©CopyrightJASSS
ScottHeckbert(2013)
MayaSim:AnAgent-BasedModeloftheAncientMayaSocial-EcologicalSystem
JournalofArtificialSocietiesandSocialSimulation 16(4)11<http://jasss.soc.surrey.ac.uk/16/4/11.html>
Received:20-Aug-2012Accepted:02-Jun-2013Published:31-Oct-2013
Abstract
ThispaperpresentsresultsfromtheMayaSimmodel,anintegratedagent-based,cellularautomata,andnetworkmodelrepresentingtheancientMayasocial-ecologicalsystem.Themodelrepresentstherelationshipbetweenpopulationgrowth,agriculturalproduction,soildegradation,climatevariability,primaryproductivity,hydrology,ecosystemservices,forestsuccession,andthestabilityoftradenetworks.Agentsrepresentingsettlementsdevelopandexpandwithinaspatiallandscapethatchangesunderclimatevariationandrespondstoanthropogenicimpacts.ThemodelisabletoreproducespatialpatternsandtimelinessomewhatanalogoustothatoftheancientMaya,althoughthisproof-of-conceptmodelrequiresrefinementandfurtherarchaeologicaldataforcalibration.Thispaperaimstoidentifycandidatefeaturesofaresilientversusvulnerablesocial-ecologicalsystem,andemployscomputersimulationtoexplorethistopic,usingtheancientMayaasanexample.Complexsystemsmodellingidentifieshowinterconnectedvariablesbehave,consideringfast-movingvariablessuchaslandcoverchangeandtradeconnections,meso-speedvariablessuchasdemographicsandclimatevariability,aswellasslow-movingvariablessuchassoildegradation.
Keywords:Social-EcologicalSystem,Archaeology,CellularAutomata,NetworkModel,TradeNetwork,Agent-BasedModel
Introduction
1.1 Fewtopicsgainasmuchcross-disciplinaryinterestastheriseandfallofancientcivilisations.Thestoryofdevelopmentanddemiseincomplexsocietiescontainsnarrativesofthehumanendeavourthreatenedbydevastatingdroughts,greedyrulers,foreignimperialism,andoveruseofnaturalresources,amongothers.Societiesare,however,asetofinteractingelementswhichasawholeexpresscharacteristicfeatures,interpretedasemergentpropertiesofunderlyingprocessesatmultiplescales.Designingaholisticapproachtounderstandingsocial-ecologicalsystemsrequiresmethodswhichsimultaneouslyobservepatternsinmanydimensions,akindofobservationforwhichvanderLeeuw(2012)arguesthattraditionalWesternscienceisnotverywellequipped.AnanalogyistheexampleofsolvingaRubik'sCube,inthatonecannotgetthecube'inorder'bydealingfirstwithoneside,thenthenext,andsoforth.Theonlywaytoarriveatorderisbylookingatthepatternsonallsidessimultaneously,andnotfavouringanyparticularoneatanytime(vanderLeeuw2012).Thispaperpresentsamethodtoidentifycandidatefeaturesofaresilientversusvulnerablesocial-ecologicalsystem,andemployscomplexsystemsscience,usingcomputersimulationtoexplorethistopicusingtheancientMayaasanexample.
1.2 Anumberofresearchquestionsarepresentedforexploration:
WhatdynamicsleadtothedevelopmentofthedenselypopulatedandinterconnectedhumangeographyoftheancientMaya?Isitpossibletousecomputationalsocialscienceto'grow'thethreeMayatemporalperiodsofthePreclassic(1000BC-AD250),Classic(AD250-900),andPostclassic(AD900-1500)?Howdoesthesimulatedsocial-ecologicalsystemdevelopandrespondtochangingconditions,andwhatmodelledindicatorswarnofvulnerability?
Inordertoexploretheseresearchquestions,asimulationmodelwasdesignedandcalibratedforthelandscapeofCentralAmerica.Modelrunsproducetemporalandspatialpatternsthatcanbeunderstoodthroughexaminingtheunderlyingassumptionsofthedifferentintegratedcomponentsofthemodel.MayaSimisacombinedagent-based,cellularautomata,andnetworkmodelthatrepresentstheancientMayasocial-ecologicalsystem.Agents,cells,andnetworksareprogrammedtorepresentelementsofthehistoricalMayacivilisation,includingdemographics,trade,agriculture,soildegradation,provisionofecosystemservices,climatevariability,hydrology,primaryproductivity,andforestsuccession.Simulatingtheseincombinationallowspatternstoemergeatthelandscapelevel,effectivelygrowingthesocial-ecologicalsystemfromthebottomup.Thisapproachconstructsanartificialsocial-ecologicallaboratorywheredifferenttheoriescanbetestedandhypothesesproposedforhowthesystemwillperformunderdifferentconfigurations.
