitsinputandvisualizestheresults,andthesupportforcommunicationandsynchronizationofpro-cessessimulatingthemodeledplantandtheenvironment.AsystemarchitecturestemmingfromthisdesignisshowninFig-ure2.WewilldescribeitfromtheperspectiveofextensionstotheformalismofL-systems. modelPlantsimulatorModel of the environment COMUICTIONCOMUICTION Communicationspecification Figure2:Organizationofthesoftwareformodelingplantsinteract-ingwiththeirenvironment.Shadedrectanglesindicatecomponentsofthemodelingframework,clearrectanglesindicateprogramsanddatathatmustbecreatedbyauserspecifyinganewmodelofaplantorenvironment.Shadedarrowsindicateinformationexchangedinastandardizedformat.4OPENL-SYSTEMSHistorically,L-systemswereconceivedasclosedcyberneticsys-tems,incapableofsimulatinganyformofcommunicationbetweenthemodeledplantanditsenvironment.IntheÞrststeptowardstheinclusionofenvironmentalfactors,RozenbergdeÞnedtableL-,whichallowforachangeinthesetofdevelopmentalrules(theproductionsetoftheL-system)inresponsetoachangeintheenvironment[31,49].TableL-systemswereapplied,forex-ample,tocapturetheswitchfromtheproductionofleavestotheproductionofßowersbytheapexofaplantduetoachangeindaylength[23].ParametricL-systems[29,46],introducedlater,madeitpossibletoimplementavariantofthistechnique,withtheenvi-ronmentaffectingthemodelinaquantitativeratherthanqualitativemanner.Inacasestudyillustratingthispossibility,weatherdatacontainingdailyminimumandmaximumtemperatureswereusedtocontroltherateofgrowthinabeanmodel[28].Environmentally-sensitiveL-systems[45]representedthenextstepintheinclusionofenvironmentalfactors,inwhichlocalratherthanglobalpropertiesoftheenvironmentaffectedthemodel.Thenewconceptwastheintroductionofquerysymbols,returningcurrentpositionorori-entationoftheturtleintheunderlyingcoordinatesystem.Theseparameterscouldbepassedasargumentstouser-deÞnedfunctions,returninglocalpropertiesoftheenvironmentatthequeriedlocation.Environmentally-sensitiveL-systemswereillustratedbymodelsoftopiaryscenes.TheenvironmentalfunctionsdeÞnedgeometricshapes,towhichtreeswerepruned.OpenL-systems,introducedinthispaper,augmentthefunctionalityofenvironmentally-sensitiveL-systemsusingareservedsymbolforbilateralcommunicationwiththeenvironment.Inshort,parametersassociatedwithanoccurrenceofthecommunicationsymbolcanbesetbytheenvironmentandtransferredtotheplantmodel,orsetbytheplantmodelandtransferredtotheenvironment.Theenvironmentisnolongerrepresentedbyasimplefunction,butbecomesanactiveprocessthatmayreacttotheinformationfromtheplant.Thus,plantsaremodeledasopencyberneticsystems,sendinginformationtoandreceivinginformationfromtheenvironment.InordertodescribeopenL-systemsinmoredetail,weneedtorecalltherudimentsofL-systemswithturtleinterpretation.Ourpresentationisreproducedfrom[45]. AnL-systemisaparallelrewritingsystemoperatingonbranchingstructuresrepresentedasbracketedstringsofsymbolswithasso-ciatednumericalparameters,calledMatchingpairsofsquarebracketsenclosebranches.Simulationbeginswithanini-tialstringcalledthe,andproceedsinasequenceofdiscretederivationsteps.Ineachstep,rewritingrulesproductionsallmodulesinthepredecessorstringbysuccessormodules.TheapplicabilityofaproductiondependsonapredecessorÕscontext(incontext-sensitiveL-systems),valuesofparameters(inproduc-tionsguardedbyconditions),andonrandomfactors(instochasticL-systems).Typically,aproductionhastheformat:lcpred&#x-190;�rcprobistheproductionidentiÞer(label),pred,andtheleftcontext,thestrictpredecessor,andtherightcontext,thecondition,isthesuccessor,andprobistheprobabilityofproductionapplication.ThestrictpredecessorandthesuccessoraretheonlymandatoryÞelds.Forexample,theL-systemgivenbelowconsistsofaxiomandthreeproductions,and:A(1)B(3)A(5):A(x)A(x+1):0.4:A(x)B(xÐ1):0.6:A(x)B(y)&#x-255;A(z):y44()Thestochasticproductionsreplacemodulebyei-,withprobabilitiesequalto0.4and0.6,respectively.Thecontext-sensitiveproductionreplacesawithleftcontextandrightcontextsupportingbranch.Theapplicationofthisproductionisguardedbycondition4.Consequently,theÞrstderivationstepmayhavetheform:form:()()Itwasassumedthat,asaresultofrandomchoice,productionwasappliedtothemodule,andproductiontothemodule.Productionwasappliedtothemodule,becauseitoccurredwiththerequiredleftandrightcontext,andthecondition4wastrue.IntheL-systemspresentedasexampleswealsouseseveraladdi-tionalconstructs(cf.[29,44]):Productionsmayincludestatementsassigningvaluestolocalvariables.Thesestatementsareenclosedincurlybracesandseparatedbysemicolons.TheL-systemsmayalsoincludearrays.ReferencestoarrayelementsfollowthesyntaxofC;forexample,MaxLenxLenorderThelistofproductionsisordered.Inthedeterministiccase,theÞrstmatchingproductionapplies.Inthestochasticcase,thesetofallmatchingproductionsisestablished,andoneofthemischosenaccordingtothespeciÞedprobabilities.FordetailsoftheL-systemsyntaxsee[29,43,46]. H / L+U ^ Figure3:ControllingtheturtleinthreedimensionsIncontrasttotheparallelapplica-tionofproductionsineachderiva-tionstep,theinterpretationoftheresultingstringsproceedssequen-tially,withreservedmodulesact-ingascommandstoaLOGO-styleturtle[46].Atanypointofthestring,theturtlestateischarac-terizedbyapositionvector interpret ... A(a1,...,ak) ?E(x1,...,xm) B(b1,...,bn) ... ... A(a1,...,ak) ?E(y1,...,ym) B(b1,...,bn) ...environment