Converse USGS Patuxent WRC Laurel MD USA Steve Morey USFWS Portland OR USA Abstract Consider the problem of protecting endangered species by selecting patches of land to be used for conservation purposes Typically the availability of patches changes ID: 31740
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(a)Rarespeciesconsideredinourcasestudy (b)Problemdomain (c)ControlledDBNFigure1:(a)Thethreetaxaconsideredinourcasestudyincludethestreakedhornedlark(left),Taylor'scheckerspot(middle)andtheMazamapocketgopher(right).(b)Illustrationoftheproblemdomain.Amapispartitionedintoparcels(whitecells),whicharegroupedintocontiguouspatches(red)ofland.Wemodelannualsurvivalandcolonizationwithinselectedpatches.(c)Illustrationofourmetapopulationmodel.speciessurvivalasacontrolledDynamicBayesianNetwork(DBN,seee.g.,KollerandFriedman[2009],andillustratedinFig.1(c))withpriorprobabilityP[Z1]andtransitionprob-abilitiesP[Zt+1jZt;R;t],i.e.,thepresenceofspeciesattimet+1dependsonthepresenceattimetaswellaswhichpatcheshavebeenselectedforconservation,andenvironmen-talconditionst(e.g.,modelingtheeffectofaharshwinter).Hereby,thetransitionprobabilityfactorizesasP[Zt+1jZt;R;t]=YiYpPhZ(i)p;t+1jZt;R;ti;(1)andtheenvironmentalconditions~=(1;:::;T)formaMarkovchain.WeneedtocapturetwoaspectswiththespeciessurvivalmodelPhZ(i)p;t+1jZt;R;ti:thefactthatapopulationmayormaynotsurviveonitsownwithinaparcel,andthefactthatotherpopulationsofthesamespeciesmaycolonizeitfromnearbyparcels.Thesedistributionscanbequitecomplex,anddependonhabitatattributesoftheparcels(e.g.,vegetation,soils,etc.)aswellaspropertiesoftheparticularreserves(i.e.,whetherthecontainedparcelsareseparatedbyroadsorwaterwayswhichhindermigration),andglobalproperties(e.g.,thelikelihoodofaharshwinter).Inx4.1,wepresentdetailsaboutthemodelsusedinourstudy.2.2StaticReserveDesignOnceweareabletomodelthepopulationdynamicsofthespecies,wewouldliketochooseareserveRtoensurelong-termpersistence.Onenaturalgoalistodeneanobjectivefunctionf(i):2P!Rsuchthatf(i)(R)=Ph9P2R;p2P:Z(i)p;T=1i;quantiestheprobabilitythatspeciesiisstillpresentinatleastoneparcelinthereserveaftersomepredictionhori-zonT(e.g.,after50years).Inordertoensurepersistenceofallspeciesi2I,anaturalobjectivefunctionis(typi-callyinyears)f(R)=Pi2Iwif(i)(R);wherewiareasetofweightsassociatedwiththedifferentspecies.Inthefollowing,w.l.o.g.,wesetwi=1forallspeciesi,andthusf(R)effectivelyquantiestheexpectednumberofspeciespersistingafterTtimesteps.Typically,eachcandidatepatchPalsohassomecostc(P)forreservation,e.g.,itsmonetarycost,ortheeffortrequiredtonegotiateforitsprotectionwiththeownersofitsparcels.ThegoalofthestaticconservationplanningproblemthenistoselectareserveR=argmaxR:c(R)bf(R);(2)thatmaximizesthepersistenceprobabilitywhilerespectingabudgetconstraintbonthetotalcostc(R)=PP2Rc(P).2.3TheDynamicReserveDesignProblemInmanynaturalreservedesignsettings,suchastheoneinourcasestudy,itisnotpossibletoconductandimplementasingleoptimization.Instead,wehavetosolveasequen-tialdecisionmakingprocesswhereovertimenewresources(patchesoflandandbudgettospend)becomeavailable,andwehavetodynamicallydeterminerecommendationsbasedonourpreviousactions.LetPtPbethesetofpatchesavailableforpotentialconservationattimet.NotethatthesequenceP1;:::;PTmaynotbeknowntousinadvance;wemaynotknowiforwhenaparticularpatchbecomesacandidateforconservation.LetRtdenotethesetofpatchesselectedintherstttimesteps.Ateverytimestep,wearegivenabudgetbt