1.3 ThemodelisabletoreproducespatialpatternsandtimelinessomewhatanalogoustothatoftheancientMaya'shistory.Thisproofofconceptmodelrequiresrefinementandfurtherarchaeologicaldataforcalibrationtoimproveresults,althoughitisnotedthatthereislittleempiricalevidencebywhichtovalidatesuchmodels,andsuchevidenceisgenerallysite-specificanddiscontinuousthroughtime.
1.4 Thepurposeofthemodelistobetterunderstandthecomplexdynamicsofsocial-ecologicalsystemsandtotestquantitativeindicatorsofresilienceaspredictorsofsystemsustainability.Anintegratedagent-based,cellularautomata,andnetworkmodelwasconstructedusingthesoftwareNetlogo(Wilensky1999).Thefullmodel,codeanddocumentationisavailableinHeckbert(2012)viathewww.openabm.orgwebsite,andfurtherdescriptionofthemodelinthecontextofMayaarchaeologicalliteratureispresentedinHeckbertetal.(inpress).
Methods
2.1 TheMayaSimmodelrepresentssettlementsasagentslocatedinagriddedlandscape.ThemodelisconstructedusingthesoftwareNetlogo(Wilensky1999).Thesoftwareinterface,showninFigure1,presentsthespatialviewofthemodelwithgraphstrackingmodeldataandauserinterfaceforinteractingwiththe
model.Theviewcanbechangedtoobservedifferentspatialdatalayerswithinthemodel.Themodeloperatesataspatialextentof516,484km2witha20km2
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resolution.Temporalextentis650timessteps,eachrepresentingroughly2years.
2.2 Uponmodelinitialisation,baseGISlayersareimportedusingtheNetlogoGISextension.Staticcellvariablesaresettherein,dynamicvariablesareresettodefaultvaluessetontheuserinterface,andsettlementagentsarerandomlyinitialisedinthespatiallandscape.Importedspatialdataincludeelevationandslope(Farretal.2007),soilproductivity(FAO2007),andtemperatureandprecipitation(Hijmansetal.2005).DataforsoilproductivitywasslightlysmoothedusingtheNetlogofunctiondiffusetomakedifferencesalongpolygonboundarieslessstark,butallotherdataisunchangedfromthecitedsources.Dataisresampledata20km2resolutionusingtheNetlogoGISextension(seeWilensky1999).AllGISdataishighresolutionanddeemedtobeofhighqualityexceptpossiblythesoilproductivitydataset.Thelatterdatasetcanbereplacedwithfinerresolutionregionalorcountry-specificdata,butbecausetheregionextendsacrossseveralmoderncountrieswithinconsistentdatasets,thebestgloballyconsistentdatasetavailablewasused.FutureworkwillaimtoreplacetheexistingdatasetwiththeHarmonizedWorldSoilDatabase(FAOetal.2009)forfinerspatialresolutionandconsistencyacrosscountryborders.
Figure1.MayaSimmodelinterfacewithinteractivecontrols,spatialview,andgraphstrackingmodeldata.Agentsoperateonacellularlandscapeandareconnectedbylinkswithinanetwork.
2.3 Thesimulationbeginswithcalculationsofbiophysicalvariablesforwaterflowandnetprimaryproductivity,andthesearefurtherusedtocalculateforestsuccession,agriculturalproduction,andecosystemservices.Settlementagentsinteractwiththespatiallandscapetogenerateagriculturalyieldthroughcropping,derivebenefitfromlocalecosystemservices,andgeneratetradebenefitswithintheirlocaltradenetwork.Thecombinedbenefitsofagriculture,ecosystemservices,andtradedrivesdemographicgrowthincludingmigration.Simulatingtheintegratedsystemrevealshowthesocial-ecologicalsystemfunctionsthroughtime.Additionalagentsincludea'migrant'agentwhosettlenewlocations,a'raindrop'agentwhichrouteshydrologicalsurfaceflow,anda'networksearchagent'whotraverseslinksbetweenconnectedsettlementnodestocalculatenetworkstatistics.Thefollowingsectiondescribesbiophysicalandanthropogenicprocessesthataretrackedinthemodel.
Biophysicalfunctions:climate,hydrology,soil,primaryproductivity,forestsandecosystemservices
2.4 Spatialdataforprecipitationandtemperature(Hijmansetal.2005)representingcurrentconditions(1950topresentday)isadjustedwithinthemodelusingassumptionsthatmimicthepaleoclimaticvariationpresentedinPruferetal.(2011).Variationinprecipitationismorepronouncedtowardsthenorthwestofthecasestudyarea.Thisdiagonalnorth-south/east-westvariationinrainfalliscreatedoverthecurrent-dayGISdataforprecipitation,usingthefunction:
(1)
whereRj,Tisprecipitation[mm]forcelljatinitialtimestepT,andCLnisalocalizedrainfalleffectduetothepresenceofclearedlandonneighbouringcellsn=1…8withweightingparameterδdeterminingthestrengthofthiseffect.DFjisthedistance[km]ofeachcellfromthetopnorthwestcornerofthemapandisthefurthestdistancedcellfromthispoint.RCtcyclesfrom+20%to-10%linearlyovera56timestepcycle,andt=1…650.Thisfunctionservestoreduceandincreaserainfallcyclically,withamorepronouncedeffectfurthertowardsthenorthwest.
