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The Implications of Asymmetric Dispersal for Metapopulation Dynamics and Conservation

Permanent Link: http://ufdc.ufl.edu/UFE0045231/00001

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Title: The Implications of Asymmetric Dispersal for Metapopulation Dynamics and Conservation
Physical Description: 1 online resource (92 p.)
Language: english
Creator: Acevedo, Miguel A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: assymetric-dispersal -- connectivity -- dispersal -- metapopulations -- modeling -- patch-dynamics
Interdisciplinary Ecology -- Dissertations, Academic -- UF
Genre: Interdisciplinary Ecology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Variations in movement have broad implications for many areas of ecology and evolution.  For instance, in metapopulation theory, movement plays an important role promoting (meta)population persistence over time.  Most metapopulation modeling approaches describe patch connectivity using pair-wise Euclidean distances resulting in the simplifying assumption of a symmetric connectivity pattern.  Symmetric connectivity may be rarely observed in nature were organisms move responding to environmental cues or advection sources through heterogeneous landscapes.  Assuming symmetric dispersal when movement is asymmetric may result in biased estimates of colonization, extinction and persistence, which has important implications for management and conservation.  Here I studied the implications of asymmetric connectivity for patch colonization and extinction, the potential mechanisms behind asymmetric connectivity and we developed a novel model for connectivity conservation that takes into consideration asymmetric connectivity. I leveraged a long-term, time-series on colonization-extinction dynamics in the wind-dispersed orchid Lepanthes rupestris and found that a patch connectivity measure that accounts for potential movement asymmetries was more accurate in modeling its metapopoulation dynamics.  I used a combination of an observational study of individual movements and translocation experiments to test for possible mechanisms of directed movement in a cactus-feeding insect, Chelinidea vittiger.  We hypothesized that three primary mechanisms may cause directed movements: (1) positive anemotaxis, (2) variations in patch size and (3) conspecific interactions and found that variations in patch size were the most important driver of directed movements in this species. Systematic conservation planning tools provide guidance for the proper management of protected area networks.  Most studies are concerned with maintaining biodiversity patterns and less about promoting ecological or evolutionary processes.  We devloped an novel application of network interdiction models to assess vulnerabilities in ecological spatial networks by assuming a worst-scenario for connectivity.  Asymmetric connectivity may be the rule more than the exception in nature with important implications for (meta)population dynamics and conservation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miguel A Acevedo.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Fletcher, Robert Jeffrey, Jr.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045231:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045231/00001

Material Information

Title: The Implications of Asymmetric Dispersal for Metapopulation Dynamics and Conservation
Physical Description: 1 online resource (92 p.)
Language: english
Creator: Acevedo, Miguel A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: assymetric-dispersal -- connectivity -- dispersal -- metapopulations -- modeling -- patch-dynamics
Interdisciplinary Ecology -- Dissertations, Academic -- UF
Genre: Interdisciplinary Ecology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Variations in movement have broad implications for many areas of ecology and evolution.  For instance, in metapopulation theory, movement plays an important role promoting (meta)population persistence over time.  Most metapopulation modeling approaches describe patch connectivity using pair-wise Euclidean distances resulting in the simplifying assumption of a symmetric connectivity pattern.  Symmetric connectivity may be rarely observed in nature were organisms move responding to environmental cues or advection sources through heterogeneous landscapes.  Assuming symmetric dispersal when movement is asymmetric may result in biased estimates of colonization, extinction and persistence, which has important implications for management and conservation.  Here I studied the implications of asymmetric connectivity for patch colonization and extinction, the potential mechanisms behind asymmetric connectivity and we developed a novel model for connectivity conservation that takes into consideration asymmetric connectivity. I leveraged a long-term, time-series on colonization-extinction dynamics in the wind-dispersed orchid Lepanthes rupestris and found that a patch connectivity measure that accounts for potential movement asymmetries was more accurate in modeling its metapopoulation dynamics.  I used a combination of an observational study of individual movements and translocation experiments to test for possible mechanisms of directed movement in a cactus-feeding insect, Chelinidea vittiger.  We hypothesized that three primary mechanisms may cause directed movements: (1) positive anemotaxis, (2) variations in patch size and (3) conspecific interactions and found that variations in patch size were the most important driver of directed movements in this species. Systematic conservation planning tools provide guidance for the proper management of protected area networks.  Most studies are concerned with maintaining biodiversity patterns and less about promoting ecological or evolutionary processes.  We devloped an novel application of network interdiction models to assess vulnerabilities in ecological spatial networks by assuming a worst-scenario for connectivity.  Asymmetric connectivity may be the rule more than the exception in nature with important implications for (meta)population dynamics and conservation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miguel A Acevedo.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Fletcher, Robert Jeffrey, Jr.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045231:00001


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THEIMPLICATIONSOFASYMMETRICDISPERSALFORMETAPOPULATIONDYNAMICSANDCONSERVATIONByMIGUELA.ACEVEDOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013MiguelA.Acevedo 2

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Tomymotherwhotaughtmethevaluesofperseveranceandhardwork 3

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ACKNOWLEDGMENTS Firstandforemost,IwouldliketothankmyadvisorDr.RobertJ.Fletcher,Jr,forbelievinginmyideasandsharingwithmehiswealthofknowledgeandforprovidinginvaluableguidance.Itwasagreathonortobehisstudent.IamgratefultoDr.RaymondTremblayandDr.ElviaMelendez-AckermanforprovidingthedataanalyzedinChapter 1 .IwouldalsoliketothankDr.J.ColeSmithandJorgeSefairfortheirpatienceandfortheirinvaluablehelpwiththemathematicalformulationsinChapter 3 .Iwouldalsoliketothankmycommittee,Dr.MadanOli,Dr.KatherineSievingandDr.RobertHoltfortheirhelpatmultiplestagesofmydegree.IwouldliketothankAndreRevell,ConorEganandKaanKermanfortheirhelpintheeld.ThisworkgreatlybenetedfromdiscussionswithDr.BenjaminBolker,Dr.SeverineVuilleumier,Dr.ChristineMillerandmembersoftheQSE3IGERT.IamgratefultoDr.JeffHostetler,AidaMiro,RayaPruner,NathanMarcy,Dr.JasonEvans,Dr.KyleMcCarthy,VarunGoswami,DivyaVasudev,RajeevPillay,ChrisRota,KristenAaltonen,EvaKneip,BinabKarmacharya,NoahBurrell,KatieHaase,LukeMcEachron,MadelonVandeKerk,IrinaSkinner,MauricioNunez-Regueiro,BrianReichert,EllenRobertson,IrinaSkinner,RebeccaWilcox,andDr.JenniferSeavey.FundingwasprovidedbytheNSFQuantitativeSpatialEcology,EvolutionandEnvironment(QSE3)IntegrativeGraduateEducationandResearchTraineeshipProgram(IGERT)grant0801544attheUniversityofFloridaandanNSFDoctoralDissertationImprovementGrant(DEB\FundingwasalsoprovidedbytheSchoolofNaturalResourcesandEnvironmentandtheDepartmentofWildlifeEcologyandConservationattheUniversityofFlorida.Iwouldalsoliketothankmyfamily,speciallyMelbaTorresandCarlosMilan,andmygoodfriendsDr.ReyRiveraandDr.MelanieMediavillafortheirsupport.IamalsoparticularlygratefultoChristineGutierrezforherinvaluablesupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1THEIMPLICATIONSOFASYMMETRICDISPERSALFORCONNECTIVITYANDPATCHOCCUPANCY ............................. 11 1.1Introduction ................................... 11 1.2Methods ..................................... 14 1.2.1StudySystem .............................. 14 1.2.2SiteOccupancyModelandParameterEstimation .......... 15 1.3Results ..................................... 20 1.4Discussion ................................... 22 1.4.1Detectability ............................... 24 1.4.2ColonizationandExtinction ...................... 24 1.4.3AsymmetricDispersalandMetapopulationModeling ........ 26 2THEPROXIMATECAUSESOFASYMMETRICMOVEMENT .......... 29 2.1Introduction ................................... 29 2.2Methods ..................................... 31 2.2.1StudyAreaandFocalSpecies .................... 31 2.2.2StudyDesign .............................. 31 2.2.2.1Mark-recapture ........................ 31 2.2.2.2Fieldexperiment ....................... 33 2.2.3Analyses ................................. 37 2.2.3.1Mark-recapture ........................ 37 2.2.3.2Fieldexperiment ....................... 38 2.3Results ..................................... 39 2.4Discussion ................................... 41 3OPTIMALSITESELECTIONSTRATEGIES:PROTECTINGAGAINSTWORST-CASEDISTURBANCESCENARIOS ........................... 47 3.1Introduction ................................... 47 3.2Methods ..................................... 49 3.2.1NetworkFlowInterdictionModels ................... 49 3.2.2ModelDescription ........................... 50 3.2.2.1Casestudies ......................... 52 5