,andcanselectacollectionR0ofadditionalpatchesfromPtofcostatmostbt,takingintoaccountwhichpatcheswehavealreadyselected,andsetRt=Rt�1[R0.Unusedbudgetfromonetimestepdoesnotcarryovertothenexttimestep.Forclarityofpresentation,hereweconsiderthesettingwhereconservationrecommendationsaremadeonafastertimescalethanthepatchdynamics.Thus,thegoalistoplantherecommendationofpatchestoprotectsuchthatthenalreserve1RTmaximizesthepersistenceprobabilityf(RT).Formally,weareinterestedinaconservationpolicy:2P2PNR+!2P,suchthat(R;P0;t;b)specieswhichpatchestorecommendattimet,giventhatwehavealreadyselectedpatchesR,havethesetP0ofpatchestochoosefrom,andbudgetbtospend.Apolicyisfeasibleif,whenever(R;P0;t;b)=R0,thenR0P0andc(R0)b. 1Ourmodelcanbeextendedsothatfalsodependsonthesequenceofselections.Wedefertoanextendedversionofthispaper. (a)ElicitedsurvivalprobabilitiesforSHL (b)FittedsurvivalprobabilitiesforSHL (c)Staticreservedesign (d)Dynamicreservedesign (e)Dynamicreservedesign (f)DynamicdesignwithfailuresFigure2:Experimentalresults.4ExperimentalResults4.1ReserveDesignCaseStudyWeareconductingacomputationalstudyincollaborationwiththeUSFishandWildlifeServiceWashingtonOfce,inWashingtonState,USA.Theeventualgoalofthiscollabora-tionistodevelopatoolthatwillfacilitatedecision-makingaboutassemblyofareserveadequatetoprotectthreeFederalCandidatetaxainhabitingaremnantprairieecosystemintheSouthPugetSoundregion.ThetargetspeciesareTay-lor'scheckerspot(TCS;Euphydryasedithataylori),Mazamapocketgopher(MPG;Thomomysmazama),andstreakedhornedlark(SHL;Eremophilaalpestrisstrigata).Aspartofthiseffort,weheldelicitationworkshopstogarnertheinputofbiologistswithexpertiseonthetargettaxaandtheSouthPugetSoundprairieecosystem.Thegoaloftheseworkshopswastoparameterizepatchdynamicsmodelsforeachofthespecies.Substantialuncertaintycurrentlyexistsabouttheecologicalprocessesgoverningthebehaviorofpopulationsofthetargettaxa.Ourintentduringtheworkshopswastoformallycapturethisuncertainty,viainter-expertvariation,sothatitcouldbereectedinthepredictivepatchdynamicsmodels,andultimatelyconservationrecommendationscouldbeobtainedthatarerobusttothisuncertainty;todosoweusedamodiedDelphiprocessforexpertelicitation(c.f.,Vose,1996).Fig.2(a)presentsanexampleannualsurvivalcurveelicitedfromanexpertforSHL,andFig.2(b)showsatbasedoninputsfrommultipleexperts.Theprimaryobjec-tiveistomaximizepersistenceprobabilityafter50yearsforeachofthecandidatetaxa.WerstidentiedthesetoflandparcelsinappropriateportionsoftheWashingtoncountiesofGraysHarbor,Lewis,Mason,Pierce,andThurston(includingFt.LewisArmyBase):Thesearelocatedatleastpartiallyonappropriateprairiesoiltypes;areclassiedbycountysurvey-orsofcesasundeveloped,agriculture,openspace,orforest;andareatleast5acresinsizeandcanbecombinedwithadjacentqualifyingparcelstoassembleacontiguouspatchthatisatleast100acresinsize.Wealsoobtainedspatialdataonsoiltypes,elevation,vegetationtype,andbarriers(selectedroadsandwaterways),whichwereprocessedus-ingArcGIS9.3(EnvironmentalSystemsResearchInstitute2008)todeterminethehabitatpropertiesofeachparcelandthebarriershinderingcolonization.WeuseparametricmodelsforthestochasticaspectsinthepatchdynamicsmodelPhZ(i)p;t+1jZt;R;ti.Duetospacelimitations,weonlyprovideanoverviewhere.1.AnnualSurvival.Annualsurvivalofapopulationinaparceldependsontheusablehabitatsize.Thisdependenceofthesurvivalprobabilityonhabitatsize,aswellasthefactorsdetermininghabitatsizeitself,wereelicited.Sinceecologicalprocessesvaryovertime,andenvironmentalconditions(e.g.,aharshwinter,thespreadofadisease)canaffectsurvival,weestimateaspatiallycorrelatedreductionorincreaseintheeffectivehabitatareabyusingaGaussianprocessmodelwithexponentialkernel;thecomponentsofthismodel(e.g.,thedegreeofannualvarianceandspatialcorrelationinvariance),werealsoelicitedfromexperts.2.Colonization.Theprobabilityofspeciescolonizationismodeledusingaparametricfunctionofthesourceparcelhabitablearea(annually-varying,asdescribedabove),the