2.5 Asurface-flowhydrologicalmodelwasdevisedtorecreatepastconditionsbasedonelevationandrainfall.Thesedataareusedtocalculatesurfaceflowandlocationofpotentialseasonalstandingwater,consistentwithReaney(2008).Eachcellgeneratesamobile'raindrop'agentcontainingprecipitationvolumeRj,T.Theraindropsfollowtheelevationdata,repeatedlymovingtotheadjacentcellwiththelowestelevation(andconsideringthesummedvolume[mm]ofraindropsalreadyatthatlocation).Ifraindropscannotmove(i.e.,alocationisflooded),theraindrops'pool'andformriverandlakepatterns,asshowninFigure2depictingsimulatedwaterflowunderclimatevariationassumptions.ThisflowpatterncanbevalidatedagainsthydrologicalprocessingfunctionsinGISsoftware.Thefunctionservestomovewaterbasedonelevation,andcangeneratethespatialdistributionandsurfacewaterflowasprecipitationvariesacrosstheclimatecycle.SeeequationsinmodelcodereadilyexploredinHeckbert(2012).Theresultoftheseequationsisafinely-resolvedspatialpatternthatisabletorecreatethepotentialriver,lake,andwetlandsystemsunderclimatevariationassumptions.Animportantcaveatofthemodelassumptions,however,isthelackofheterogeneityininfiltrationratesacrosstheregion.Inreality,thesurfacewaterinmanyareasdrainsrapidlytotheaquiferowingtothekarstlimestonebedrock,howeverspatialdatatoinformthisprocesswasnotabletobesourced,andassuchaconstantrateofinfiltrationisassumedacrossthelandscape,andisatopicforfurtherresearcheffort.
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Figure2.Simulatedsurfacewaterflow[m3](blueonwhitebackground,darkerbluebeinggreaterflow)basedonclimatevariationassumptionsofa)-15%,b)0%,andc)+15%changeinannualprecipitationinequation1.
2.6 ThebaseGISlayersforrainfallandtemperatureareusedtocalculatenetprimaryproductivity,whichinturnisusedtocalculateforestsuccession.GISsoildataisusedtocalculateagriculturalproductivity,andallthesecombinetocalculateprovisionofecosystemservices.Whilewerecognizethattheserelationshipsarecomplex,simplifyingassumptionsallowinitialrepresentationsasmethodologiesarefurtherrefined.
2.7 Forestsuccessionoperatesasacellularautomatamodel,wherethestateofacellisdependentoninternalconditionsandisinfluencedbytheconditionofneighbouringcells.Inthisproof-of-conceptstage,cellstakeononeofthreegeneralforeststatesthatrepresentclimaxforest,secondaryregrowth,andcleared/croppedland,referredtoasstate1,2and3respectively.Theforeststateisdecrementedfor3.5%ofrandomlyselectedcells,torepresentnaturaldisturbance.Thedisturbancerateislinearlyamplifiedbypopulationdensityofnearbysettlementstorepresentlocalwoodharvesting,toamaximumof15%.Cellsadvanceintheirforeststatebasedonthetimesincelastdisturbanceandtherelativenetprimaryproductivityofthecell.Oncethetimesincelastdisturbanceisaboveathreshold(40*NPPmaxj,t /NPPj,tyearsforsecondaryregrowthand100*NPPmaxj,t /NPPj,tyearsforclimaxforest,toaccountforspatialvariationinnetprimaryproductivity)theforestconvertstothenewstate.Forconversiontoclimaxforest,acellularautomatafunctionisappliedthatrequiresanumberofneighbouringcellstoalsocontainclimaxforest.Thisrulerepresentstheneedtohavelocalvegetationforseeddispersal.
2.8 NetprimaryproductivityNPPj,t[gCm2-1yr-1]isafunctionofprecipitationandtemperature,calculatedbasedontheMiamimodel(Lieth1972)as:
(2)
whereRj,tisprecipitation[mm]andTjistemperature[degreesC].
2.9 Foreachcell,agriculturalproductivityAGj,tiscalculatedas:
(3)
whereSPjissoilproductivity(FAO2007)[index1-100],Sjisslope[%],WFjiswaterflowcalculatedasthesumvolumeofwateragentstraversinganygivencellj,asdepictedinFigure2,andSDj,tissoildegradation[%lossofproductivity].Soildegradationoccursataconstantrateof1.5%pertimestepforeachcroppedcell.Thisconstantrateofsoildegradationisobviouslyasimplifyingassumption,assoilmanagementandmaintenanceofsoilproductivityisinitselfacomplexsocial-ecologicalsystem.Theβparametersareweightingsforcalibration,andpresentedinHeckbert(2012).