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3.2.2.2Connectivityassessment .................. 54 3.2.2.3Disturbance:simulatingaworst-casescenario ...... 55 3.2.2.4Protection ........................... 56 3.2.2.5Generalmodelassumptions ................ 57 3.2.3Scenarios ................................ 57 3.2.4SensitivityAnalysis ........................... 58 3.3Results ..................................... 59 3.3.1RoseateTerns:Step-by-Step ..................... 59 3.3.2SkipjackTuna .............................. 59 3.3.3SensitivityAnalysis ........................... 59 3.4Discussion ................................... 61 3.4.1Modications .............................. 64 3.4.2ConservationPlanningforaWorst-caseScenarioforConnectivity 65 3.4.3UncertaintyandWorst-caseScenarioPlanning ........... 67 APPENDIX AMODELSUMMARIESANDADDITIONALFIGURES ............... 68 BCONNECTIVITYASSESSMENT:MATHEMATICALDETAILS .......... 73 CDISTURBANCEOPTIMIZATION .......................... 74 DPROTECTIONOPTIMIZATION ........................... 77 EMODELITERATIONSFORWEIGHTEDLIFEEXPECTANCYOPTIMIZATION 80 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 92 6

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LISTOFTABLES Table page 1-1ModelselectionofoccupancymodelspredictingcolonizationandextinctionofLepanthesrupestris ................................ 22 2-1Modelselectionforeachexperimentaltreatment ................. 42 A-1Parameterestimatesandstandarderrors(SE)forthemulti-seasonoccupancymodelthatresultedinthebestt .......................... 68 A-2Parameterestimatesandstandarderrors(SE)forthemodelthatincludedasymmetricdispersalkernel(Ssym)asasitecovariate ............... 69 E-1ModeliterationsshowingtheinteractionbetweendisturbanceandprotectionstageswhenpT=[3773,2467,350,172] ...................... 80 7

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LISTOFFIGURES Figure page 1-1Variousapproachestomodelpatchconnectivity ................. 13 1-2Calculationoftheparameter ........................... 18 1-3DistributionofijintheLepanthessystem ..................... 21 1-4Partialrelationshipsbetweenextinction,colonizationandconnectivitymeasures(SsymandSasym) .................................... 23 2-1Observedinter-patchtransitionsofC.vittigerina3030mplot ........ 32 2-2Diagramshowingthestudydesign ......................... 34 2-3Linearregressionanalysis .............................. 40 2-4Partialrelationshipsofthebesttmodelexplainingtherateofmovementintheareatreatmentsoftheeldexperiment .................... 41 2-5Movementrate(SE)atthepatchlevelintheexperimentaltreatments. .... 45 3-1Schematicdiagramofthethreestagesofthemodelingprocess:assessment,disturbanceandprotection ............................. 52 3-2SpatialnetworkrepresentationofRoseateTerncoloniesinnortheastUSAandSkipjackTunaintheWesternPacic ..................... 54 3-3Exampleoftheiterativeprocessbetweenthedisturbanceandprotectionmodelsforaprotectionbudgetofu=2andallowingb=2patchestobedisturbed .. 60 3-4Sensitivityoftheoptimallifeexpectancyintheprotectionstagegivenchanges()inprobabilityofstayinginthesamepatch(qii). ................. 61 3-5Optimallifeexpectancy(z)givenallpossibleprotectionbudgets(u)andnumberofpatchesallowedtobedisturbed(b) ....................... 62 A-1Spatialrepresentationofpatches(bouldersandtreetrunks)sampledforLep-anthesrupestrisoccurrenceintheLuquilloExperimentalForest. ........ 69 A-2Theoreticaldistributionsoftheparameterij .................... 70 A-3Predictedpartialrelationshipsbetweencolonization,extinctionandpatchareaforbothphorophytesforthebest-supportedmodel ................ 71 A-4Partialrelationshipbetweenpatchareaanddetectability ............. 72 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEIMPLICATIONSOFASYMMETRICDISPERSALFORMETAPOPULATIONDYNAMICSANDCONSERVATIONByMiguelA.AcevedoMay2013Chair:RobertJ.Fletcher,JrMajor:InterdisciplinaryEcologyVariationsinmovementhavebroadimplicationsformanyareasofecologyandevolution.Forinstance,inmetapopulationtheory,movementplaysanimportantrolepromoting(meta)populationpersistenceovertime.Mostmetapopulationmodelingapproachesdescribepatchconnectivityusingpair-wiseEuclideandistancesresultinginthesimplifyingassumptionofasymmetricconnectivitypattern.Symmetricconnectivitymayberarelyobservedinnaturewhereorganismsmoverespondingtoenvironmentalcuesoradvectionsourcesthroughheterogeneouslandscapes.Assumingsymmetricdispersalwhenmovementisasymmetricmayresultinbiasedestimatesofcolonization,extinctionandpersistence,whichhasimportantimplicationsformanagementandconservation.HereIstudiedtheimplicationsofasymmetricconnectivityforpatchcolonizationandextinction.Ileveragedalong-term,time-seriesoncolonization-extinctiondynamicsinthewind-dispersedorchidLepanthesrupestrisandfoundthatmodelsthataccountedfortheseasymmetrieshadbettert.Theimplicationsofasymmetricconnectivitytometapopulationdynamicsmaybecontingentonthedrivingmechanismbehindthem.Iusedacombinationofanobservationalstudyofindividualmovementsandtranslocationexperimentstotestforpossiblemechanismsofdirectedmovementinthecactus-feedinginsect,Chelinideavittigerandfoundthatvariationsinpatchsizewerethemostimportantdriverofdirectedmovementsinthisspecies.Ialsodeveloped 9

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anovelsiteselectionmodelthatoptimallyselectspatchesthatwillbestpreserveconnectivitygivenaworst-casedisturbancescenario. 10

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CHAPTER1THEIMPLICATIONSOFASYMMETRICDISPERSALFORCONNECTIVITYANDPATCHOCCUPANCY 1.1IntroductionVariationinmovementhasbroadimplicationsforevolutionarybiology( KaweckiandHolt 2002 ),communityecology(e.g., Salomonetal. 2010 ; Tilmanetal. 1994 ),andpopulationdynamics( ArmsworthandRoughgarden 2005 ; Revillaetal. 2004 ; Wiegandetal. 1999 ).Inmetapopulationecology,movementisafundamentalprocessformetapopulationdynamicsandlocalpopulationpersistenceovertime( Hanski 1998 ).Often,thefocusonmovementemphasizesdispersalfromnatalenvironmentsorpreviousbreedinglocations( Colbertetal. 2001 ).Here,webroadlyusethetermdispersaltoreectthesemovementsandrelatedmovementsthatgeneratevariationinemigration,immigration,andcolonizationandextinctionrates(sensu Vuilleumieretal. 2010 ).Mostmetapopulationapproachesdescribecolonization-extinctiondynamicsfollowingthearea-isolationparadigm( Dias 1996 ; Hanski 1998 ; Pelletetal. 2007 ).Underthisparadigm,extinctionisnegativelyrelatedtopatchareaassumingthatpopulationsizeincreaseswithpatcharea;( Hanski 1998 ),andcolonizationisnegativelyrelatedtoisolationfromotherpatches.Patchisolationisoftenquantiedusingpatchdistance-dependentconnectivitymeasures(i.e.,theinverseofisolation)thatassumetheprobabilityofcolonizationdeclineswithdistancetosurroundingoccupiedpatches,whichactassourcesofpropagules.Todoso,thesemeasurestypicallyuseanassumeddispersalkernelweightedbytheoccupancystateandareaofsurroundingpatches( Hanski 1994 ).Whiletheseconnectivitymetricshaveprovenuseful,thesemetricsmakeassumptionsregardingthemovementbehaviorofthespecies.Forinstance,anegativeexponentialfunctionofEuclideandistanceiscommonlyassumedinmanypatchconnectivitymeasures( Hanski 1998 ; MoilanenandNieminen 2002 ).Assumingakernelbased 11

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onEuclideandistanceresultsinthesimplifyingassumptionofsymmetricdispersalinwhichthelikelihoodofanindividualdispersingfrompatchitopatchjisthesameasthelikelihoodofanindividualdispersingfrompatchjtoi.Nevertheless,severalfactorsinnatureoftencauseanasymmetricpatternofdispersal,suchasspatialheterogeneity(e.g., Ferreras 2001 ; PrevedelloandVieira 2010 )andadvectionsources(e.g.,wind,oceanandmarinecurrents;( Keddy 1981 ; Tremletal. 2008 ).Anasymmetricpatternofdispersalmayalsoariseduetovariationsinpatchorinter-patchattributes(Figure 1-1 ).Atthepatchlevel,variationinpatchareamayresultinagreaterlikelihoodofdispersaltowardslargepatchesbecauseactivedispersersmaybetterdetectorpreferlargepatches.Forpassivedispersers,largepatchesmaysimplyactaslargetargetsorbecausethesearelargetargetsforpassivedispersers(i.e.targeteffects;( GilpinandDiamond 1976 ; Lomolino 1990 );Figure 1-1 a).Quantitatively,theseasymmetriesduetopatchlevelvariationshavebeentraditionallyincorporatedinconnectivitymeasuresbyweightingadispersalkernelbypatcharea.Inter-patchattributes,suchasadvectionsources(e.g.,wind,oceanorrivercurrents),mayresultinagreaterlikelihoodofdispersalinthedirectionoftheadvectionsourcethanintheoppositedirection(Figure 1-1 d).Thatis,theeffectivedistanceinthedirectionoftheadvectionsourcemaybelessthantheeffectivedistanceintheoppositedirection(Figure 1-1 d).Connectivitymeasurescurrentlyusedinmetapopulationmodelinggenerallylackaformalwaytoincorporatethiskindofasymmetryineffectivedistance. MoilanenandHanski ( 1998 )providewaystoaccountforhabitattype-speciceffectivedistances(h);however,theeffectivedistancemeasureisstillsymmetric(i.e.dhij=dhji;Figure 1-1 c).Otheralternateparameterizationsemployleast-costpaths(LCP)betweenpatches( Chardonetal. 2003 ).EventhoughtheyhavebeenshowntobebetterdescriptorsofeffectivedistancethanEuclidiandistance( Sawyeretal. 2011 ),inmostapplicationstheseLCParealsosymmetric(LCPij=LCPji;Figure 1-1 b). 12