Figure4:Informationßowduringthesimulationofaplantinter-actingwiththeenvironment,implementedusinganopenL-systemthreemutuallyperpendicularorientationvectors,and,indi-catingtheturtleÕsheading,theupdirection,andthedirectiontotheleft(Figure3).Modulecausestheturtletodrawalineinthecur-rentdirection.Modules,&,rotatetheturtlearoundoneofthevectors,or,asshowninFigure3.ThelengthofthelineandthemagnitudeoftherotationanglecanbegivengloballyorspeciÞedasparametersofindividualmodules.Duringtheinterpretationofbranches,theopeningsquarebracketpushesthecurrentpositionandorientationoftheturtleonastack,andtheclosingbracketrestorestheturtletothepositionandorientationpoppedfromthestack.Aspecialinterpretationisreservedforthemodule%,whichcutsabranchbyerasingallsymbolsinthestringfromthepointofitsoccurrencetotheendofthebranch[29].Themeaningofmanysymbolsdependsonthecontextinwhichtheyoccur;forexample,denotearithmeticoperatorsaswellasmodulesthatrotatetheturtle.TheturtleinterpretationofL-systemsdescribedabovewasde-signedtovisualizemodelsinapostprocessingstep,withnoeffectontheL-systemoperation.Positionandorientationoftheturtleareimportant,however,whileconsideringenvironmentalphenom-ena,suchascollisionswithobstaclesandexposuretolight.Theenvironmentally-sensitiveextensionofL-systemsmakestheseat-tributesaccessibleduringtherewritingprocess[45].Thegeneratedstringisinterpretedaftereachderivationstep,andturtleattributesfoundduringtheinterpretationarereturnedasparameterstore-servedquerymodules.Syntactically,thequerymoduleshavetheform?x;y;z,whereP;H;U;.Dependingontheactualsymbol,thevaluesofparameters,andrepresentapositionoranorientationvector.OpenL-systemsareageneralizationofthisconcept.cationmodulesoftheform?areusedbothtosendandreceiveenvironmentalinformationrepresentedbythevaluesof;:::;x(Figure4).Tothisend,thestringresultingfromaderivationstepisscannedfromlefttorighttodeterminethestateoftheturtleassociatedwitheachsymbol.Thisphaseissimilartothegraphicalinterpretationofthestring,exceptthattheresultsneednotbevisualized.Uponencounteringacommunica-tionsymbol,theplantprocesscreatesandsendsamessagetotheenvironmentincludingallorapartofthefollowinginformation:theaddress(positioninthestring)ofthecommunicationmodule(mandatoryÞeldneededtoidentifythismodulewhenareplycomesfromtheenvironment),valuesofparametersthestateoftheturtle(coordinatesofthepositionandorientation vector,aswellassomeotherattributes,suchascurrentlinethetypeandparametersofthemodulefollowingthecommuni-cationmoduleinthestring(notusedintheexamplesdiscussedinthispaper).TheexactmessageformatisdeÞnedinacommunicationspeciÞ-cationÞle,sharedbetweentheprogramsmodelingtheplantandtheenvironment(Figure2).Consequently,itispossibletoincludeonlytheinformationneededinaparticularmodelinthemessagessenttotheenvironment.Transferofthelastmessagecorrespondingtothecurrentscanofthestringissignaledbyareservedend-of-transmissionmessage,whichmaybeusedbytheenvironmentalprocesstostartitsoperation.Themessagesoutputbytheplantmodelingprogramaretransferredtotheprocessthatsimulatestheenvironmentusinganinterprocesscommunicationmechanismprovidedbytheunderlyingoperatingsystem(apairofUNIXpipesorsharedmemorywithaccesssyn-chronizedusingsemaphores,forexample).Theenvironmentpro-cessesthatinformationandreturnstheresultstotheplantmodelusingmessagesinthefollowingformat:theaddressofthetargetcommunicationmodule,valuesofparameterscarryingtheoutputfromtheenvironment.Theplantprocessusesthereceivedinformationtosetparameterval-uesinthecommunicationmodules(Figure4).Theuseofaddressesmakesitpossibletosendrepliesonlytoselectedcommunicationmodules.DetailsofthemappingofmessagesreceivedbytheplantprocesstotheparametersofthecommunicationmodulesaredeÞnedinthecommunicationspeciÞcationÞle.Afterallrepliesgeneratedbytheenvironmenthavebeenreceived(afactindicatedbyanend-of-transmissionmessagesentbytheenvironment),theresultingstringmaybeinterpretedandvisualized,andthenextderivationstepmaybeperformed,initiatinganothercycleofthesimulation.Notethat,byprecedingeverysymbolinthestringwithacommuni-cationmoduleitispossibletopasscompleteinformationaboutthemodeltotheenvironment.Usually,however,onlypartialinforma-tionaboutthestateofaplantisneededasinputtotheenvironment.Properplacementofcommunicationmodulesinthemodel,com-binedwithcarefulselectionoftheinformationtobeexchanged,provideameansforkeepingtheamountoftransferredinformationatamanageablelevel.WewillillustratetheoperationofopenL-systemswithinthecon-textofcompletemodelsofplant-environmentinteractions,usingexamplesmotivatedbyactualbiologicalproblems.5AMODELOFBRANCHTIERSBackground.Apicalmeristems,locatedattheendpointsofbranches,areenginesofplantdevelopment.Theapicesgrow,con-tributingtotheelongationofbranchsegments,andfromtimetotimedivide,spawningthedevelopmentofnewbranches.Ifallapicesdividedperiodically,thenumberofapicesandbranchsegmentswouldincreaseexponentially.Observationsofrealbranchingstruc-turesshow,however,thattheincreaseinthenumberofsegmentsislessthanexponential[8].Hondaandhiscollaboratorsmod-eledseveralhypotheticalmechanismsthatmaycontroltheextentofbranchinginordertopreventovercrowding[7,33](seealso[4]).Oneofthemodels[33],supportedbymeasurementsandearliersimulationsofthetropicaltreeTerminaliacatappa[19],assumesanexogenousinteractionmechanism.Terminaliabranchesformhorizontaltiers,andthemodelislimitedtoasingletier,treatedasatwo-dimensionalstructure.Inthiscase,thecompetitionforlighteffectivelyamountstocollisiondetectionbetweentheapicesandleafclusters.Wereproducethismodelasthesimplestexampleillustratingthemethodologyproposedinthispaper.Communicationspecication.Theplantcommunicateswiththeenvironmentusingcommunicationmodulesoftheform?Messagessenttotheenvironmentincludetheturtlepositionandthevalueofparameter,interpretedasthevigorofthecorrespondingapex.Onthisbasis,theenvironmentalprocessdeterminesthefateofeachapex.Aparametervalueof0returnedtotheplantindicatesthatthedevelopmentofthecorrespondingbranchwillbeterminated.Avalueof1allowsforfurtherbranching.Themodeloftheenvironment.Theenvironmentalprocesscon-siderseachapexornon-terminalnodeofthedevelopingtierasthecenterofacircularleafcluster,andmaintainsalistofallclusterspresent.Newclustersareaddedinresponsetomessagesreceivedfromtheplant.Allclustershavethesameradius,speciÞedintheenvironmentaldataÞle(cf.Figure2).Inordertodeterminethefateoftheapices,theenvironmentcomparesapexpositionswithleafclusterpositions,andauthorizesanapextogrowifitdoesnotfallintoanexistingleafcluster,orifitfallsintoaclustersurroundinganapexwithasmallervigorvalue.Theplantmodel.TheplantmodelisexpressedasanopenL-system.Thevaluesofconstantsaretakenfrom[33].