llllllllllllllllllllllll 0100200300400500 0.00.20.40.60.81.0 Pa Size (Acres)Annual Patch Survival Probability llllll distancebetweensourceandtargetparcel,andenviron-mentalconditionsusingmodelsfromtheliteraturewhereavailable(fortheTaylor'scheckerspot;seeHanskietal.[1996])orbasedonexpertelicitation.Barriers(interstates,majorhighways,andwaterbodies)reducemigrationprob-abilitytovaryingdegreesforTCSandMPG.Priordistributionsontheparametersofthesestochasticcom-ponentswereelicitedfromtheexpertpanel.Inordertocap-turethevariation,weconductmultiplesimulations,withpa-rameterssampledfromtheestimatedpriordistributions.Forconservationcost,weusethesize(inkm2)ofeachparcel.4.2ExperimentalSetupWegeneratecontiguouscandidatepatchesfromtheparcelsbyaregiongrowingprocess,whichpicksarandomparcelasseed,andtheniterativelygrowsthepatchuptoarandomsize.Thisgrowthprocessisrandomlybiasedtoavoidcomplexboundaries.Usingthisprocedurewegenerate10,000candi-datepatchesforselection.Toevaluatetheobjectivefunction,wegenerate100randomsamplesfromtheDBN(1).Toavoidovertting,twothirdsofthoseareusedforoptimization(asdone,forexamplebySheldonetal.[2010]forasimilarprob-lem),andthequalityofthesolutionsareevaluatedagainsttheremainingonethird.AsnotedbySheldonetal.[2010],theadvantageofthisprocedureisthatpreprocessingcanbeusedtodrasticallyspeedupcomputationandboundsonthegeneralizationerrorcanbeobtained.Further,insteadofusingthealgorithmdescribedbySviridenko[2004]forsolvingProblem(2),weuseafasteralgorithmofLeskovecetal.[2007]thatalsocarriestheoreticalguarantees.4.3OptimizationResultsInourexperiments,wemainlyaimtoinvestigatethefollow-ingquestions:1.Howmuchbetterdooptimizedsolutionsperformcom-paredtosimplebaselines?2.Howmuchcanbegainedfromdynamicoptimization?Werstconductexperimentsonthestaticreservedesignproblem(2).Wevarythebudgetfrom0to60km2,andcom-paretheoptimizedreserveswithrandomselection,aswellasselectingpatchesaccordingtodecreasingarea.Fig.2(c)presentstheresults.Notethatoptimizedselectiondrasticallyoutperformsthebaselines.Runningtimeforthisprobleminstanceislessthan45secondsonastandard2.66GHz,4GBconguration.Fig.3showsasolutionobtainedforb=10km2.Wethenevaluateournear-optimalpolicyfordynamicconservationplanning.WerandomlypartitionthesetofallpatchesintoT=10differentsubsetsP1;:::;PT.Inourexperiment,wevarythebudgetbtwhichismadeavailableineachroundfrom0to60km2.Wethenopportunisticallyse-lectpatcheseachround,eitherbyoptimization,indecreasingorderofarea,oratrandom.Allexperimentsarerepeated,andresultsaveraged,over10randomtrials.Inordertoestimatethebenetofdynamicselection,wealsocompareagainstanotherbaseline,whereweapriori(approximately)optimizeaxedreserve(havingaccesstoallpatchesandtheentirebudget),andthen,forthisxedsolution,pickpatchesintherstroundinwhichtheybecomeavailable. Figure3:Selectedpatches(red)foronesolution,withb=10km2.InFig.2(d)weplottheexpectednumberofpersistentspecies(after50years)aftertenroundsofselection.Notethatthedynamicallyoptimizedsolutionoutperformsthebaselines.Notethatevenafteralltenrounds(i.e.,afterallpatchesweremadeavailable)thesequentialsolutionoutperformstheapriorisolution.Thereasonisthatthestaticapriorioptimizationisnotawareoftheper-roundbudgetconstraints,andthereforemaynotbeabletoselectsomepatchesastheybecomeavailable.Fig.2(e)plots,foraxedbudgetperround,thevalueofthereservesf(Rt)duringthetenroundsofselection.Notethatthedynamicapproachesdrasticallyoutperformtheaprioriselection.Lastly,wealsoperformanexperiment,wherethealgo-rithmsateachroundattempttorecommendsomepatchesforconservation.However,theserecommendationsmayfail(i.e.,cannotbeimplementedduetoexternalconstraints).Hereweconsiderfailuresthathappenrandomly,withprobability0:5independentlyforeachpatch4.Fig.2(f)presentstheresultofthisexperiment.NotethatincontrasttoFig.2(e),herethedynamicapproachesachievemuchbetterperformancethanthestaticbaseline.Thereasonforthisisthatthedynamicapproachesmaybeabletosubstitutean“important”failedselectionbyasimilaralternativethatbecomesavailableinalaterround.5RelatedWorkConservationplanning.Thereareseveralpowerfultoolsavailableforconservationplanning,includingMarxan(Ball,Possingham,andWatts2009)andZonation(MoilanenandKujala2008).However,noneofthosetoolscurrentlyim-plementscomplexpatchdynamicsmodelsofspeciespersis-tence.Also,theydonotprovideguaranteesof(near-)opti-mality.PerhapsclosestinspirittoourworkisanapproachbySheldonetal.[2010].Theyproposeanetworkoptimiza-tionapproachwithapplicationstoconservationplanning.Intheirapproach,theymodelthepopulationbehaviorusing 4Ourtheoreticalanalysisholdseveninthismoregeneralsetting,relyingonageneralizationofsubmodularitytoadaptivepolicies(GolovinandKrause2010).Weomitdetailsduetospacelimitations. theindependentcascademodelofGoldenberg,Libai,andMuller[2001].Oneparticularaspectthattheyconsideristhenon-submodularityofthereservedesignprobleminabsenceofCondition4.TheyproposeanapproachbasedonMixed-IntegerProgramming(MIP)toovercomelimitationsofthegreedyalgorithm.However,theirworkdoesnothandlethedynamicaspectsofconservationplanningthatarethefocusofthispaper.Theindependent-cascademodelisessentialtotheirapproach,anditseemsdifculttouseittomodelcomplexinteractionsbetweenspecies(suchassymbiosisorpredator-preyrelationships),ormorecomplexpopulationdynamics(beyondpresence-onlypopulationmodels),whichallcanbehandledbyourapproach.Furthermore,inmostoftheirexperimentsconductedonarealreservedesigncasestudy,thenon-greedynetworkdesignapproachbasedonMIPperformscomparablytothegreedyapproach,providingfurtherevidenceabouttheappropriatenessofCondition4.WeconsiderthedevelopmentofprincipleddynamicplanningapproachesthatdonotrelyonCondition4aninterestingdirectionoffuturework.Submodularoptimization.Theproblemofadaptivelyop-timizingsubmodularfunctionshasbeenstudiedbyGolovinandKrause[2010].However,theirapproachisnotknowntoprovidecompetitivenessguaranteessuchasthoseofTheo-rem2,wherethesetofavailableactionschangesovertime.StreeterandGolovin[2008]provideanalgorithmfortheproblemofonlinemaximizationofsubmodularfunctions.Intheirsetting,thedecisionmakerchoosesadifferentsetateachround,maximizingthesumofobjectivevaluesat-tainedovertime.Theconservationplanningproblemdoesnottintothisframework,sinceweareinterestedinbuildingasinglesetofreservesbyaddingpatchesovertime.An-otherrelatedproblemissubmodularoptimizationoverdatastreams,asstudiedbyGomesandKrause[2010].However