2.10 Ecosystemservicesaremodelledbyquantifyingtheavailabilityprovisioningservices(asdefinedinMA2005;TEEB2010)relatingtowater,food,andrawmaterials.Thisisasubsetofecosystemservicesanddoesnotincludeafullsetofindicatorswhichwouldincorporatesupportingservices(forexampleerosionprevention),habitatservices(suchasmaintenanceofgeneticdiversity)orculturalservices(suchasinspirationforculture,art,anddesign).Thecurrentecosystemservicesequationincorporatesasubsetoffourimportantservicesprovisionbasedonarablesoils,precipitation,accesstoavailablefreshwater,andtimberresources.EcosystemservicesESjarecalculatedas:
(4)
whereAGj,tistakenfromequation3,Rj,tistakenfromequation1,WFj,tisthesimulatedwaterflowvolume,andFj,tistheforeststate[1-3],ESDj,tisacatch-allproxyvariableforallotherecosystemservicesdegradation[%]asafunctionofpopulationdensity.
Anthropogenicfunctions:agriculture,trade,anddemographics
2.11 Eachsettlementagentimaintainsatleastonecelljforgeneratingagriculturalyield.Settlementsperformanagriculturebenefit-costassessmentconsideringthecostsofproduction,travelcostgiventhedistanceofthecellfromthesettlementsite,andwithlargersettlementsachievingeconomiesofscale,modelledas;
(5)
whereBCAj,tisthetotalbenefitprovidedfromagricultureyield,κj,α,andφ,arecropyieldandslopeparameters,AGj,tisagaintakenfromequation3,γistheestablishmentcostofagriculture(annualvariablecosts),Ojistheagriculturetravelcostasafunctionfordistancefromthecityandaperkmcostparameter,andPj,tispopulationofthesettlement.
2.12 Thebenefit-costofagriculturefunctiongeneratesyieldsthatarespatiallydistributedbasedonindividualconditionsofthecellsandthelocationofsettlements.Costsofproduction,includingdistancefromsettlements,resultsinaddingcroppedcells,generatingyieldandincreasingpopulation,whichinturnaddmore
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croppedcells,butcausessoildegradation.Thesystemadjustsovertimeinresponsetothespatially-explicitagriculturalbenefit-cost.
2.13 Aseriesoffunctionsrepresenttradewithinaspatiallyconnectednetworkofagents.Itisassumedthatthroughtheprocessofspecialization,settlementsthatareconnectedtooneanotherwithinanetworkwillgeneratebenefitsfromtrade.Itisassumedalargernetworkproducesgreatertradebenefits,andalsothemorecentralasettlementiswithinthenetwork,thegreaterthetradebenefitsforthatindividualsettlement.Tomodelthesebenefits,settlementsareconnectedviaanetworkoflinksthatrepresenttraderoutes.Asasimplifyingassumptionofhowtheyconnecttogether,itisassumedwhenasettlementreaches(ordropsbelow)acertainsize,theywilladdroutes(orallowroutestodegrade)tonearbysettlementswithina40kmradius.Ateachtimestep,thesizeofthelocalnetworkiscalculatedaswellaseachsettlement'scentralitywithinthatlocalnetwork,furtherdiscussedbelow.
2.14 Combiningthefunctionsforagriculture,ecosystemservices,andtradebenefit,totalrealincomepercapitaRIiiscalculatedas:
(6)
whereNj,tisthenetworksize[#nodes],Cj,tisthecentrality[degree]andTCiisthetravelcost,andϑparametersarepricesforagriculture,ecosystemservices,andtrade,respectively.BenefitsfromagriculturearecalculatedonlyforcellsundercroppingproductionAGJi,t=1…nwhereasecosystemservicesarecalculatedencompassingtheentire'areaofinfluence'ofeachsettlementIJi,t=1…mwhichisbasedonthepopulationsizeofthesettlement,increasinglinearlytoamaximumof40kmindiameterforthemostpopuloussettlements(thosewithpopulationsgreaterthan15000people),astakenfromHeckbertetal.(inpress)andinterpretedfromChaseandChase(1998).Travelcostmeasurestherelative'friction'ofdifferentlandcovertypes,andisrepresentedas:
(7)
whereSjisslopeandWFj,tissimulatedwaterflowvolume,bothdescribedinpreviousequations,resultinginareasofhigherslopebeingrelativelymorecostlytotravelthrough,mitigatedbythepresenceofflowingwaterforcanoetransport.