#deÞner0.94/*contractionratioandvigor1*/#deÞner0.87/*contractionratioandvigor2*/24.4/*branchingangle1*/36.9/*branchingangle2*/138.5/*divergenceangle*/:Ð(90)[F(1)?E(1)A(1)]+(Ð(90)[F(1)?E(1)A(1)]+(()()()+(')[F(1)?E(1)A(1)]+(')[F(1)/?E(1)A(1)]+(')[F(1)?E(1)A(1)]p1:?E(x)A(v):x==11( 2)F(v*r2)?E(r2)A(v*r2)]Ð(:?E(x)TheaxiomspeciÞestheinitialstructureasawhorlofÞvebranchsegments.Thedivergenceanglebetweenconsecutivesegmentsisequalto138.Eachsegmentisterminatedbyacommunicationsymbol?followedbyanapex.Inaddition,twobranchesincludemodule,whichchangesthedirectionsatwhichsubsequentbrancheswillbeissued(leftright)byrotatingtheapex180aroundthesegmentaxis.describestheoperationoftheapices.Ifthevalueofreturnedbyacommunicationmodule?isnot1,theassociatedapexwillremaininactive(donothing).Otherwisetheapexwillproduceapairofnewbranchsegmentsatangleswithrespecttothemothersegment.Constantsthelengthsofthedaughtersegmentsasfractionsofthelengthoftheirmothersegment.Thevaluesarealsopassedtotheprocesssimulatingtheenvironmentusingcommunicationmodules.Communicationmodulescreatedinthepreviousderivationsteparenolongerneededandareremovedbyproduction Figure5:CompetitionforspacebetweentwotiersofbranchessimulatedusingtheHondamodelFigure5illustratesthecompetitionforspacebetweentwotiersdevelopingnexttoeachother.TheextentofbranchingineachtierislimitedbycollisionsbetweenitsapicesanditsownortheneighborÕsleafclusters.Thelimitedgrowthofeachstruc-tureinthedirectionofitsneighborillustratesthephenomenonofmorphologicalplasticity,oradaptationoftheformofplantstotheirenvironment[17].6AMODELOFFORAGINGINCLONALPLANTSBackground.Foraging(propagation)patternsinclonalplantspro-videanotherexcellentexampleofresponsetocrowding.Aclonalplantspreadsbymeansofhorizontalstemsegments(spacerswhichformabranchingstructurethatgrowsalongthegroundandconnectsindividualplants(ramets)[3].Eachrametconsistsofaleafsupportedbyanuprightstemandoneormorebuds,whichmaygiverisetofurtherspacersandramets.Theirgradualdeath,afteracertainamountoftime,causesgradualseparationofthewholestructure(the)intoindependentparts.Followingthesurfaceofthesoil,clonalplantscanbecapturedusingmodelsoperatingintwodimensions[5],andinthatrespectresem-Terminaliatiers.Weproposeamodelofahypotheticalplantthatrespondstofavorableenvironmentalconditions(highlocalin-tensityoflight)bymoreextensivebranchingandreducedsizeofleaves(allowingformoredensepackingoframets).IthasbeeninspiredbyacomputermodelofcloveroutlinedbyBell[4],theanalysisofresponsesofclonalplantstotheenvironmentpresentedbyDong[17],andthecomputermodelsanddescriptionsofveg-etativemultiplicationofplantsinvolvingthedeathofinterveningconnectionsbyRoom[47].Communicationspecication.Theplantsendsmessagestotheen-vironmentthatincludeturtlepositionandtwoparametersassociatedwiththecommunicationssymbol,?type;x.TheÞrstparam-eterisequalto0,1,or2,anddeterminesthetypeofexchangedinformationasfollows:Themessage?representsarequestforthelightintensity(irradiance[14])atthepositionofthecommunicationmodule.Theenvironmentrespondsbysettingtotheintensityofincom-inglight,rangingfrom0(nolight)to1(fulllight).Themessage?notiÞestheenvironmentaboutthecreationofarametwithaleafofradiusatthepositionofthecommu-nicationmodule.Nooutputisgeneratedbytheenvironmentinresponsetothismessage.Themessage?notiÞestheenvironmentaboutthedeathofarametwithaleafofradiusatthepositionofthecommunica-tionmodule.Again,nooutputisgeneratedbytheenvironment.Themodeloftheenvironment.Thepurposeoftheenvironmentprocessistodeterminelightintensityatthelocationsrequestedbytheplant.Thegroundisdividedintopatches(speciÞedasarasterimageusingapaintprogram),withdifferentlightintensitiesassignedtoeachpatch.Intheabsenceofshading,theseintensitiesarereturnedbytheenvironmentalprocessinresponsetomessagesoftype0.Toconsidershading,theenvironmentkeepstrackofthesetoframets,addingnewrametsinresponsetoamessagesoftype1,anddeletingdeadrametsinresponsetomessagesoftype2.Ifasamplingpointfallsinanareaoccupiedbyaramet,thereturnedlightintensityisequalto0(leavesareassumedtobeopaque,andlocatedabovethesamplingpoints).Theplantmodel.TheessentialfeaturesoftheplantmodelarespeciÞedbythefollowingopenL-system.45/*branchingangle*/#deÞneMinLight0.1/*lightintensitythreshold*/#deÞneMaxAge20/*lifetimeoframetsandspacers*/#deÞneLen2.0/*lengthofspacers*/#deÞneProb(x)(0.12+x*0.42)#deÞneProb(x)(0.03+x*0.54)#deÞneRadius(x)(sqrt(15Ðx*5)/:A(1)?E(0,0):A(dir)�?E(0,x�):x=:A(dir)�?E(0,x:B(x,dir)B(x,dir)( *dir)F(Len,0)A(Ðdir)?E(0,0)]:Prob:B(x,dir):1ÐProb:R(x)[@o(Radius(x),0)?E(1,Radius(x))]:Prob:R(x):1ÐProb:@o(radius,age):ageMaxAge:@o(radius,age):age==MaxAge:F(len,age):ageMaxAge:F(len,age):age==MaxAge:?E(type,x)TheinitialstructurespeciÞedbytheaxiomconsistsofanapexfollowedbythecommunicationmodule?.IftheintensityoflightreachinganapexisinsufÞcient(belowthethresholdMinLighttheapexdies(production).Otherwise,theapexcreatesarameti.e.,amodulethatwillyieldaramet),abranchinitial,aspacer,andanewapexterminatedbycommunicationmodule?).Theparameter,valuedeither1or-1,controlsthedirectionofbranching.Parametersofthespacermodulespecifyitslengthandage. Abranchinitialmaycreatealateralbranchwithitsownapexandcommunicationmodule?),oritmaydieanddisappearfromthesystem(production).TheprobabilityofsurvivalisanincreasinglinearfunctionProbofthelightintensitythathasreachedthemotherapexinthepreviousderivationstep.Asimilarstochasticmechanismdescribestheproductionofarametbytherametinitial),withtheprobabilityoframetformationcontrolledbyanincreasinglinearProb.Therametisrepresentedasacircle@;itsradiusisadecreasingfunctionRadiusofthelightintensity.Asinthecaseofspacers,thesecondparameterofarametindicatesitsage,initiallysetto0.TheenvironmentisnotiÞedaboutthecreationoftherametusingacommunicationmodule?Thesubsequentproductionsdescribetheagingofspacers()andramets().UponreachingthemaximumageMaxAge,arametisremovedfromthesystemandamessagenotifyingtheenvironmentaboutthisfactissentbytheplant().Thedeathofthespacersissimulatedbyreplacingspacermoduleswithinvisiblelineseg-ofthesamelength.Thisreplacementmaintainstherelativepositionoftheremainingelementsofthestructure.Finally,produc-removescommunicationmodulesaftertheyhaveperformedtheirtasks. Figure6:DivisionofthegroundintopatchesDivisionofthegroundintopatchesusedinthesim-ulationsisshowninFigure6.Ara-bicnumeralsindicatetheintensityofincominglight,andRomannu-meralsidentifyeachpatch.Thede-velopmentofaclonalplantassum-ingthisdivisionisillustratedinFig-ure7.Asanextensionofthebasicmodeldiscussedabove,thelengthofthespacersandthemagnitudeofthebranchinganglehavebeenvar-iedusingrandomfunctionswithanormaldistribution.Rametshavebeenrepresentedastrifoliateleaves.Thedevelopmentbeginswithasinglerametlocatedinrelativelygood(lightintensity0.6)patchIIatthetoprightcornerofthegrowtharea(Figure7,step9ofthesimulation).TheplantpropagatesthroughtheunfavorablepatchIIIwithoutproducingmanybranchesandleaves(step26),andreachesthebestpatchIatthebottomleftcorner(step39).Afterquicklyspreadingoverthispatch(step51),theplantsearchesforfurtherfavorableareas(step62).TheattempttoreachpatchIIfails(step82).Theplanttriesagain,andthistimesucceeds(steps101and116).LightconditionsinpatchIIarenotsufcient,however,tosustainthecontinuouspresenceoftheplant(step134).Thecolonydisappears(step153)untilthepatchisreachedagainbyanewwaveofpropagation(steps161and182).ThesustainedoccupationofpatchIandtherepetitiveinvasionofpatchIIrepresentanemergingbehaviorofthemodel,difculttopredictwithoutrunningsimulations.Variantsofthismodel,includ-ingotherbranchingarchitectures,responsestotheenvironment,andlayoutsofpatchesintheenvironment,wouldmakeitpossibletoanalyzedifferentforagingstrategiesofclonalplants.Afurtherextensioncouldreplacetheempiricalassumptionsregardingplantresponseswithamoredetailedsimulationofplantphysiology(forexample,includingproductionofphotosynthatesandtheirtrans-portandpartitionbetweenramets).Suchphysiologicalmodelscouldprovideinsightintotheextenttowhichtheforagingpatternsoptimizeplantsaccesstoresources[17]. Figure7:DevelopmentofahypotheticalclonalplantsimulatedusinganextensionofL-system3.Theindividualimagesrepresentstructuresgeneratedin9,26,39,51,62,and82derivationsteps(top),followedbystructuresgeneratedin101,116,134,153,161,and182steps(bottom).7AMODELOFROOTDEVELOPMENTBackground.Thedevelopmentofrootsprovidesmanyexamplesofcomplexinteractionswiththeenvironment,whichinvolveme-chanicalproperties,chemicalreactions,andtransportmechanismsinthesoil.Inparticular,themainrootandtherootletsabsorbwaterfromthesoil,locallychangingitsconcentration(volumeofwaterperunitvolumeofsoil)andcausingwatermotionfromwater-richtodepletedregions[24].Thetipsoftheroots,inturn,followthegra-dientofwaterconcentration[12],thusadaptingtotheenvironmentedbytheirownactivities.BelowwepresentasimpliedimplementationofthemodelofrootdevelopmentoriginallyproposedbyClausnitzerandHopmans[12].Weassumeamorerudimentarymechanismofwatertransport,namelydiffusioninauniformmedium,assuggestedbyLiddellandHansen[37].Theunderlyingmodelofrootarchitectureissim-ilartothatproposedbyDiggle[16].Forsimplicity,wefocusonmodeloperationintwo-dimensions.Communicationspecication.Theplantinteractswiththeen-vironmentusingcommunicationmodules?c;locatedattheapicesoftherootsystem.Amessagesenttotheenvironmentin-cludestheturtleposition,theheadingvector,thevalueof representingtherequested(optimal)wateruptake,andthevalueofparameterrepresentingthetendencyoftheapextofollowthegradientofwaterconcentration.Amessagereturnedtotheplantspeciestheamountofwateractuallyreceivedbytheapexasthevalueofparameter,andtheanglebiasingdirectionoffurthergrowthasthevalueof inCCout Figure8:DenitionofthebiasingangleThemodeloftheenvironment.Theenvironmentmaintainsaofwaterconcentrations,repre-sentedasanarrayoftheamountsofwaterinsquaresamplingareas.Wateristransportedbydiffusion,simulatednumericallyusingdifferencing[41].Theenviron-mentrespondstoarequestforwa-terfromanapexlocatedinanareai;jbygrantingthelesserofthevaluesrequestedandavailableatthatlocation.Theamountofwaterinthesampledareaisthendecreasedbytheamountreceivedbytheapex.Theenvironmentalsocalculatesalinearcombinationtheturtleheadingvectorandthegradientofwaterconcentration(estimatednumericallyfromthewaterconcentrationsinthesampledareaanditsneighbors),andreturnsananglebetweenthevectors(Figure8).Thisangleisusedbytheplantmodeltobiasturtleheadinginthedirectionofhighwaterconcentration.Therootmodel.TheopenL-systemrepresentingtherootmodelmakesuseofarraysthatspecifyparametersforeachbranchingorder(mainaxis,itsdaughteraxes,).TheparametervaluesarelooselybasedonthosereportedbyClausnitzerandHopmans[12].neN3/*max.branchingorder+1*/Req[N]=0.1,0.4,0.05,/*requestednutrientintake*/MinReq[N]=0.01,0.06,0.01,/*minimumnutrientintake*/ElRate[N]=0.55,0.25,0.55,/*maximumelongationrate*/MaxLen[N]=200,5,0.8,/*maximumbranchlength*/Sens[N]=10,0,0,/*sensitivitytogradient*/Dev[N]=30,75,75,/*deviationinheading*/*/Ð1]=30,60,/*delayinbranchgrowth*/*/Ð1]=90,90,/*branchingangle*/*/Ð1]=1,0.5/*distancebetweenbranches*/:A(0,0,0)?E(Req[0],Sens[0]):A(n,s,b)�?E(c,):(s�MaxLen[n])||(cMinReq[n]):A(n,s,b)&#x-255;?E(c,(n&#x-255;=N1)||(bBrSpace[n])BrSpace[n])g!