,theiralgorithmrequiresthatitispossibletounselectalreadyselecteditems(patches),whichmaynotbepossibleintheconservationplanningproblem.Lastly,theproblemstudiedinthispaperisalsorelatedtothesubmodularsecretaryprob-lem,studiedbyBateni,Hajiaghayi,andZadimoghaddam[2010].However,theirguaranteesrequirethatthesequenceofelementsthatbecomeavailableovertimeisrandomlyper-muted.Ourapproachdoesnotmakeanyassumptionabouttheorderinwhichpatchesbecomeavailable.6ConclusionWeconsideredtheproblemofprotectingrarespeciesbyrec-ommendingpatchesoflandforconservation.Ourapproachemploysadetailedprobabilisticpatchdynamicsmodelinordertoensurelong-termpersistenceoftaxaintheselectedreserves.Ourmodelcanhandlecomplexannualsurvivalandcolonizationpatterns,aswellasinteractionsamongspecies(thoughnotdemonstratedhere).Inordertocopewithchang-ingavailabilityofpatches,weproposedanopportunisticpolicyfordynamicallymakingrecommendations.Weprovedthesurprisingresultthatthissimpleopportunisticpolicyiscompetitivewithaclairvoyantsolutionthatisinformedinadvanceofthebudgetineachtimestepandwheneachpatchwillbeavailable.Weconductedadetailedcasestudyofcon-servationplanningforthreeraretaxainthePacicNorthwestoftheUnitedStates.Ourresultsindicatethattheoptimizedsolutionsdrasticallyoutperformsimplebaselines,andthatsignicantbenetcanbeobtainedfromdynamicplanning.Webelievethatourresultsprovideinterestinginsightsfordynamic/adaptiveoptimization,andcouldbeusefulforotherapplicationssuchasinuencemaximizationovernetworks.Acknowledgments.ThisresearchwaspartiallysupportedbyONRgrantN00014-09-1-1044,NSFgrantsCNS-0932392andIIS-0953413,theCaltechCenterfortheMathematicsofInformation,andbytheUSFishandWildlifeService.WethankJ.Bakker,J.Bush,M.Jensen,T.Kaye,J.Kenagy,C.Langston,S.Pearson,M.Singer,D.Stinson,D.Stokes,andT.Thomasfortheircontributions.ReferencesBall,I.;Possingham,H.;andWatts,M.2009.Spatialconservationprioritisation:Quantitativemethodsandcomputationaltools.Ox-fordUniversityPress.chapterMarxanandrelatives:Softwareforspatialconservationprioritisation.Bateni,M.H.;Hajiaghayi,M.;andZadimoghaddam,M.2010.Thesubmodularsecretaryproblemanditsextensions.InAPPROX.EnvironmentalSystemsResearchInstitute.2008.ArcGIS9.3usersmanual.Technicalreport,ESRI.Goldenberg,J.;Libai,B.;andMuller,E.2001.Talkofthenetwork:Acomplexsystemslookattheunderlyingprocessofword-of-mouth.MarketingLetters12(3):211–223.Golovin,D.,andKrause,A.2010.Adaptivesubmodularity:The-oryandapplicationsinactivelearningandstochasticoptimization.CoRRabs/1003.3967v3.Gomes,R.,andKrause,A.2010.Budgetednonparametriclearningfromdatastreams.InICML.Goundan,P.R.,andSchulz,A.S.2007.Revisitingthegreedyapproachtosubmodularsetfunctionmaximization.Technicalreport,MassachusettsInstituteofTechnology.Hanski,I.A.;Moilanen,A.;Pakkala,T.;andKuussaari,M.1996.Thequantitativeincidencefunctionmodelandpersistenceofanendangeredbutterymetapopulation.ConservationBiology10.Koller,D.,andFriedman,N.2009.ProbabilisticGraphicalModels.TheMITPress.Leskovec,J.;Krause,A.;Guestrin,C.;Faloutsos,C.;VanBriesen,J.;andGlance,N.2007.Cost-effectiveoutbreakdetectioninnetworks.InKDD.Moilanen,A.,andKujala,H.2008.ZONATION:Spa-tialconservationplanningframeworkandsoftware.www.helsinki./bioscience/ConsPlan.Nemhauser,G.L.;Wolsey,L.A.;andFisher,M.L.1978.Ananalysisofapproximationsformaximizingsubmodularsetfunctions-I.Math.Prog.14(1):265–294.Sheldon,D.;Dilkina,B.;Elmach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