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Figure1-1. Variousapproachestomodelpatchconnectivity.PatchconnectivityinIncidenceFunctionModels(a)takesintoconsiderationvariationsinpatcharea.Leastcostpaths(b)replaceEuclideandistancebytheleastcostpathbetweentwopatches.Otherproposedmeasures(c)scaleEuclideandistancedependingonhabitattype(orland-cover/usetype).Eventhoughthesemodications,dij=djiinallthesemodelingapproaches,whichmakestheminapplicabletomodelorganismsthatdisperseasymmetricallydrivenbyadvectionsources(d). Recentmetapopulationtheoryhypothesizesthatfailingtoacknowledgedispersalasymmetriesleadstoanoverestimationofpatchconnectivity,resultinginbiasedestimatesofcolonizationandextinction( Bodeetal. 2008 ; Vuilleumieretal. 2010 ; VuilleumierandPossingham 2006 ).Thereareveryfewempiricaltestsonthesehypotheses,inpartbecausewelackanadequatewaytoquantitativelyincorporateasymmetricdispersalduetointer-patchattributesinmetapopulationmodels.Here,weleveragealong-term,time-seriesoncolonization-extinctiondynamicsinthewind-dispersedorchidLepanthesrupestristotestforasymmetricdispersal.Weassessfortheroleofasymmetriesforcolonizationandextinctiondynamicsbyextendingtheconnectivityfunctionof Hanski ( 1994 )toaccountfordispersalasymmetryduetowindadvection.Weexpectedasymmetricdispersalwouldbeprevalentinthis 13

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metapopulationandthatsuchasymmetrieswouldaltercolonizationandextinctionestimatesrelativetomodelsassumingdispersalsymmetries.Giventhewind-dispersednatureofthespecies,wealsoexpectedthattargetandrescueeffectswillplayacrucialroleintheestimationofcolonizationandextinctionprobabilities. 1.2Methods 1.2.1StudySystemOrchids,suchasLepanthesrupestris,thatgrowonrock(lithophytichabitat)and/ortrees(epiphytichabitat)areappropriatetostudymetapopulationswithasymmetricdispersalbecausetheyoftenliveinspatiallydiscreteephemeralhabitatsandarepassivelydispersedbydirectedsourcessuchaswind(e.g., Snalletal. 2005 ; Tremblayetal. 2006 ).Moreover,manyepiphyticandlithophyticorchidsaresubjecttocolonizationandextinctiondynamicsduetotheirsmallpopulationsizesandstochasticreproductivesuccessdriven,inpart,bydispersalandpollinatorlimitation( Ackerman 1995 ; Olaya-Arenasetal. 2011 ; Tremblay 1997 ).Lepanthesrupestrisisasmall,wind-dispersedorchid(leafs1.34.3cm,shoots15cminheightandowersof<6mm)commonlyfoundalongtheriverbedsoftheLuquilloMountainsinPuertoRico( Ackerman 1995 ).Thispatchilydistributedorchidanchorsitsrootstothesubstrateandrootsareoftencoveredbymosslivingonthesurfacesoftreesorrockyboulders.Theseedsaremicroscopicwithameandispersaldistanceof4.8m( Tremblay 1997 ).Giventhesmallsizeofitsseeds,littleisknownaboutthefateofLepanthesseedsafterdispersalandthepresenceofseedbanksinthisspecieshasbeendebated. Whighametal. ( 2006 )foundthattemperateorchidseedsinexperimentalconditionswereviableevenafter7years;however,Lepanthesseedsareexpectedtodieiftheyfallintheriver.Giventheooddynamicsoftropicalrivers( Johnsonetal. 1998 ),theeffectofseedbanksinthemetapopulationdynamicsofthisorchidislikelynegligible. 14

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ApermanentplotforthestudyofthemetapopulationdynamicsofL.rupestriswasestablishedinQuebradaSonadoraintheLuquilloExperimentalForest(LEF;1818N,6547W)in1999.Thispermanentplotiscomposedof1000occupiedandunoccupiedbouldersandtreetrunks(patcheshereafter; Tremblayetal. 2006 ).Mostpatches(920)weremappedtowithinapproximately10cm(x,y,zposition)relativetothecenterofeachpatchusingmetalrulers,asightingcompassandaclinometer(Figure A-1 ).Thepresence/absenceofL.rupestriswascensusedtwiceayearfrom1999(17censuses).Additionaldescriptionsofthesiteandsamplingprocedurecanbefoundin Tremblayetal. ( 2006 ).Patchsize(i.e.,totalmossarea),wasestimatedastheperimeteroccupiedbytheindividualsinapatchusinga150cm2mesh.Theaveragewinddirectionwascalculatedfromdailymeasuresfromthenearestweatherstationlocated(1kmfromstudysite)inElVerdeBiologicalStation( RamrezandMelendez-Colom 2003 ). 1.2.2SiteOccupancyModelandParameterEstimationIncidencefunctionmodels(IFM; Hanski 1994 ; MoilanenandNieminen 2002 )areprobablythemostcommonoccupancymodelingapproachesusedtoestimatecolonizationandextinctiondynamicsinametapopulationcontext.AlthoughIFMshavebeenappliedsuccessfullytoawiderangeofspecies,theymaketwoassumptionsthataredifculttomeetinmostapplications.First,theyassumethatthespeciesisalwaysdetectedwhereitispresent(i.e.perfectdetection).Thisisrarelythecase,whereanobservedabsencemaybeeitheratrueabsence(thepopulationislocallyextinct)orsimplythatthespecieswasnotdetected,whichmayresultinanoverestimationofextinction( MoilanenandNieminen 2002 ).Second,IFMsassumethatthemetapopulationsareataMarkovianpseudo-equilibriuminwhichtheoccupancystatusofeachpatchattimetisgivenonlybythepatchstatusattimet)]TJ /F8 11.955 Tf 13.08 0 Td[(1.Thisassumptionisdifculttotestandsomestudiesarguethatitisappropriatetoassumenothingabouttheequilibriumstateofthemetapopulation( Erwinetal. 1998 ; Pellet 15

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etal. 2007 ).Inthisstudy,weappliedanalternatemethod,adynamicoccupancymodel,inwhichthesetwoassumptionsarerelaxed( MacKenzieetal. 2003 ).Thedynamic(i.e.,multiple-season)occupancymodelingapproachresemblesPollocksrobustdesign( Pollock 1982 )inthattherearetwotypesofsamplingperiods.Primaryperiodsareusedtoestimatecolonizationandextinctionparameters.Thepopulationisassumedtobeopenbetweentheseprimaryperiods.Withintheprimaryperiods,sitesaresurveyedmultipletimes(secondarysampling).Thepopulationisassumedtobeclosed(noimmigration,emigration,births,ordeaths;see Rotaetal. ( 2009 )foranassessmentofthisassumption)betweenthesesecondarysamplingperiods,whichareusedtoestimatedetectionprobabilities.Foreachsecondarysamplingperiod,therearethreeoccupancypossibilities:presence,absence(whichmaycorrespondtoimperfectdetectionoranactualabsence),ormissingdata( MacKenzieetal. 2003 ).Ourprimarysamplingperiodsareyears(1999-2008,n=10).Thesecondarysamplingperiodsconsistedoftwocensusesthatwereperformedeachyear.Onecensuswasconductedatthebeginningoftheyear(January-February)andthesecondinthesummer(July-August).Thismodelformulationallowsthesystemtobeopentocolonizationsandextinctionsduringthewetseason(August-December).Tropicalstormsarecommonthroughoutthewetseasoncausingashoods,whichmayberesponsibleformostlocalextinctionsandanomalousstrongwinds,whichmayincreasethemagnitudeofdispersaleventspotentiallyresultinginmorelocalcolonizations.Eventhoughsomecolonizationsandextinctionsmayhappeninthedryseason,ourconclusionsarebasedonrelativecomparisons(symmetricvs.asymmetricmodels;seebelow)andthepotentialviolationoftheclosureassumptionwillaffectbothinstancesequally,withminimaleffectinthenalrelativecomparisons.Giventhesmallsizeoftheorchid,weexpectedvariationindetectabilityespeciallyinpatcheswithsmallareas 16