+(nran(,Dev[n]))F(h)A(n,s+h,b+h)?E(Req[n],Sens[n]):A(n,s,b)&#x-255;?E(c,(n1)&&(b&#xN000;=BrSpace[n])BrSpace[n])g!+(nran(,Dev[n]))B(n,0)F(h)v[n]))B(n,0)F(h))p4:B(n,tB(n,t!B(n,t+1)p5:B(n,t&#xN000;):t==![+(BrAngle[n])A(n+1,0,0)?E(Req[n+1],Sens[n+1])]:?E(c,Thedevelopmentstartswithanapexfollowedbyacommunica-tionmodule?.Theparametersoftheapexrepresentthebranchorder(0forthemainaxis,1foritsdaughteraxes,),currentaxislength,anddistance(alongtheaxis)tothenearestbranchingpoint. Figure9:Atwo-dimensionalmodelofarootinteractingwithwaterinsoil.Backgroundcolorsrepresentconcentrationsofwaterdiffus-inginsoil(blue:high,black:low).Theinitialandboundaryvalueshavebeensetusingapaintprogram. Figure10:Athree-dimensionalextensionoftherootmodel.Waterconcentrationisvisualizedbysemi-transparentiso-surfaces[55]surroundingtheroots.Asaresultofcompetitionforwater,therootsgrowawayfromeachother.Thedivergencebetweentheirmainaxesdependsonthespreadoftherootlets,whichgrowfasterontheleftthenontheright.describepossiblefatesoftheapexasdescribedbelow.IfthelengthofabranchaxisexceedsapredenedmaximumvalueMaxLenxLenn]characteristictothebranchorder,ortheamountofwaterreceivedbytheapexisbelowtherequiredminimumMinReqqn],theapexdies,terminatingthegrowthoftheaxis(pro-Ifthebranchorderisequaltothemaximumvalueassumedinthemodel(1),orthedistancetotheclosestbranchingpointontheaxisislessthanthethresholdvalueBrSpacepacen],theapexaddsanewsegmenttotheaxis(production).ThelengthistheproductofthenominalgrowthincrementElRateen]andtheratiooftheamountofwaterreceivedbytheapextotheamountReqqn].Thenewsegmentisrotatedwithrespecttoitspredecessorbyananglenran;Devvn]),wherenranisarandomfunctionwithanormaldistribution.Themeanvalue,returnedbytheenvironment,biasesthedirectionofgrowthtowardsregionsof higherwaterconcentration.ThestandarddeviationDevvn]char-acterizesthetendencyoftherootapextochangedirectionduetovariousfactorsnotincludedexplicitlyinthemodel.Ifthebranchorderislessthanthemaximumvalueassumedinthemodel(1),andthedistancetotheclosestbranchingpointontheaxisisequaltoorexceedsthethresholdvalueBrSpacepacen],theapexcreatesanewbranchinitial).OtheraspectsofapexbehaviorarethesameasthosedescribedbyproductionAfterthedelayofDelln]steps(production),thebranchinitialistransformedintoanapexfollowedbythecommunicationmod-ule?),givingrisetoanewrootbranch.Productionremovescommunicationmodulesthatarenolongerneeded.Asampletwo-dimensionalstructureobtainedusingthedescribedmodelisshowninFigure9.Theapexofthemainaxisfollowsthegradientofwaterconcentration,withsmalldeviationsduetorandomfactors.Theapicesofhigher-orderaxesarenotsensitivetothegradientandchangedirectionatrandom,withalargerstandarddeviation.Theabsorptionofwaterbytherootandtherootletsdecreaseswaterconcentrationintheirneighborhood;aneffectthatisnotfullycompensatedbywaterdiffusionfromthewater-richareas.Lowwaterconcentrationstopsthedevelopmentofsomerootletsbeforetheyhavereachedtheirpotentialfulllength.Figure10presentsathree-dimensionalextensionofthepreviousmodel.Asaresultofcompetitionforwater,themainaxesoftherootsdivergefromeachother(left).Iftheirrootletsgrowmoreslowly,theareaofinuenceofeachrootsystemissmallerandthemainaxesareclosertoeachother(right).Thisbehaviorisanemergentpropertyofinteractionsbetweentherootmodules,mediatedbytheenvironment.8MODELSOFTREESCONTROLLEDBYLIGHTBackground.Lightisoneofthemostimportantfactorsaffect-ingthedevelopmentofplants.Intheessentiallytwo-dimensionalstructuresdiscussedinSection5,competitionforlightcouldbeconsideredinamannersimilartocollisiondetectionbetweenleavesandapices.Incontrast,competitionforlightinthree-dimensionalstructuresmustbeviewedaslong-rangeinteraction.Specically,shadowscastbyonebranchmayaffectotherbranchesatsignirstsimulationsofplantdevelopmentthattakethelocallightenvironmentintoaccountareduetoGreene[25].Heconsideredtheentireskyhemisphereasasourceofilluminationandcomputedtheamountoflightreachingspecicpointsofthestructurebycastingraystowardsanumberofpointsonthehemisphere.AnotherapproachwasimplementedbyKanamaruetal.[35],whocomputedtheamountoflightreachingagivensamplingpointbyconsideringitacenterofprojection,fromwhichallleafclustersinatreewereprojectedonasurroundinghemisphere.Thedegreetowhichthehemispherewascoveredbytheprojectedclustersindicatedtheamountoflightreceivedbythesamplingpoint.Inbothcases,themodelsofplantsrespondedtotheamountandthedirectionoflightbysimulatingheliotropism,whichbiasedthedirectionofgrowthtowardsthevectorofthehighestintensityofincominglight.Subsequently,Chibaetal.extendedthemodelsbyKanamaruusingmoreinvolvedtreemodelsthatincludedamechanismsimulatingtheowofhypotheticalendogenousinformationwithinthetree[10,11].Abiologicallybetterjustiedmodel,formulatedintermsofproductionanduseofphotosynthatesbyatree,wasproposedbyTakenaka[52].Theamountoflightreachingleafclusterswascalculatedbysamplingaskyhemisphere,asintheworkbyGreene.BelowwereproducethemainfeaturesoftheTakenakamodelusingtheformalismofopenL-systems.Dependingontheunderlyingtreearchitecture,itcanbeappliedtosynthesizeimagesofdeciduousandconiferoustrees.Wefocusonadeciduoustree,whichrequiresaslightlysmallernumberofproductions.Communicationspecication.Theplantinteractswiththeenvi-ronmentusingcommunicationmodules?.Amessagesentbytheplantincludesturtleposition,whichrepresentsthecenterofasphericalleafcluster,andthevalueofparameter,whichrepresentstheclustersradius.Theenvironmentrespondsbysettingtotheux[14]oflightfromtheskyhemisphere,reachingthecluster.Themodeloftheenvironment.Onceallmessagesdescribingthecurrentdistributionofleavesonatreehavebeenreceived,theenvironmentalprocesscomputestheextentofthetreeinthedirections,encompassesthetreeinatightgrid(3232voxelsinoursimulations),andallocatesleafclusterstovoxelstospeedupfurthercomputations.Theenvironmentalprocessthenestimatesthelightfromtheskyhemispherereachingeachcluster(shadowscastbythebranchesareignored).Tothisend,thehemisphereisrepresentedbyasetofdirectionallightsources(9inthesimulations).Theuxdensities(radiosities)ofthesourcesapproximatethenon-uniformdistributionoflightfromthesky(cf.