2.15 ThetermsNJi,tnetworksize,andCi,tcentralityinequation6arecalculatedusinganetworksearchalgorithmcustom-builtforthisapplication.Thenetworksearchalgorithmisthemostdetailedprocedureinthemodel,andusesarecursivesearchfunctionusing'networkwalkingagents'whotraveleverypossiblecombinationofroutesalonganetworkpath.Thesereporttothenetworknodes(settlements)thetotalsizeofthelocalcluster,andthepositionofthenodewithinthatcluster(degree).Thenetworksearchalgorithmisthelengthiestprocedureintermsofamountofcodewritten,isthemostcomputationallyexpensiveprocedureinthemodel(morethandoublesthetotalruntimeofasimulationof650timesteps).TheoutcomeoftheprocedureisdepictedinFigure3fortimestep250,withthenetworklinkscolouredblackandtherangeofvaluesonthevisualscalehalvedcomparedtofollowingfiguresofthesameindicator,forvisualpurposes.
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Figure3.Simulated'tradestrength'[lasttermofequation6](redonwhitebackground,darkerredbeinggreatertradestrength)basedoncentrality,sizeoflocalnetworkcluster,andtravelcostofeachsettlement(templeicon),connectedviatraderoutes(blacklinks).Alocalclusterisidentifiedforvisualpurposeswithablue
ovaloverthesamecluster,witha)andb)beingzoomed-inviewsofthewidersystemdepictedinc).
2.16 WeseeinFigure3anareahighlightedbyablueoval,zoomed-inatthreedifferentscales.InFigure3a),weseethatthislocalclusterisnotconnectedtoadjacentclusters,andNi,tnetworksize[#ofnodesinnetwork]issmall.InFigure3b)weseethatthetradestrength(redcolouring)isgreaterforlarger-sizedlocalclustersgiventhelargervalueofNi,t.Lastly,inFigure3c)weobservethatcentralnodesinlargerclusters(thoselocatedinthedarkestredregions),havegreatertradestrengthcomparedtonodesattheperipheryofthesamecluster.
2.17 ThevalueofeachcontributiontoRIi,tisdeterminedbytheweightingϑ,whichisstaticforϑAGandϑES;however,thevalueoftradeϑTRtisdynamic.
Specifically,ϑTRtincreaseseachtimerainfalldecreases,accordingtotheclimatevariationassumptions.Theassumptionhereisthatsettlementsspecialize
productionwithinanoveralltradenetworktoincreasethevalueoftradegoodsrelativetoothercommodities.Theeffectistolinearlyincreasethevalueoftradeeachtimetheclimatecycleisindecline.Thisassumptionisobviouslyrudimentary,andfurtherformulationsareexplored.
2.18 AfterdeterminingRIi,t,settlementdemographicsaccountforbirths,deaths,andmigration.Thebirthrateisassumedtoremainconstantat15%,whiledeathrateandout-migrationdecreaselinearlywithincreasedRIi,tpercapita,withamaximumout-migrationrateof15%andamaximumdeathrateof25%perannum.Settlementswithapopulationbelowaminimumnumberrequiredtomaintainsubsistenceagriculturearedeleted.Settlementsthatregisterout-migrationaboveaminimumthresholdofthenumberofpeoplerequiredtomaintainsubsistenceagriculturecreatea'migrantagent'.Themigrantagentusesautilityfunction(Heckbertetal.2010)toselectlocationstocreateanewsettlement.Themigrationutilityfunctioniscalculatedas:
(8)
whereλparametersareweightingsfortravelcostandecosystemservices,andESj,tistakenfromequation4,andDjisthedistancefromtheoriginsettlementtothepotentialnewsettlementsite.
2.19 Modeloutcomesaredependentonparameterizationofequations1-8.Thisstageofmodeldevelopmentisofferedforproofofconcept,fortransparency,andtoreceivefeedbackonmodelassumptions.Modelcodeandparameterizationisavailablefordownloadviathehttp://www.openabm.orgwebsiteinHeckbert(2012).
Results
3.1 Thissectiondescribesresultsandsensitivityanalysisfromsimulationruns.Thisproof-of-conceptstagemodelrevealsfamiliarpatternsanalogoustotheMayahistoricaltimelineandoffersaquantitativebasisfortheseassessments.Themodelwasinitialisedaccordingtodefaultvalues(Heckbert2012).Simulationruns
wereconductedwithatemporalextentof650timestepswithaspatialresolutionof20km2.Figures5-7describemodelresults,depictingmeanvaluesfor20runs(withconfidenceintervals[a=0.05]whererelevant),andreportingonsocial-ecologicalindicators.