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ofmosswhereitmaybedifculttodiscernbetweenmossandasmallL.rupestrisindividual.Theeffectsofarea(Aj)andpatchconnectivity(Si)wereaddedascovariatestomodelcolonization,extinctionand/ordetectionprobabilities.Wecalculatedpatchconnectivityintwoways,oneofwhichtreatsthedistanceeffectinthedispersalkernelassymmetricandasecondthatadjustseffectivedistancetoallowforasymmetricmovement.PatchconnectivitywithasymmetricdispersalkernelwascalculatedusingthegeneralapproachappliedinmanyIncidenceFunctionModels( MoilanenandNieminen 2002 ): Ssymi=NXj6=iexp()]TJ /F9 11.955 Tf 9.3 0 Td[(dij)Aj(1)whereNisthetotalnumberofpatchesinthelandscape,1=istheaveragedispersaldistanceofthespecies,dijistheEuclideandistancebetweenpatchesiandjandAjistheareaofthetargetpatchj( Hanski 1994 1998 ).Notethattheeffectivedistancebetweenpatchesinthismodelissymmetric(i.e.dij=dji).Thisformulationcanresultinasymmetricpatchconnectivityduetovariationinpatcharea,butnotduetovariationsineffectivedistanceamongpatches(i.e.inter-patchattributes).AlsonotethattheconnectivitymeasuresweemployeddonotincorporateoccupancydataasistraditionallydoneinsomeIFMsmeasuresofisolationwheretheoccupancystateofpatchjisincorporatedasabinaryindicatorvariabletorestrictpotentialre-colonizationonlyfrompreviouslyoccupiedsites.Abinaryincorporationofoccupancystateassumesperfectdetection,whichwefoundisnotthecaseinoursystem(Figure A-4 ).Giventhatoccupancystatewasnotincorporatedineithersymmetricorasymmetricmeasuresofconnectivity,thismodicationdidnotaffectthegeneralconclusions.WemodiedEquation 1 toincorporatetheeffectofaveragewinddirectionintheestimationofconnectivity.Patchconnectivitywithanasymmetricdispersalkernelwascalculatedas: 17

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Sasymi=NXj6=iexp)]TJ /F9 11.955 Tf 9.29 0 Td[(dij 1)]TJ /F9 11.955 Tf 11.96 0 Td[(ijAj(1)whereijdescribesthedifferencebetweentheangleofwinddirectionandtheanglebetweenpatchesiandjinradianswithrespecttothehorizontalaxis(Figure 1-2 ).Thismodelingapproachresemblesthestronglyasymmetricdistance-dependenttheoreticalmodelfrom( Vuilleumieretal. 2010 )inwhichisincorporatedasamodierinsidetheexponentialdecayfunction.Forsimplicity,ijisscaledbytoconstrainij2[0,1], ij=jij)]TJ /F9 11.955 Tf 11.96 0 Td[(windj .(1) Figure1-2. Calculationoftheparameter.Theijparameteriscalculatedasthedifferenceoftheanglesbetweenpatchesiandj,andtheangleofwinddirectionwithrespecttothehorizontalaxis Forbothmeasuresofpatchconnectivity(SsymandSasym),connectivitydecreaseswithincreasingdistancebetweenpatches.InSasymwhenij=0,theeffectivedistancebetweenpatchesisthesameasEuclideandistance,andhenceSasym=Ssym;however, 18

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asijincreases,theeffectivedistancebetweenpatchesincreasesresultingindecreasedconnectivitywhencomparedtoSsym.Thismodicationallowsanasymmetricdispersalkernelbecausetheeffectivedistancebetweenpatchesinthedirectionofwindislessthantheeffectivedistancebetweenthesamepatchesintheoppositedirection(againstwinddirection;Figure 1-1 d).Thedistributionoftheparameterwilldependonthespatialarrangementofthepatcheswithrespecttotheangleoftheadvectionsource.Forinstance,ifpatchesarerandomlyarrangedinspace,wewillexpectauniformdistributionofs(Figure A-2 a, A-2 b).Incontrast,ifpatchesarespatiallyalignedparalleltotheangleoftheadvectionsource(e.g.,alongariverwhenwindadvectionisgoingdownstream),wewillexpectabimodaldistribution(Figure A-2 c, A-2 d).Thebimodalnatureofthedistributionisexpectedbecausethedifferenceintheangleswasscaledby,andthusij+ji=1.Asthedifferencebetweenijandjiincreases,effectivedistancebecomesmoreasymmetric,andthedistributionwillbecomemoreskewedtowardsthelimitsofthedistribution.Wettedsixkindsofoccupancymodels.EachrepresentedahypothesisoftheunderlyingmechanismgoverningcolonizationandextinctionsintheL.rupestrismetapopulation.Traditionally,patchoccupancymodelsinametapopulationcontextoriginatefromthearea-isolationparadigm.Hence,whatwenamedthe(1)area/isolationmodel,includespatchconnectivityasasitecovariateforcolonizationandpatchareaasasitecovariateforextinction(i.e.(S),(A);whereSrepresentsameasureofconnectivityeitherwithasymmetricorasymmetricdispersalkernel,andAdescribespatcharea).Connectivitymayalsodecreaseextinctionprobabilitybytherescueeffectofnearbypatches(e.g., Hanski 1998 ).The(2)rescueeffectsmodelincludesconnectivityasasitecovariateforcolonizationandbothconnectivityandpatchareaareincorporatedasinteractingsitecovariatesforextinctioni.e.(S),(AS).Giventhewind-dispersednatureofLepanthesrupestris,weexpectthatlargerpatchesmayhave 19

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agreaterlikelihoodofreceivingseedsbecausetheyarelargertargets.Hence,wealsotteda(3)targeteffectsmodelinwhichpatchconnectivitytogetherwithpatchareaareincorporatedasacovariateforcolonizationandpatchareaasacovariateforextinctioni.e.(SA),(A).Wetteda(4)target/rescueeffectsmodel,whichincludedbothtargetandrescueeffectsi.e.(SA),(SA).Foreachofthesemodels(1)wettedamodelwithSsymandanotherwithSasymasasitecovariate.Becauseconnectivitymaynotnecessarilybeanimportantfactorpredictingcolonizationandextinctiondynamics( Pelletetal. 2007 ),wealsotteda(5)modelthathadmossareaassitecovariatesforcolonizationandextinction,butnoconnectivitymetricwereincluded.Previousanalysesforthisspecieshavefoundthatcolonizationandextinctiondynamicsmaybedifferentdependingonphorophytetype(treeorrockyboulder; Tremblayetal. 2006 ).Moreover,apreliminaryanalysisshowedthatmodelsthatincludedphorophytetypeasasitecovariateforcolonization,extinctionanddetectabilityhadabettertthanmodelsthatdidnotincorporatedit.Hence,forallofthesemodels(1)weincludephorophytetypeasaninteractingsitecovariateforcolonization,extinctionanddetectability.Wealsoincludedpatchareaasasitecovariatefordetectabilityinallmodels,becausethispreliminaryanalysisalsoshowedthatmodeltincreased.Finally,wealsotteda(6)null(intercept-only)modelwithnocovariatestocomparewithmorecomplexmodels.Wettedeachmodelusingmaximumlikelihoodwithcovariatesscaledandcentered,andrankedeachoccupancymodelbasedonAkaikeInformationCriterion(AIC; BurnhamandAnderson 2002 ).ModelswiththelowestAICwereconsideredmostparsimonious.Occupancymodelswerettedusingthepackageunmarked( FiskeandChandler 2011 )inR( Teametal. 2011 ). 1.3ResultsThecalculatedvaluesforrangedbetween1.1110)]TJ /F3 7.97 Tf 6.58 0 Td[(6and9.910)]TJ /F3 7.97 Tf 6.59 0 Td[(1.Thedistributionoftheijparametershowedabimodaldistribution,skewedtowardsthelimits 20

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ofthedistribution.Thisdistributionwascalculatedbasedonanaveragewinddirectionof2.02rad(Figure 1-3 ). Figure1-3. DistributionofijintheLepanthessystem Thetarget/rescueandtargeteffectsmodelshadbettertthanthenoconnectivitymodelsuggestingthatpatchconnectivityisanimportantpredictorofpatchdynamicsinthissystem.Themostparsimoniousmodelwasthetarget/rescueeffectsmodelthatincludedpatchareaandpatchconnectivitywithanasymmetricdispersalkernel(Sasym)assitecovariatesforbothcolonizationandextinction(asymmetricmodelhereafter).Theasymmetricalmodelhadconsiderablymoresupportthanthesimilarmodel(target/rescueeffects)thatincludedthepatchconnectivitymeasurewithasymmetricdispersalkernelasasitecovariate(Ssym;symmetricmodelhereafter;Table 21