[52]).Foreachleafclusterandeachlightsource,theenvironmentdeterminesthesetofleafclustersthatmayshade.Thisisachievedbycastingarayfromthecenterofinthedirectionofandtestingforintersectionswithotherclusters(thegridacceleratesthisprocess).Inordernottomissanyclustersthatmaypartiallyocclude,theradiusofeachclusterisincreasedbythemaximumvalueofclusterradiusTocalculatetheuxreachingcluster,thisclusterandallclustersthatmayshadeitaccordingtothedescribedtestsareprojectedonaplaneperpendiculartothedirectionoflightfromthesource.Theimpactofaclusteronthereachingclusterthencomputedaccordingtotheformula:istheareaoftheprojectionofistheareaoftheintersectionbetweenprojectionsof,andisthelighttransmittancethroughleafcluster(equalto0.25inthesimulations).Ifseveralclusters,theirinuencesaremultiplied.ThetotaluxreachingclusteriscalculatedasthesumoftheuxesreceivedfromeachlightsourceTheplantmodel.Inadditiontothecommunicationmodule?theplantmodelincludesthefollowingtypesofmodules:Apexvig;del.Parametervigrepresentsvigor,whichdeter-minesthelengthofbranchsegments(internodes)andthediam-eterofleafclustersproducedbytheapex.Parameterisusedtointroduceadelay,neededforpropagatingproductsofphoto-synthesisthroughthetreestructurebetweenconsecutivestagesofdevelopment(years).vig;p;age;del.Parametersdenotetheleafradiusvigtheamountofphotosynthatesproducedinunittimeaccordingtotheleafsexposuretolight,thenumberofyearsforwhichaleafhasappearedatagivenlocationage,andthedelaywhichplaysthesameroleasintheapices.vig.Consistentwiththeturtleinterpretation,the vigindicatestheinternodelength.Branchwidthsymbol!w;p;n,alsousedtocarrytheendoge-nousinformationow.Theparametersdetermine:thewidthofthefollowinginternode,theamountofphotosynthatesreach-ingthesymbolslocation,andthenumberofterminalbranchsegmentsabovethislocationThecorrespondingL-systemisgivenbelow.137.5/*divergenceangle*/5/*directionchange-nobranching*/20/*branchingangle-mainaxis*/32/*branchingangle-lateralaxis*/neW0.02/*initialbranchwidth*/neVD0.95/*apexvigordecrement*/neDel30/*delay*/neLS5/*howlongaleafstays*/neLP8/*fullphotosynthateproduction*/neLM2/*leafmaintenance*/nePB0.8/*photosynthatesneededforbranching*/nePG0.4/*photosynthatesneededforgrowth*/neBM0.32/*branchmaintenancecoefcient*/neBE1.5/*branchmaintenanceexponent*/neN25/*thresholdforshedding*/Consider:?E[]!L/*forcontextmatching*/:!(W,1,1)F(2)L(1,LP,0,0)A(1,0)[!(0,0,0)]!(W,0,1):A(vig,del):del:L(vig,p,age,del):(age)(:L(vig,p,age,del):(age)(:L(vig,p,age,del)&#xLS&&;Þl=;&#x=Del;?E(r):(age)&&&#xLS-2;瀀(r*LP=LM)&&(del==Del):L(vig,p,age,del)&#xLS-2;瀀?E(r):((age==LS)||(r*LP))&&(del==Del):?E(r)A(vig,del):r*LPr*LP( 2)!(W,PB,1)F(vig)L(vig,LP,0,0)A(vig,0)[!(0,0,0)]!(W,0,1)])!(W,0,1)F(vig)L(vig,LP,0,0)/A(vig,0):?E(r)A(vig,del):r*LPLM&#x-255;PGG()!(W,PG,1)F(vig)L(vig,LP,0,0)A(vig,0):?E(r)A(vig,del):r*LPLMPG:?E(r):!(w)&#x=-27;�L(vig,p,age,del)[!(w0.5;p=p&#x=-27;�(p0)||(n!(w,p,n:!(w)&#x=-27;�L(vig,p,age,del)[!(wThesimulationstartswithastructureconsistingofabranchsegment,supportingaleafandanapex).Therstbranchwidthsymbol!denesthesegmentwidth.Twoadditionalsymbols!followingtheapexcreatevirtualbranches,"neededtoprovidepropercontextforproductions.Thetreegrowsinstages,withthedelayofDel1derivationstepsbetweenconsecutivestagesintroducedbyproductionfortheapicesandfortheleaves.Immediatelybeforeeachnewgrowthstage,communicationsymbolsareintroducedtoinformtheenvironmentaboutthelocationandsizeoftheleafclusters().Ifthereturnedbytheenvironmentresultsintheproductionofphotosynthates 4816122024 Figure11:Thenumberofterminalbranchsegmentsresultingfromunrestrictedbifurcationofapices(continuousline),comparedtothenumberofsegmentsgeneratedinasimulation(isolatedpoints)exceedingtheamountneededtomaintainacluster,itremainsinthestructure().Otherwiseitbecomesaliabilitytothetreeanddies().Anotherconditiontoproductionpreventsaleaffromoccupyingthesamelocationformorethanalsodeterminesthefateoftheapex,capturedbypro-.Iftheamountofphotosynthatestransportedfromthenearbyleafexceedsathresholdvalue,theapexproducestwonewbranches().Thesecondparameterinrstbranchsymbol!issetto,tosubtracttheamountofphotosynthatesusedforbranchingfromtheamountthatwillbetransportedfurtherdown.Thelengthofbranchsegmentsvigreducedwithrespecttothemothersegmentbyapredenedfactor,reectingagradualdecreaseinthevigorofapiceswithage.Thebranchwidthmodules!followingtherstapexareintro-ducedtoprovidecontextrequiredbyproductions,asintheaxiom.Iftheamountofphotosynthatestransportedfromtheleafisinsufcienttoproducenewbranches,butabovethethreshold,theapexaddsanewsegmenttothecurrentbranchaxiswithoutcreatingalateralbranch().Again,avirtualbranchcontainingthebranchwidthsymbol!isbeingaddedtoprovidecontextforproductionsIftheamountofphotosynthatesisbelow,theapexremainsdor-mant().Communicationmodulesnolongerneededareremovedfromthestructure(capturestheendogenousinformationowfromleavesandterminalbranchsegmentstothebaseofthetree.First,itdeterminestheradiusofthemotherbranchsegmentasafunctionoftheradiiofthesupportedbranches: Thus,acrosssectionofthemothersegmenthasanareaequaltothesumofcrosssectionsofthesupportedsegments,aspostulatedintheliterature[40,46].Next,productioncalculatesthephotosynthatesintothemothersegment.Itisdenedasthesumofowsreceivedfromtheassociatedleafandfrombothdaughterbranches,decreasedbytheamountw=Wrepresentingthecostofmaintainingthemothersegment.Finally,0calculatesthenumberofterminalbranchsegmentssupportedbythemothersegmentasthesumofthenumbersofterminalsegmentssupportedbythedaughterbranches,takeseffectiftheispositive(thebranchisnotaliabilitytothetree),orifthenumberofsupportedterminals