3.2 Figure4presentsspatialoutcomesforfiveindicators,at200timestepintervals.Populationdensity,forestcondition,settlementtradestrength,soildegradation,theconditionofecosystemservicesandforestseachcontainanarrativedescribingthedevelopmentandreorganisationofthesimulatedsocial-ecologicalsystem.Bytimestep200,settlementshaveexpandedintoallregions,firstoccupyingareaswithgreaterecosystemservicesandprogressivelygrowingwithagriculturedevelopment.Populationdensitiesarehigherinareaswheresettlementshaveclusteredandformedlocaltradeconnections.Bytimestep400,asthevalueoftradeincreases,thepopulationdramaticallyincreases,extendinglocaltradeconnectionsto'global'connectivity.Anotablefringeexistsbetweentheconnectednetworkandtheperiphery.ThecentreoftheglobaltradenetworkisapproximatelyinthebroadregionwhereancientMayacaptialsofTikalandCaracolexisted.Theconditionoftheforestismarkedlychanged,withonlysmallpatchesofclimaxforestremaininginagriculturallyunsuitableareas,formingecologicalrefugiawithinthenear-completelysettledlandscape.Dramatically,bytimestep600thetradenetworkhasdisintegrated,thecentreofthemostdenselypopulatedareasisnearlyentirelyabandoned,leavingonlyasmallnumberoflocallyconnectedsettlementsofanynotablesizeinwhatwasoncethefringeofthegloballyconnectednetwork.Abandonedcroplandandsignificantlydecreasedfuelwoodharvestingallowsbroad-levelsecondaryregrowth,andclimaxforesteventualyexpandsoutfromitsrefugiatoanextentsimilartoprepopulationexpantionlevels.However,soilsandecosystemservicesremainseverelyimpactedandlargescaleresettlementofdegradedareasisnotpossible.
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Figure4.Populationdensity,forestcondition,settlementtradestrength,soildegradationandtheconditionofecosystemservicesareshownat200timestepintervals.Darkercolouringshowsincreaseda)populationdensity(blue),c)tradestrength(red),d)soildegradation(red),ande)ecosystemservices(green).Forest
conditionb)depictsthreestatesofcleared/croppedcells(yellow),secondaryregrowth(lightgreen)andclimaxforest(darkgreen).
3.3 Themodelreportsquantitiativeindicatorsthroughtimeandcanbeusedtodrilldowninordertoexplorethedynamcisofthedevelopmentandreoganisationobserved.Figure5presentsthetotalpopulationofallsimulatedsettlementsandcontributionstorealincomebyecosystemservices,agricultureandtrade,respectively.Inthefirst_ofthesimulation,ecosystemsservicesprovidethemajorityofvalue.Ecosystemservicesvaluesaresupersededbyagriculturebytimestep150,andbotharesupercededbytradearoundtimestep350.Neitherrecoverastradevaluesdecreaseinthelatterhalfofthesimulationrun,andpopulationadjustsaccordingly.
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Figure5.Totalpopulation[#people,primaryaxis]ofallsimulatedsettlementsovertime,andcontributionstorealincomebyecosystemservices,agricultureandtrade[#proxyvalueunits,secondaryaxis].Ecosystemservicesvaluesaresupercededbyagriculturebytimestep150,andbotharesupercededbytradearound
timestep350.Neitherrecoverastradevaluesdecreaseinthelatterhalfofthesimulationrun.
3.4 TherapidchangeinthevalueoftradecanbeexplainedbyexaminingFigure6,whichdepitcsthetotalnumberofsettlementnodes,thenumberofnodeswithinthelargestcluster,andtotalnaturalcapital,whichisrepresentedasthetotalsumofecosystemservicestakenfromequation4.Thenetworkgrowsfromlocalclusterstoanear-globallyconnectedsystem.Periodicperturbationsgivetheclustersstructurewhichinevitablyformthe'skeleton'oftheglobalstructure.However,whennaturalcapitalhasreacheditslowestlevel,perturbationsresultincascadingfailureinthenetwork.Althoughnaturalcapitalrecovers,thenumberofsettlementsdoesnotduetothelackoftradenetworkstructure.
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Figure6.Totalnumberofsettlementnodesandnumberofnodeswithinthelargestcluster[primaryaxis],andtotalnaturalcapital[totalsumofecosystemservicesvalues,secondaryaxis].Thenetworkgrowsfromlocalclusterstoanear-globallyconnectedsystemthroughgrowthinlinkconnectionsandperiodicperturbations
whichgivetheclustersstructure.However,whennaturalcapitalhasreacheditslowestlevel,perturbationsresultincascadingfailureinthenetwork.
3.5 Figure7depictssoildegradationandforestconditionbythreestatesofcleared/croppedcells(state1),secondaryregrowth(state2)andclimaxforest(state3).Theinitialperiod,correspondingtoroughlythefirstthirdofthesimulationrun,isdescibedbyaccelerateddeclineofclimaxforestandinhibitedregrowthisaresultofcroppingandtimberharvestingforconstructionandfuelwoodasaresultofincreasingpopulationlevels.Thefollowingperiod,correspondingtoroughlythesecondthirdofthesimulationrun,showscontinuedrepressedregrowth,andincreasedareasofcleared/croppedlandaspopulationpressureresultsinmarginallandsbeingputunderagriculturalproduction.Asaresult,soildegradationincreasesatitshighestrateduringthisperiod.Thelastthirdofthesimulationrunshowsarapiddeclineincleared/croppedlandwithcorrespondinglargescalesecondaryregrowth,andeventualsuccessionintoclimaxforestwhichrecoverstonearpre-developmentlevels.