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1-1 ).Inallvemodel-types,theasymmetricformofthemodelhadarelativebetterthanthecorrespondingsymmetricmodel. Table1-1. ModelselectionofoccupancymodelspredictingcolonizationandextinctionofLepanthesrupestris.Notation: ,occupancy;,colonization;,extinction;p,detectability.Covariates:Ssym,connectivitywithsymmetrickernel;Sasym,connectivitywithasymmetrickernel;A,mossarea;Ph,phorophytetype ModelAICAICweight 1 (.)(SasymAPh)(SasymAPh)p(APh)09.70E-012 (.)(SasymAPh)(APh)p(APh)7.572.20E-023 (.)(SsymAPh)(SsymAPh)p(APh)10.864.30E-034 (.)(SsymAPh)(APh)p(APh)20.643.20E-055 (.)(APh)(APh)p(APh)24.464.80E-066 (.)(SasymPh)(SasymAPh)p(APh)30.022.90E-077 (.)(SsymPh)(SsymAPh)p(APh)31.321.50E-078 (.)(SasymPh)(APh)p(APh)38.933.40E-099 (.)(Ph)(APh)p(APh)41.897.80E-1010 (.)(SPh)(APh)p(APh)42.635.40E-1011 (.)(.)(.)p(.)106.577.00E-24 Bothmodels(asymmetricandsymmetric)predictedsimilarpartialrelationshipsbetweencolonization,extinctionandconnectivity(Figure 1-4 ).Bothmodelspredictedapositiverelationshipbetweenincreasingconnectivityandcolonizationprobabilityforbothphorophytes.Also,bothmodelspredictthatextinctionprobabilitydecreaseswithincreasingconnectivityfortherockphorophyte.However,modelsdifferedinthepredictionofextinctionprobabilitiesfortreephorophyte.Theasymmetricmodelpredictsthatextinctionprobabilitydecreasedwithincreasingconnectivity,whilethesymmetricmodelpredictsaxedrelationship(Figure 1-4 ).Inbothmodels,mossareawaspositivelyrelatedtocolonizationanddetection(Figure A-2 A-3 ),butitwasnegativelyrelatedtoextinction.Detectabilitywashigh(p>0.80)andpositivelyrelatedwithpatchareainbothphorophytes(Figure A-3 ). 1.4DiscussionAsymmetricdispersalmaybetherulemorethantheexceptioninnature( GustafsonandGardner 1996 ).Nevertheless,mostparameterizationsofmetapopulationmodels 22

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Figure1-4. Partialrelationshipsbetweenextinction,colonizationandconnectivitymeasures(SsymandSasym).Panels(a)and(b)showthepartialrelationshipsfortherockphorophyte,while(c)and(d)showpartialrelationshipsfortreephorophytes.Shadeareasrepresent95%condenceintervaloftheestimates.NotethattherangeoftheconnectivityaxisrepresentthevaluesofSsymandSasymscaledandcentered. 23

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assumesymmetriceffectivedistancesbetweenpatches.Herewedevelopedanovelmodicationoftheconnectivityformulationtraditionallyusedinincidencefunctionmodelsthatallowsforasymmetriceffectivedistancesbetweenpatcheswhendispersalisdirectedbyanadvectionsource.Ourresultsonthewind-dispersedorchidLepanthesrupestrissuggeststhatasymmetricdispersalwasprevalentinthissystemandthatitscolonization-extinctiondynamicswerebetterdescribedbyametapopulationmodelthatincorporatethismodiedasymmetricconnectivityformulation. 1.4.1DetectabilityEventhoughSasymwasarelativebetterdescriptorofconnectivitythanSsym(modelsthatincorporatedSasymhadalwaysbettertthanmodelsthatincorporatedSsym),thismodelpotentiallyoverestimatesconnectivity.Theconnectivitymeasuresused(Equations 1 and 1 )donotincorporatetheoccupancystateofpatchj,whichiscommonlyusedtorestrictpotential(re)colonizationonlyfromoccupiedpatches.Thisincorporationimplicitlyassumesperfectdetection,whichwefoundisnotthecaseinthissystem.Imperfectdetectionisoftenregardedasanimportantfactorinthestudyandmanagementofanimalpopulations(e.g., MacKenzieetal. 2003 ).Nevertheless,thereisincreasingevidencethattheprobabilityofdetectioninplantoccupancystudiesishighbutoftenp<1( KeryandGregg 2003 ; Keryetal. 2006 ).Ourstudyshowedthatdetectabilityincreasedwithincreasingmossarea.L.rupestrisisaparticularlysmallorchid;hencethechancesofdetectingasingleindividual(particularlyifitismissingleaves)areexpectedtoincreasewithincreasingmossarea.Also,generallyinepiphyticspecies,largerphorophyteswillholdmoreindividuals,whichwillalsoincreasetheprobabilityofdetection( Fedrowitzetal. 2012 ; Snalletal. 2005 ; Tremblayetal. 2006 ). 1.4.2ColonizationandExtinctionOurresultssuggestthepotentialforbothtargetandrescueeffectsasimportantmechanismspredictingcolonizationandextinctions( BrownandKodric-Brown 1977 ; 24

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GilpinandDiamond 1976 ; Lomolino 1990 ).Aspatchesbecomelargerandmoreconnectedtheyhaveanincreasedlikelihoodofbeingcolonizedbecausetheybecomebiggertargets.Uncertaintyalsoincreasedwithincreasingconnectivityandarea(Figure 1-3 andFigure A-3 ).Inthissystem,patchesarespatiallyclusteredinspacewithveryfewreallyisolatedpatches,andtherearealsoveryfewreallylargepatches,whichmayaccountforthelackofvariationattheendoftheconnectivityandareaspectrum.Eventhoughwinddispersedplantspeciesarecharacterizedforhavinghighgeneowestimates( Hamricketal. 1995 ),L.rupestrishasarelativelyshortmeandispersaldistanceandmostoftheseedsmayfallbelowthematernalplant( Ackermanetal. 1996 ).Thismaybeduetotherelativelylowwindspeedintheforestunderstoryandtherelativelowfrequencyofhighwinds.Theimportanceoftheinteractionbetweenconnectivityandpatchsizeforcolonizationhasbeenalsoreportedforotherepiphyticspecies.Forinstance,treediameterandconnectivityhavebeenpositivelyrelatedtocolonizationoftheepiphyticlichenLobariapulmonaria( Gustafssonetal. 1992 ; Snalletal. 2005 )andtheepiphyticbryophytesNyholmiellaobtusifolia,Orthortichumspeciosum,Pylaisiapolyantha,andRadulacomplanata( Hazelletal. 1998 ).Theimportanceoftargeteffectstopredictcolonizationhasalsobeendescribedinactivedispersers,suchasthebutteryMaculineanausithous( Hovestadtetal. 2011 ).Thepartialrelationshipsbetweenconnectivityandextinctionforthesymmetricmodelshowedarelativelyconstantrelationshipwhilethebest-rankedmodel(asymmetric)showedanegativerelationshipfortreephorophyte.Treephorophytesarelocatedrelativelyhigherinelevationthanrockphorophytes(Figure A-1 )andtheiroccupancystatusmaydependmoreonwind-dispersedseeds.Metapopulationmodelsoftenmaketheplausibleassumptionthatpopulationsizeincreaseswithincreasingpatcharea( HanskiandHanski 1999 ).Lowerextinctionratesinlargepatchareasmaybetheresultofpositivedensity-dependentregulation,whichhavebeenpreviouslyreportedforL.rupestris( RiveraGomezetal. 2006 ).Previous 25

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researchonotherepiphyticplantsidentieslossofsubstrate(mostlytreemortality)asoneofthemostimportantdriveroflocalextinction( Snalletal. 2003 ; Tremblay 2008 ).Inoursystemoodingisthemostimportantdriverofextinction.Eventhoughtherehasbeensubstratelossbyoodingwhichalwaysresultedinlocalorchidextinction,thesewererelativelyinfrequent(15treesand2boulders;( Tremblayetal. 2006 ).Treeshad,ingeneral,higherextinctionratesthanboulders.ThisisconsistentwithapreviousstudyonL.rupestristhatfoundthattreeshadalmostdoubleextinctionratesasboulders( Tremblayetal. 2006 ).Substratelossduetoashoodingwashigherfortrees,whichmaysuggestthatoodingaffectsextinctionintreephorophytesgreaterthanboulders.Theremaybeotherfactors(covariates)thatmayaffectthecolonizationandextinctiondynamicsofL.rupestris,inadditiontoconnectivity,areaandphorophytetype.Forinstance,seedestablishmentinorchidsislimitedbytheirassociationwithmycorrhizae( Dressler 1993 ; RasmussenandWhigham 1993 ).Also,theamountofmossmoisturemayalsoaffectseedestablishment( Tremblayetal. 2006 ).Thesevariableswerenotincludedinthismodelduetothedifcultyofincorporatingtheseintoalong-termmetapopulationmonitoringprogram.Nevertheless,theirpotentialinteractionwithasymmetricconnectivitytopredictcolonizationandextinctiondynamicsremainsunexplored. 1.4.3AsymmetricDispersalandMetapopulationModelingThereareseveralexamplesoforganisms,includingbothactiveandpassivedispersersthatoccurasmetapopulationswithasymmetricconnectivity.InthecloselyrelatedspeciesL.eltoroensis,individualsmorefrequentlycolonizedbouldersandtreesbetween270and90( TremblayandCastro 2009 ).Inaquaticsystems,oceanandrivercurrentsareimportantdriversofasymmetricdispersalforbothvertebratesandinvertebrates( Lutscheretal. 2007 ; Tremletal. 2008 ; Watsonetal. 2011a ).AsymmetricdispersalpatternhavealsobeenfoundinactivedisperserssuchasEvergladesSnailKites(Rostrhamussociabilisplumbeus),cactusbugs(Chelinidea 26