Figure12:Atreemodelwithbranchescompetingforaccesstolight,shownwithouttheleaves Figure13:Aclimbingplantgrowingonthetreefromthepreviousisaboveathreshold.IftheseconditionsarenotsatisÞed,removes(sheds)thebranchfromthetreeusingthecutsymbol%.Thecompetitionforlightbetweentreebranchesismanifestedbytwophenomena:reducedbranchingordormancy Figure14:Amodelofdeciduoustreescompetingforlight.Thetreesareshowninthepositionofgrowth(top)andmovedapart(bottom)torevealtheadaptationofcrowngeometrytothepresenceoftheneighbortree.ofapicesinunfavorablelocallightconditions,andsheddingofbrancheswhichdonotreceiveenoughlighttocontributetothewholetree.Bothphenomenalimittheextentofbranching,thuscontrollingthedensityofthecrown.ThispropertyofthemodelissupportedbythesimulationresultsshowninFigure11.Ifthegrowthwasunlimited(productionwasalwayschosenover),thenumberofterminalbranchsegmentswoulddoubleeveryyear.Duetothecompetitionforlight,however,thenumberofterminalsegmentsobservedinanactualsimulationincreasesmoreslowly.Forrelatedstatisticsusingadifferenttreearchitecturesee[52].AtreeimagesynthesizedusinganextensionofthepresentedmodelisshowninFigure12.Thekeyadditionalfeatureisagradualreductionofthebranchingangleofayoungbranchwhosesisterbranchhasbeenshed.Astheresult,theremainingbranchassumestheroleoftheleadingshoot,followingthegeneralgrowthdirectionofitssupportingsegment.Branchsegmentsarerepresentedastexture-mappedgeneralizedcylinders,smoothlyconnectedatthebranchingpoints(cf.[6]).Thebarktexturewascreatedusingapaintprogram.Asanillustrationoftheßexibilityofthemodelingframeworkpre-sentedinthispaper,Figure13showstheeffectofseedingahypo-theticalclimbingplantnearthesametree.Theplantfollowsthesurfaceofthetreetrunkandbranches,andavoidsexcessivelydense Figure15:Amodelofconiferoustreescompetingforlight.Thetreesareshowninthepositionofgrowth(top)andmovedapartcolonizationofanyparticulararea.Thus,themodelintegratessev-eralenvironmentally-controlledphenomena:thecompetitionoftreebranchesforlight,thefollowingofsurfacesbyaclimbingplant,andthepreventionofcrowdingasdiscussedinSection6.Leavesweremodeledusingcubicpatches(cf.[46]).InthesimulationsshowninFigure14twotreesdescribedbythesamesetofrules(youngerspecimensofthetreefromFigure12)competeforlightfromtheskyhemisphere.Movingthetreesapartaftertheyhavegrownrevealstheadaptationoftheircrownstothepresenceoftheneighbortree.Thissimulationillustratesboththenecessityandthepossibilityofincorporatingtheadaptivebehaviorintotreemodelsusedforlandscapedesignpurposes.Thesamephenomenonappliestoconiferoustrees,asillustratedinFigure15.ThetreemodelissimilartotheoriginalmodelbyTakenaka[52]andcanbeviewedasconsistingofapproximatelyhorizontaltiers(asdiscussedinSection5)producedinsequencebytheapexofthetreestem.ThelowertiersarecreatedÞrstandthereforepotentiallycanspreadmorewidelythentheyoungertiershigherup(thephaseeffect[46]).Thispatternofdevelopmentisaffectedbythepresenceoftheneighboringtree:thecompetitionforlightpreventsthecrownsfromgrowingintoeachother.ThetreesinFigure15retainbranchesthatdonotreceiveenoughlight.Incontrast,thetreesinthestandpresentedinFigure16shed Figure16:Relationshipbetweentreeformanditspositioninabranchesthatdonotcontributephotosynthatestotheentiretree,usingthesamemechanismasdescribedforthedeciduoustrees.Theresultingsimulationrevealsessentialdifferencesbetweentheshapeofthetreecrowninthemiddleofastand,attheedge,oratthecorner.Inparticular,thetreeinthemiddleretainsonlytheupperpartofitscrown.Inlumberindustry,thelossoflowerbranchesisusuallyadesirablephenomenon,asitreducesknotsinthewoodandtheamountofcleaningthattreesrequirebeforetransport.Simulationsmayassistinchoosinganoptimaldistanceforplantingtrees,whereself-pruningismaximized,yetthereissufÞcientspacebetweentreestooallowforunimpededgrowthoftrunksinheightanddiameter.9CONCLUSIONSInthispaper,weintroducedaframeworkforthemodelingandvisu-alizationofplantsinteractingwiththeirenvironment.Theessentialelementsofthisframeworkare:asystemdesign,inwhichtheplantandtheenvironmentaretreatedastwoseparateprocesses,communicatingusingastan-dardinterface,andthelanguageofopenL-systems,usedtospecifyplantmodelsthatcanexchangeinformationwiththeenvironment.Wedemonstratedtheoperationofthisframeworkbyimplementingmodelsthatcapturecollisionsbetweenbranches,thepropagationofclonalplants,thedevelopmentofrootsinsoil,andthedevelopmentoftreecrownscompetingforlight.Wefoundthattheproposedframeworkmakesitpossibletoeasilycreateandmodifymodelsspanningawiderangeofplantstructuresandenvironmentalpro-cesses.Simulationsofthepresentedphenomenawerefastenoughtoallowinteractiveexperimentationwiththemodels(Table1).Therearemanyresearchtopicsthatmaybeaddressedusingthesimulationandvisualizationcapabilitiesoftheproposedframework.Theyinclude,forinstance:Fundamentalanalysisoftheroleofdifferentformsofinforma-tionßowinplantmorphogenesis(inparticular,therelationship Numberof Derivation Time Fig. branch leaf steps yrs sim. render. segments clusters 5 138 140 5 5 1s 786 229 182 50s 4194 34b 186 67s 37228 448b 301 15min 70s 12 22462 19195 744 24 22min 13s 15 13502 3448 194 15 4min SimulationandrenderingusingOpenGLona200MHz/64MBIndigoactiveapiceswithoutgeneralizedcylindersandtexturemappingbranchingstructurewithoutneedlesTable1:Numbersofprimitivesandsimulation/renderingtimesforgeneratingandvisualizingselectedmodelsbetweenendogenousandexogenousßow).ThisisacontinuationoftheresearchpioneeredbyBell[4]andHondaetal.[7,33].Developmentofacomprehensiveplantmodeldescribingthecyclingofnutrientsfromthesoilthroughtherootsandbranchestotheleaves,thenbacktothesoilintheformofsubstancesreleasedbyfallenleaves.DevelopmentofmodelsofspeciÞcplantsforresearch,cropandforestmanagement,andforlandscapedesignpurposes.Themodelsmayincludeenvironmentalphenomenanotdiscussedinthispaper,suchastheglobaldistributionofradiativeenergyinthetreecrowns,whichaffectstheamountoflightreachingtheleavesandthelocaltemperatureofplantorgans.Thepresentedframeworkitselfisalsoopentofurtherresearch.Tobegin,theprecisefunctionalspeciÞcationoftheenvironment,im-pliedbythedesignofthemodelingframework,issuitableforaformalanalysisofalgorithmsthatcapturevariousenvironmentalprocesses.Thisanalysismayhighlighttradeoffsbetweentime,memory,andcommunicationcomplexity,andleadtoprogramsmatchingtheneedsofthemodeltoavailablesystemresourcesinanoptimalmanner.Adeeperunderstandingofthespectrumofprocessestakingplaceintheenvironmentmayleadtothedesignofamini-languageforenvi-ronmentspeciÞcation.AnalogoustothelanguageofL-systemsforplantspeciÞcation,thismini-languagewouldsimplifythemodelingofvariousenvironments,relievingthemodelerfromtheburdenoflow-levelprogramminginageneral-purposelanguage.FleischerandBarrÕsworkonthespeciÞcationofenvironmentssupportingcollisionsandreaction-diffusionprocesses[20]isaninspiringstepinthisdirection.Complexityissuesarenotlimitedtotheenvironment,butalsoariseinplantmodels.Theybecomeparticularlyrelevantasthescopeofmodelingincreasesfromindividualplantstogroupsofplantsand,eventually,entireplantcommunities.Thisraisestheproblemofselectingtheproperlevelofabstractionfordesigningplantmodels,includingcarefulselectionofphysiologicalprocessesincorporatedintothemodelandthespatialresolutionoftheresultingstructures.Thecomplexityofthemodelingtaskcanbealsoaddressedatthelevelofsystemdesign,byassigningvariouscomponentsofthemodel(individualplantsandaspectsoftheenvironment)todifferentcomponentsofadistributedcomputingsystem.Thecommunicationstructureshouldthenberedesignedtoaccommodateinformationtransfersbetweennumerousprocesseswithinthesystem.Insummary,webelievethattheproposedmodelingmethodologyanditsextensionswillproveusefulinmanyapplicationsofplantmodeling,fromresearchinplantdevelopmentandecologytoland-scapedesignandrealisticimagesynthesis.AcknowledgementsWewouldliketothankJohannesBattjes,CampbellDavidson,ArtDiggle,HeinjoDuring,MichaelGuzy,NaoyoshiKanamaru,BrunoMoulia,ZbigniewPrusinkiewicz,BillRemphrey,DavidReid,andPeterRoomfordiscussionsandpointerstotheliteraturerele-vanttothispaper.WewouldalsoliketothankBrunoAndrieu,MarkHammel,JimHanan,LynnMercer,ChrisPrusinkiewicz,Pe-terRoom,andtheanonymousrefereesforhelpfulcommentsonthemanuscript.MostimageswererenderedusingtheraytracerbyCraigKolb.ThisresearchwassponsoredbygrantsfromtheNaturalSciencesandEngineeringResearchCouncilof[1]AGRAWAL,P.Thecellprogramminglanguage.ArtiÞcialLife2(1995),37Ð77.[2]ARVO,J.,,D.Modelingplantswithenvironment-sensitiveautomata.InProceedingsofAusgraphÕ88(1988),pp.27Ð33.[3]B,A.Plantform:Anillustratedguidetoßoweringplants.UniversityPress,Oxford,1991.[4]B,A.D.Thesimulationofbranchingpatternsinmodularor-Philos.Trans.RoyalSocietyLondon,Ser.B313[5]B,A.D.,ROBERTS,D.,,A.Branchingpatterns:thesimulationofplantarchitecture.JournalofTheoreticalBiology81(1979),351Ð375.[6]B,J.ModelingtheMightyMaple.ProceedingsofSIG-GRAPHÕ85(SanFrancisco,California,July22-26,1985),inputerGraphics,19,3(July1985),pages305Ð311,ACMSIGGRAPH,NewYork,1985.[7]BORCHERT,R.,ONDA,H.Controlofdevelopmentinthebi-furcatingbranchsystemofTabebuiarosea:Acomputersimulation.BotanicalGazette145,2(1984),184Ð195.[8]BORCHERT,R.,,N.Bifurcationratiosandtheadaptivegeometryoftrees.BotanicalGazette142,3(1981),394Ð401.[9]C,S.G.,C,R.,,I.AfractalbasedPopuluscanopystructuremodelforthecalculationoflightinterception.ForestEcologyandManagement[10]CHIBA,N.,OHKAWA,S.,MURAOKA,K.,,M.Visualsim-ulationofbotanicaltreesbasedonvirtualheliotropismanddormancyTheJournalofVisualizationandComputerAnimation5[11]CHIBA,N.,OHSHIDA,K.,MURAOKA,K.,M,M.,AITO,N.Agrowthmodelhavingtheabilitiesofgrowth-regulationsforsimulatingvisualnatureofbotanicaltrees.ComputersandGraphics18,4(1994),[12]CLAUSNITZER,V.,,J.Simultaneousmodelingoftran-sientthree-dimensionalrootgrowthandsoilwaterßow.PlantandSoil(1994),299Ð314.[13]C,D.Computersimulationofbiologicalpatterngenerationpro-Nature216(October1967),246Ð248. 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VisualModelsofPlantsInteractingwithTheirEnvironmentrMechandPrzemyslawPrusinkiewiczUniversityofCalgaryABSTRACTInteractionwiththeenvironmentisakeyfactoraffectingthedevel-opmentofplantsandplantecosystems.Inthispaperweintroduceamodelingframeworkthatmakesitpossibletosimulateandvisualizeawiderangeofinteractionsatthelevelofplantarchitecture.ThisframeworkextendstheformalismofLindenmayersystemswith DepartmentofComputerScience,UniversityofCalgary,Cal-gary,Alberta,CanadaT2N1N4(mechpwp@cpsc.ucalgary.ca)mayerproposedtheformalismofL-systemsasageneralframeworkforplantmodeling[38,39],andHondaintroducedtheÞrstcomputermodeloftreestructures[32].Fromtheseorigins,plantmodeling Visual Models of Plants Interacting with TheirEnvironmentRadomir Mech and Przemyslaw PrusinkiewiczDepartment of Computer ScienceAbstractInteraction with the environment is a key factor affecting the development of plants and plantecosystems. In this paper we introduce a modeling framework that makes it possible toKeywords: scientific visualization, realistic image synthesis, software design, L-system,modeling, simulation, ecosystem, plant development, clonal plant, root, tree.ReferenceRadomir Mech and Przemyslaw Prusinkiewicz. Visual Models of Plants Interacting with Their Environment.Proceedings of SIGGRAPH 96 (New Orleans, Louisiana, August 4-9, 1996). In Computer Graphics