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Figure7.Forestcondition[%oftotalarea,primaryaxis]bythreestatesofcleared/croppedcells(state1),secondaryregrowth(state2)andclimaxforest(state3),andsoildegradation[totalproxysoilproductivityunitslost,secondaryaxis].Accelerateddeclineofclimaxforestandinhibitedregrowthisaresultofcroppingandtimberharvesting.Increasedcleared/croppedarearesultsinhighersoildegradationrates,withsoilsbeginningtorecoverafterclimaxforestagainbecomes
dominant.3.6 ThetimelineforeachindicatorpresentedinFigures5-7revealstheunderlyingdynamicsofthemodelledsocial-ecologicalsystem.Table1presentsan
integratednarrativeofmodelindicatorsat100timestepintervals,akintotheanalogyofexaminingaRubik'sCubebylookingatthepatternsonallsidessimultaneously(vanderLeeuw2012).Thiscomplexsystemsperspectivehighlightsthatareductionistmethodofidentifyingfactorscontributingtoresilienceandvulnerabilitycannotbeusedtoidentifyasinglecause(oralinearprogressionofsinglecauses)oftheunravellingofthesocial-ecologicalsystem.
Table1:Narrativedescriptionofmodelledindicatorsforpopulation,forests,soils,andtradein100timestepintervals
TimeStep
100 200 300 400 500 600
Population Numberofsettlementsgrowsbasedonmigrationtounsettledareaswithhighecosystemservices
Populationincreaseswithagriculturedevelopment
Populationincreaseswithtradevalue
Populationreachesheight,significantperturbationswithclimatevariability
Precipitousdeclineinpopulation
Populationdoesnotrecovertopriorlevelsandsettlementsofanysignificantsizearelocatedattheformerfringeofthemostdenselypopulatedareas
Forests Gradualdeclineofclimax-dominatedforestandgradualincreaseofsecondaryregrowthandcleared/croppedareas.
Roughlyevendistributionofthreeforeststates.
Climaxforestreducedto10%oflandscapearea,timberharvestingandcroppingsuppressessecondaryregrowth.
Cleared/croppedlandaccountsfor70%oflandsurface,climaxforestreducedtorefugia.
Broad-scalesecondaryregrowth.
Climaxforestbeginstoemergefromrefugiatonearlyregainpre-developmentlevels.
SoilsandAgriculture
Agriculturalexpansionintoprimesoils.
Increaseofcroppedareasinbroadlysettledlandscape.
Expansionofagricultureintomarginalandmoredistantlocations,soildegradationrateincreases.
Broad-scalecroppingalbeitwithdiminishingreturnsrelativetolandareaunderproduction.Soildegradationreachesheight.
Abandonmentofcropsandlossofeconomiesofscaleinagricultureproduction,legacysoildegradationinhibitsre-sowingcrops.
Soilsremainunproductiveandareassuitableforagriculturearepredominantlyintheformerfringeofdenselycroppedareas.
Tradeandnetworkstructure
Littleconnectivityoftraderoutes,withsmallnetworkclustersizesof<10settlements.
Localconnectivityoftraderoutesintowell-definedclustersof<20settlements.
Regional-scaleconnectivityofpreviouslocalclusters,withkeysettlementsbecomingcriticalnetworklinks.
Near-globalnetworkemergesandexpandstofringes.Tradevaluesupersedescontributionsfromagricultureorecosystemservices.
Globalnetworkfullyconnectedthroughkeylocations,overallnetworkstructurevulnerabletoperturbance.
Cascadingnetworkfailureresultsinexistenceofonlysmalllocalnetworksattheformerfringe.
Discussionandconclusions
4.1 TheMayaSimmodelpresentsaproof-of-conceptagent-basedmodelrepresentingkeyelementsoftheancientMayasocial-ecologicalsystem.Agents,cells,andnetworksareprogrammedtorepresentdemographics,trade,agriculture,soildegradation,ecosystemservices,climatevariability,hydrology,primaryproductivity,andforestsuccession.Simulatingeachoftheseincombinationallowspatternstoemergeatthelandscapelevel,effectivelygrowingthesocial-ecologicalsystemfromthebottomup.ThisallowsinvestigationofwhatconditionsleadtothedevelopmentofdenselypopulatedandhighlyinterconnectedhumangeographyoftheancientMaya,andrevealsdynamicsofhowpastsocietiesimpactedtheirenvironmentandviceversa.