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vittiger Fletcheretal. 2011 )andtheendangeredIberanlynx(Lynxpardinus; Ferreras 2001 ).Moreover,anindividualbasedmodeldevelopedby GustafsonandGardner ( 1996 )showedthatalteringlandscapeheterogeneityresultedinasymmetricratesofimmigrationandemigrationamongresourcepatches.Hence,asymmetricconnectivitymaybetherulemorethantheexception,giventhatsymmetricconnectivitymayonlybeapplicablefororganismswhichdispersalisnotaffectedbyadvectionsources,spatialvariationinresourcesorthatliveinhomogeneouslandscapes.Suchexamplesareuncommoninnature.Wefoundthatconnectivitywasimportanttopredictcolonizationsandextinctions,andthatasymmetricdispersalisprevalentinthesystem.Thisisshownbytheskeweddistributionoftheparametertowardsthelimitsofthedistributionandbecausethebest-rankedmodelincludedSasym.Apreviousanalysison10speciesofbirds,amphibiansandbutteriesshowedthataddingconnectivityasasitecovariatedidnotimprovedmodeltcomparedtoconstantcolonizationparameters( Pelletetal. 2007 ).Thisstudyarguesthatusingthesymmetricconnectivitymeasuretraditionallyusedinincidencefunctionmodelsmaynotbeadequatetomodelmostspeciesbecauseitdoesnotaccountfordensity-independentmovementsamongpatches(e.g.,responsetoanadvectionsourceortaxis)whichmaysignicantlyaffectdispersalrates.TheSasymconnectivitymeasureaccountsforsomeofthesedensity-idependentmovements.Itsignicantlyimprovedmodelt,evenwhenitonlytookintoconsiderationaveragewinddirection,whichisabroadwaytodescribeadvectioneffects.Weexpectthatifweincorporatepatch-specicmeasuresofwinddirection,modeltwouldimproveevenmore.Inthissystemwhichiscomposedof920patches,itisunfeasibletocollectpatch-specicwinddirection(orotheradvection);however,inotherapplicationsinsmallermetapopulations,thismaybefeasible.Thelackofrecognitionofasymmetricconnectivityinempiricalmetapopulationstudiesmaybedueinparttotheabsenceofanestimationframeworkthatcan 27

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incorporateasymmetricdispersal.Hereweprovideageneralframeworkusingmulti-seasonoccupancymodelingusingconnectivityasacovariate.Theasymmetricconnectivitymeasurethatweapplied(Equation 1 )canbegenerallyappliedtoanyothersimpleadvectionsources(e.g.,watercurrentdirection)oritcanbemodiedtoadaptittoothersystems.Forexample,winddirectioncanbereplacedbytheangleofriverineormarinecurrents.Theincorporationofasymmetricconnectivityintometapopulationmodelingwillpotentiallyresultinanincreaseintheaccuracyofconservationandmanagementdecisions( Begeretal. 2010 ). 28

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CHAPTER2THEPROXIMATECAUSESOFASYMMETRICMOVEMENT 2.1IntroductionVariationinmovementhasbroadimplicationsformanyareasofecologyandevolution,includingalteringlocaladaptationviageneow( KaweckiandHolt 2002 ),communitystructure( Liebholdetal. 2004 ; Salomonetal. 2010 ; Tilmanetal. 1994 ),andpopulationdynamics( ArmsworthandRoughgarden 2005 ; Revillaetal. 2004 ; Wiegandetal. 1999 ).Inmetapopulationtheory,movementisparticularlycriticalbecauseprovidesthemeansfor(meta)populationpersistenceovertime( Hanski 1998 ).Mostmetapopulationmodelingapproachesdescribeinter-siteconnectivityusingpair-wiseEuclideandistancebetweenpatches( Dias 1996 ; HanskiandHanski 1999 ).Usingthesedistancestypicallyresultsinthesimplifyingassumptionofasymmetricmovementpatterninwhichtheprobabilityofmoving,p,frompatchitopatchjisthesameasmovingintheoppositedirection(i.e.pij=pji).Asymmetricpatternofmovementmaybeexpectedwhenallthefactorsthataffectdispersalareidenticalinalldirections( Bodeetal. 2008 ; GustafsonandGardner 1996 );however,suchsymmetrymightberarelyobservedinnature.Instead,anasymmetricpatternofmovement(e.g.pij6=pji)maymorelikelyemergefromorganismsmovingacrossspatiallyheterogeneousenvironments( Ferreras 2001 ; PrevedelloandVieira 2010 ).Recentmodelsthatincorporatethiscomplexitysuggestthatmovementasymmetriescanhavesubstantialeffectsonmetapopulationdynamics( ArmsworthandRoughgarden 2005 ; Vuilleumieretal. 2010 ; VuilleumierandPossingham 2006 ).Avarietyofmechanismsmaygiverisetoasymmetricalpatternofmovement.Forinstance,asymmetricmovementmaybetheresultofenvironmentalheterogeneitysuchasvariationinhabitatqualityorpatchsize( Ferreras 2001 ; GustafsonandGardner 1996 ; Holt 1996 ; Pulliam 1988 ).Individualsmayhaveadaptedtoseekbetterhabitatsbyactivelydispersingfromlow-tohigh-qualitypatches(oftenlargerinsize;e.g., 29

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Ferreras 2001 ; GustafsonandGardner 1996 ),orhighqualitypatchesmaybenetproducersofemigrantstolow-qualitypatchesduetodensitydependence(e.g.,insource-sinkdynamics; Holt 1985 ; Pulliam 1988 ).Othermechanismsmaylikelydifferbetweenpassiveandactivedispersers.Inpassivedispersers,asymmetricmovementmaybetheresultofadvectionthroughadirectionalmedium( ArmsworthandBode 1999 ; Keddy 1981 ; SchickandLindley 2007 ; Tremletal. 2008 ).Inactivedispersers,asymmetriesmaybetheresultofchangesintheorganismsmovementbehaviorduetoenvironmentalcues(e.g.taxis; Compton 2002 ).Forinstance,manyspeciesofinsectsareattractedtovisualorchemicalcuesthatdirecttheirmovement( ProkopyandOwens 1983 ).SomeHomoptera,HymenopteraandHemipteradirecttheirmovementupwindwhiletrackingvolatilechemicalsignalsfromvegetation(anemotaxis;e.g. Compton 2002 ; Moseretal. 2009 ; Williamsetal. 2007 ).Thelikelihoodofanindividualtomovetoaparticularpatchmayalsodependonthepatchoccupancystateortheabundanceconspecics( SerranoandTella 2003 ; Serranoetal. 2001 ; SmithandPeacock 1990 ).Hereweuseacombinationofanobservationalandexperimentalapproachtotestformechanismsofasymmetricmovementinthecactus-feedinginsect,Chelinideavittiger.Thisspeciesisentirelydependentonpatchypricklypearcactus(Opuntiaspp.),whereitfeeds,breedsandaggregates( Milleretal. 2012 ).Recently, Fletcheretal. ( 2011 )foundthatmovementinthisspeciesacrossapatchnetworkwashighlydirectional,whichalteredassessmentsoflandscapeconnectivity.Wehypothesizedthatthreeprimarymechanismsmaycausedirectedmovements:(1)positiveanemotaxis(movementupwind),(2)movementtowardlargerpatches,and(3)conspecicattraction.WindmayhelpC.vittigerdetectvolatileolfactorycuesofOpuntia,potentiallyresultinginindividualsdirectingtheirmovementstowardspatchesthatarelocateddownwind( SchooleyandWiens 2004 ).Similarly,C.vittigermayactivelysearchforlargerpatchesofOpuntia,becauseofenhancedresourceavailability(see SchooleyandWiens 2005 ). 30

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Becauseoftheaggregativebehaviorsofthisspecies( FletcherandMiller 2008 ; Milleretal. 2012 ),conspecicattractionmayoccurandalsocausedirectedmovements.Toaddressthesemechanisms,werstre-analyzeddatafromamark-recapturestudyacrossa56patchnetwork( Fletcheretal. 2011 )totestforthemagnitudeofmovementasymmetryandifwinddirectionorpatchareamayexplainobservedvariationinmovementacrossthepatchnetwork.Second,weconductedatranslocationexperimentinwhichwemanipulatedpatcharea,windadvectionandthepresenceofconspecicstodetermineif,andtheextenttowhich,C.vittigerbiasesmovementsinresponsetothesefactors. 2.2Methods 2.2.1StudyAreaandFocalSpeciesThestudywasconductedattheOrdway-SwisherBiologicalStation(29.4N,82.0W),inPutnamCounty,Florida,USA.Inthisarea,C.vittigerusesOpuntiahu-mifusa,primarilyoccurringinold-eldhabitats,whicharecommonatthestation.O.humifusaisnativetotheeasternU.S.andtypicallygrowsindry,sandysoilsinsandhills,oldelds,prairies,orscrub.WefocusedourresearchonadultmovementsofC.vittiger.AdultsofC.vittigerarewinged,butrarelyy;instead,adultstypicallywalkbetweencactuspatchesthroughanunsuitablematrix( DeVolandGoeden 1973 ; SchooleyandWiens 2004 ).Mediandispersalrangesfrom1.5m/day( SchooleyandWiens 2004 2005 ),whichmakesmovementtractableandanidealsystemtostudyshort-termdispersal(Figure 2-1 b). 2.2.2StudyDesign 2.2.2.1Mark-recaptureWere-analyzeddatapublishedby Fletcheretal. ( 2011 )inthecontextofconnectivitymodeling.ThedataconsistedofindividualmovementsofmarkedC.vittigerinallpatches(n=56)insidea3030mplot(Figure 2-1 a).Patchesweredenedfollowing SchooleyandWiens ( 2004 ),wherecladodes(i.e.,cactuspads)<25cmapartwere 31

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Figure2-1. Observedinter-patchtransitionsofC.vittigerina3030mplot.(a)Greencirclesrepresentpatchesthatincludedmovementwhileblackcirclespatchesthatdidnot.Notehowmostoftheconnectionsgoinonlyonedirectionshowingapatternofhighasymmetricmovement.(b)Distributionofmovementdistancesdividedbythenumberofpatchesavailableformovement(availability).Circlesizeisproportionaltopatchsize. consideredthesamepatch.CensuseswereconductedfromSeptember2008toNovember2009every2weeks(exceptduringthewinter)foratotalof21censuses.Duringeachcensus,allindividualsoneachpatchwerecounted(bothnymphsandadults),andalladultsweresexedandindividuallymarkedonthepronotumwithnontoxicpermanentmarker.Eventhoughthisplotwasnotaclosedsystem,additionalrecaptureratesinthesurroundingareawerelow.Toaddressthemechanismsforasymmetricmovement,weestimatedpatchareausingtheequationforanellipse(area=majorradiusminorradius; SchooleyandWiens 2005 ).Themajorandminorradiiweremeasuredusingaruler.Weuseddatafromthenearestweatherstationtothesite(<1km),toestimatewinddirection.WeaverageddailywinddirectionfromJanuary1,2009toNovember27,2009toproduceasingleestimateofaveragewinddirectionrelativetoeachpair-wisecombinationofpatchesinthenetwork. 32

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2.2.2.2FieldexperimentWeusedarandomblockdesigninwhicheachblockcontainedfourplots,oneforeachtreatment(wind,patcharea,andconspecics)andacontrol(Figure 2-2 ).EachplotincludedfourpatchesofOpuntiahumifusacladodeslocatedinthefourcardinaldirections.Thereleasepatchwaslocatedinthecenteroftheplotandconsistedofasinglecladode(Figure 2-2 ).Weusedasmallreleasepatchtocatalyzemovement( CastellonandSieving 2006 )becauseasinglecladodeisbelowtheminimumpatcharearequirementsforC.vittiger( SchooleyandWiens 2005 ).Allexperimentalpatcheswerelocated1-mawayfromthereleasepatch,whichiswithintheperceptualrangeofthisspecies( Fletcheretal. 2013 ).Allplotsconsistedof32cladodesthatweredistributedamongfourpatches,arrangedindifferentproportionsbetweenthepatchesdependingonthetreatment.Inthewindandconspecictreatments,andinthecontrolplot,eachexperimentalpatchwascomposedofeightcladodesperpatch(Figure 2-2 a,b,c).Intheareatreatment,onepatchconsistedof16cladodes(largepatch);anotherconsistedofeightcladodes(mediumpatch)andtwooffourcladodes(smallpatches;Figure 2-2 d).Thesecladodeswerefresh,withnovisiblefeedingmarksandwerecollectedinnearbyold-elds(<100m).ThewindtreatmentconsistedofplacingabatteryoperatedCaframoKonablackcolor(7cmdiameter)fanlocated5cmawayfromarandomlyselectedexperimentalpatchand10cmfromthesoilwithwinddirectedtowardsthereleasepatch(Figure 2-2 b).Thefanwasoperatedbya12Vbatteryandcontinuouslyblowingduringthesamplingperiod(48hrs).Windspeedatthewindtreatmentexperimentalpatchwas14.8km/hand4.7km/hatthereleasepatch.ThesewindvariationincludedspeedspreviouslyreportedintheliteratureinastudythatfoundastrongrelationshipbetweenwinddirectionandC.vittigermovement(0.5.75km/h; SchooleyandWiens 2005 ). 33

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Figure2-2. Diagramshowingthestudydesign.Experimentalblockswerecomposedoffourplots,oneforeachtreatment(a)control,(b)wind,(c)conspecics,and(d)variationsinpatcharea.Thestudyincluded3blocksofthisdesignandalltreatmentswereassignedrandomly. 34

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Theconspecicstreatmentconsistedofplacingoneindividualmaleinonepatchandoneindividualfemaleinanotherpatchinthesameplot(Figure 2-2 c).Weincludedmalesandfemalesseparatelybecausepreviousstudieshavefounddifferencesinaggregativebehaviorformalesandfemales( Milleretal. 2012 ).Treatmentswereassignedtothepatchesrandomly.Theindividualcladodesthatcontainedtheconspecicswereenclosedwithnemeshtopreventtheindividualfromleavingthepatch( Milleretal. 2012 ).Themeshallowedforvisualand/orpheromonalcuestobepotentiallydetectedbythereleasedindividuals.Weconsideredcontrolpatchestobeincludedinthetreatmentplotsthatwerenotsubjecttothetreatment.Forexample,inthewindtreatmentplots,onlyoneoutofthefourpatcheshadafan,suchthattheotherthreepatchesintheplotwereconsideredcontrolplots.Similarly,intheconspecictreatmentsthetwopatchesthatdidnothaveconspecicswereconsideredcontrolpatches.Intheareatreatment,patcheswitheightcladodeswereconsideredcontrolalso.Inadditiontothecontrolpatches,weincludedintheblockdesignacontrolplotinwhicheachpatchwascomposedof8cladodesandreleasesandcensusesweredoneinthesamewayasinthetreatmentplots.Wealsoperformedtwotypesofproceduralcontrols,onefortheconspecicsandanotherforthewindtreatment,toensuretheobservedratesofimmigrationforeachtreatmentweredrivenbythetreatmentsandnotanartifactofincorporatingafanormeshintheplot.Weconductedtheproceduralcontrolexperimentforthewindtreatmentinthesameplotplots(i.e.samelocationandsamecactuspatchesineachblock)wherethewindexperimenttookplace;theonlydifferencewasthatthefanwasplacedblowinginthedirectionawayfromthereleasepatch.Bydoingthis,wekeepthepotentialeffectsofhavinganon-naturalobjectinthelandscape(i.e.blackfanproducingnoise),butremovedtheeffectofwind.Similarly,weconductedtheproceduralcontrolexperimentfortheconspecicstreatmentinthesameplotswheretheconspecictreatmentexperimentstookplace(i.e.differentplotsthanthewindproceduralcontrol 35

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experiments).Weenclosedthesametwopatchesatusedfortheconspecictreatmentswithmesh,withtheonlydifferencebeingthataconspecicindividualwasnotincluded.Samplingwasdoneinthesamewasasintheotherexperimentaltreatments(seebelow).Atthebeginningofthestudy,plotsweredepletedofanyothercactusandindividualsofC.vittigerwithin10mofthereleasepatch.Plotswerelocatedatleast35mapartfromthenearestexperimentalplot.Locationsofthetreatmentsatboththeplotandpatchlevelwereassignedatrandom.Individualswerecapturedinnearbycactuspatches(<100m).Thesewerebredandrearedindelicontainersinascreenhouse.NymphswereraisedingroupswithfreshandhealthyO.humifasacladodes.Assoonastheybecameadults,theyweremarkedontheirpronotumusingnon-toxicpermanentmarkerandtransferredtoindividualcontainersuntiltheywerereleasedintheexperimentalplots.Agreenhouseexperimentfoundthatmarkingindividualsinthiswaydoesnotaltersurvivalrates(P>0.5;Fletcher,unpublished).Ineachtrial,twomalesandtwofemaleswerereleasedatthereleasedpatch.Thelocationofeachoftheseindividualsintheplotwascensusedafter24andthenafter48hoursofbeingreleased.Eachobservedindividualwasremovedfromtheplotassoonasitslocationwasnoted.Weconducted7(5forexperimentaltreatmentsand2fortheproceduralcontrol)trialsineachofthethreeblocksbetweenJulyandNovember2012.PreviousstudieshavefoundthatmatrixresistancemaybeanimportantdeterminantinthemovementofC.vittiger( SchooleyandWiens 2005 ).Tocontrolforthiseffect,wemeasuredthestructureofthematrixfromthereleasepatchtoeachexperimentalpatch.Wemeasuredmatrixstructureat15pointswithina10050cmstriptransect(5measuresineachedgeofthetransectand5inthecenterspacedevery20cm)betweenthereleasepatchandtheexperimentalpatch.Foreachpoint,wequantiedthenumberoftimesvegetationtouchedagradedpoleineach10cmcategory(vertically) 36