4.2 Themodelverificationprocessinvolveddebuggingandindividuallytestingmoduleswithintheoverallmodel.Throughthisprocess,itwasobservedthatmodeloutcomeswereparticularlysensitivetotherelationshipbetweensoildegradationandtherateofincreaseintradevalue,againwithintherangeofnon-extremeclimatevariation.Therelationshipliesinthefactthatsettlementswithhigherpercapitarealincome,duetotradeflows,increaseinpopulationandlocalmarginallandsareputunderproduction.Areaswithbettersoilsareabletomaintainpopulationsandtradeconnectionsovermultiplecyclesofdisturbanceandintheendbecomecriticalnodesinthetradenetwork.Thisinturnincreaseslocalsoildegradation.Whensoildegradationbecomessevere,acriticalnodeintheoverallglobaltradenetworkmaynotbeablesustainsufficientlocalpopulationtomaintaintradeconnections.Thedemiseofagloballysignificantnodecanthenresultincascadingfailureinthenetwork,witheffectsnotnecessarilylocatednearby,buteverywhereonthenetwork.
4.3 Theverificationprocessandmanualdebuggingoverthousandsofrunsproducedaninformalsensitivityanalysis,whichrevealedthatthatthemodelledMayasocial-ecologicalsystem,undergivenassumptions,doesnotalwaysreachalargepopulation'peak'.Itisonlyunderanarrowsetofconditionsthatthepopulationisabletogrowlargeenough,andwiththeproperspatialdistribution,toresultinthegloballyconnectedtradenetworkwhichinturnallowsforfurther
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populationgrowth.Inordertogetthepeakwhichisanalogoustotheobservedprehistoricalrecord,arelativebalanceinthesetofparametersrelatingtotrade,ecosystemservices,andagriculturalrealincomemustoccur.Inotherwords,theequationsrelatingtodemographics,soildegradation,andtradevaluemustallbewithinaspecificrangeforthepopulationtorisesubstantiallyenoughtocreatepatternsthatarerecognisableasPreclassic,ClassicandPostClassictemporalperiods.Thetwoprimaryparametersthatseemtorequirebalancearetheincreaseintradevalueandtherateofsoildegradation.
4.4 Furtherresearchwillfocusoncalibratingmodelinputsfromagronomicmodels,inputtingfinerresolutionsoilandvegetationdata,andrunningthemodeloverageneratedpaleoclimatereconstruction.Rudimentaryassumtionsonhowtradevaluechangescanbereplacedwithothercandiateassumtions.FuturevalidationispossibleusingGIS-basedarcheologicalsitedataforsettlementlocationsovertime,comparisonofsimulatedbuiltareastoLIDARdata,comparingpollencorecontenttosimulatedcatchmentlandcover,aswellasnetworkconnectivityofsettlementsasevidencedthroughceramicinteractionsovertime.
4.5 Revisitingresearchquestionsinlightofthemodelresults,wecanidentifycriticaldynamicsofhowtheMayacametodevelopintoahighlyinterconnectedanddenselypopulatedsocial-ecologicalsystem.Theseprocessescanbeinterpretedintoatemporalpatternofdevelopmentandreorganisation,thatinthesimulatedmodelproducesoutcomesroughlyanalogoustothearchaeologicaldescriptionoftheMayaPreclassic(<modeltimestep200),Classic(roughlytimestep200-450),andPostclassicperiods(>modeltimestep450).Fromacomplexsystemsscienceperspective,thereisnoonecauseofthedramaticre-organisationwhichoccurstomarktheendoftheClassicperiod,andnotablyinthismodelthereisno'drought',merelycyclicalclimatevariabilitywhoseeffectcompletelydependsontheentiretyoftherestofthestatevariablesinthesocial-ecologicalsystem.Climatevariability,soildegradation,deforestation,demographicpressure,andthephysicalconfigurationoftradenetworksareallfactorswhichconcurrentlycontributetoresilienceorvulnerability,andcannotbe'ordered'or'prioritised'intoanygivensequenceofeventswhichcausesthere-organisationintothePostclassicperiod.Complexsystemsmodellingcanhoweveridentifyhowthesevariablesinteract,andprovidesapictureofhowaninterconnectedsystemrespondsgivenitsembeddedfast-movingvariablessuchaslandcoverchangeandtradeconnections,meso-speedvariablessuchasdemographicsandclimatevariability,aswellasslow-movingvariablessuchassoildegradation.
Acknowledgements
TheauthorwouldliketoacknowledgethecontributionsofChristianIsendahl,JoelGunn,AndrewReeson,SimonBrewer,TimBaynes,VernonScarborough,ArlenChase,DianeChase,RobertCostanza,JohnMurphy,DerekRobinson,NicholasDunning,CarstenLemmen,LaelParrot,TimothyBeach,SherylLuzzadder-Beach,DavidLentz,PaulSinclair,CaroleCrumleyandSandervanderLeeuw.ThisprojectwassupportedbyAlbertaInnovatesTechnologyFutures,PortlandStateUniversity,ArizonaStateUniversity,UppsalaUniversity,andUniversityofCincinnati.
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