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toamaximumof90cm.Weaveragedthenumberoflayersandthemaximumheightofeachsetof15measurestocharacterizethematrixbetweenthereleasepatchandeachexperimentalpatch.Bothmeasures(numberoflayersandmaximumheight)werehighlycorrelated(rp=0.85);therefore,herewepresentonlythedataformaximumheighttobeconsistentwithotherinvestigationsonthisspecies( SchooleyandWiens 2005 ). 2.2.3Analyses 2.2.3.1Mark-recaptureWerepresentedthemovementsofC.vittigerasamatrixWwerewijrepresentsthetotalnumberofmovementsfrompatchitopatchj.WecalculatedtheproportionofsymmetriclinksasthenumberoflinksinWinwhichwij)]TJ /F5 11.955 Tf 12.49 0 Td[(wji=0(i.e.thenumberofmovementswasthesameinbothdirections)dividedbythetotalnumberoflinksthathadmovement.While Fletcheretal. ( 2011 )notedthatmovementappearedhighlydirectional,theydidnotassessifthisputativepatternwasgreaterthanwhatwouldbeexpectedbasedonchance.Totestifthisproportionwasdifferentfromtheproportionofsymmetriclinksexpectedbychance,wecomparedtheproportionofsymmetriclinksinthismatrixofmovementsW,totheproportionofsymmetriclinksin1000randomlygeneratedmovementmatrices.TheserandommatriceshadthesamenumberofmovementsasW,butthepositionsofthelinkswererandomlyassignedamongthelinksthathadmovement(i.e.shufingtheweightsofthematrixwhilekeepingthesametopology).Weusedalinearregressionmodeltotestfortherelationshipbetweenapatternofasymmetricmovementandvariationsinpatchsizeandtheangleofwinddirection.Theresponsevariablesforbothofthesemodelswasthedifferenceinmovementbetweeniandj,wij=jwij)]TJ /F5 11.955 Tf 13.04 0 Td[(wjij.Similarly,wecalculatedthedifferenceinpatchareaasAij=jAi)]TJ /F5 11.955 Tf 12.33 0 Td[(AjjwhereAiistheareaofpatchi.Wealsocalculatedthedifference(ij),betweentheangleofwinddirectionandtheanglebetweenpatchesiandj,inradians 37

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withrespecttothehorizontalaxis.Forsimplicity,ijisscaledbytoconstrainij2[0,1].SeeChapter 1 2.2.3.2FieldexperimentTotestfortheeffectofthetreatments(variationsinpatcharea,windandconspecicpresence)drivingthedirectedmovementsofC.vittiger,wemodeledtherateofmovementfromthereleasetoeachoftheexperimentalpatches.Thisrateofmovementdescribedtheproportionofthereleasedindividualsthatmovedfromthereleasepatchtoanexperimentalpatchinaperiodof48hrs.Thisisanappropriateresponsegiventhatexperimentalpatcheswerelocated1-mawayfromthetargetpatchandtheaverageindividualmovementrateis<1m/day(Figure 2-1 b).Eachtreatmentwasanalyzedindependentlybecausevariationwithinplotsfordifferenttreatmentsmadepoolingdataacrosstreatmentsimpractical.Intheconspecictreatment,wettedseparatemodelstodescribetherateofmovementofmales,femalesandanotherforbothsexestogether,becausepreviousresearchhasshowndifferentaggregativebehaviorsbetweenmalesandfemalesC.vittiger( Milleretal. 2012 ).Foreachtreatmentconsideration(area,wind,andconspecicsmale,femalesandboth),wettedvetypesofmodelsthatrepresenteddifferenthypothesesexplainingtherateofmovement.An(1)intercept-onlymodelrepresentingthenullhypothesisthattheexperimentaltreatmentsarenotdrivingthemovementsofC.vittiger.We,alsottedamodelrepresentingthe(2)treatment.Becauseeachtreatmentwasmodelindependently,thevariableusedwasdifferentforeachtreatment.Intheareatreatment,variationinpatchareawasrepresentedbythenumberofcladodesineachpatch(4,8or16cladodes).Inthewindtreatment,thepresenceofwindwasrepresentedbyabinaryvariableindicatingthepresenceornotofthefanintheexperimentalpatch.Intheconspecictreatment(responseofmales,femalesandboth),thepresenceofaconspecicwastreatedascategoricalvariablewiththreefactors:male,femaleandcontrol.Theeffectofthematrix,representedbythemaximumaveragevegetationheight 38

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betweenthereleasepatchandtheexperimentalpatch,wasincorporatedintomodelsbothasa(3)singlemaineffect,andalsoasan(4)additiveand(5)interactingcovariatewiththetreatment.Weusedanegativebinomialregressiontomodeltherateofmovementtowardexperimentalpatcheswithanoffsetrepresentingthetotalnumberofindividualsreleased(4individuals,exceptfortheresponseofmalesandfemaleconspecicsinwhichtwoindividualswerereleased).Thenegativebinomialdistributionwasanappropriatechoicebecausethecountdatashowedover-dispersion.Inaddition,wecomparedmodelttoaPoissondistributionandthenegativebinomialresultedinbettert(allnegativebinomialmodelsresultedinAICvalues<2unitsthanthecorrespondingPoissonmodel).Inapreliminaryanalysis,weincorporatedthedateofthecensus,blockandplotasrandomeffects,buttheirvarianceresulted<10)]TJ /F3 7.97 Tf 6.59 0 Td[(8andthustheirincorporationdidnotsignicantlyincreasemodelt.NegativebinomialregressionswerettedbymaximumlikelihoodusingtheMASSpackageinR.WeusedAkaikesInformationCriterion(AIC)tocomparemodeltandchoosethemostparsimoniousmodel. 2.3ResultsWeobservedatotalof70movementsinthe3030mpatchnetwork(see Fletcheretal. 2011 ,formoredetails).Theproportionofsymmetricconnectionsinthisnetwork(0.04)waslowerthantheaverageintherandomlygeneratedmovementnetworks(0.100.04;z=)]TJ /F8 11.955 Tf 9.3 0 Td[(1.77,P=0.04).Patcharearangedfrom1.25cm2.Averagewinddirectionwas13199,withaverageprevailingwindspeedsof4.073.03km/hr.Wefoundaweaklysignicant,positiverelationshipbetweenthedifferenceinmovement(wij)anddifferenceinpatcharea(Aij)betweenpatches(R2=0.09,P=0.05;Figure 2-3 ).Incontrast,therewasnosignicantrelationshipbetweenthedifferenceinmovementandincreasingdeviationfromtheangleofwinddirection(R2=0.03,P=0.25;Figure 2-3 ). 39

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Figure2-3. Linearregressionanalysis.Figureshowstherelationshipbetweenthedifferenceinmovement(wij),and(a)thedifferenceinpatcharea(Aij)and(b)deviationfromaveragewinddirection(ij)inthemark-recapturestudy Intheeldexperiment,atotalof240individualsweremarkedandreleasedofwhich86wererecaptured.Werecaptured19individualsintheareatreatment,16individualsinthewindtreatment,25(14femalesand11males)intheconspecicstreatmentand26inthecontrolplot.Themostparsimoniousmodelexplainingtherateofmovementintheareatreatmentincludedthenumberofcladodes(patchsize)interactingwiththemaximumheightofthevegetationinthematrixascovariates(Table 2-1 ).Thismodelpredictedarelativesmallandconstantrateofmovementtoallsizepatcheswhenthematrixwaslow,butahigherrateofmovementtowardsbiggerpatcheswhenthematrixvegetationwashigherthanthemedian(Figure 2-4 ).Thenullmodel(interceptonly)wasthemostparsimoniousmodelexplainingtherateofmovementinthewindtreatmentandcontrolplots.Anullmodelwasalsothemostparsimoniousexplainingtherateofmovementintheconspecictreatmentswhenmodelingtheresponseofbothmalesandfemalestogetherandwhenmodelingthemovementoffemales.Whenmodelingthe 40

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rateofmovementofmalesintheconspecicstreatment,themostparsimoniousmodelincorporatedthepresenceofconspecics;however,thismodelhadnobettertthanthenull(intercept-only)model(AIC<2;Table 2-1 ).Thenull(intercept-only)modelwasthemostparsimoniousmodelexplainingtherateofmovementinbothproceduralcontrols(Table 2-1 ). Figure2-4. Partialrelationshipsofthebesttmodelexplainingtherateofmovementintheareatreatmentsoftheeldexperiment.Thismodelincludedpatcharea(numberofcladodes)interactingwithvegetationheight.Linesrepresent1stquartile(lowmatrixheight),median(moderatematrixheight)and3rdquartile(highmatrixheight)ofmaximumvegetationheight.Shadedareasrepresentthe95%condenceintervals 2.4DiscussionAsymmetricdispersalmaybetherulemorethantheexceptioninnature.Whilerecentmetapopulationmodelssuggestthatmovementasymmetriescanhavesubstantialeffectsonmetapopulationdynamics,empiricaltestsofdispersalasymmetriesacrosslandscapesremainrarewithevenfewertestsofthemechanismsdrivingsuchpatterns.WefoundthatmovementsofC.vittigerweregenerallyasymmetricandbetterexplainedbymovementsfromsmall-to-largepatches. 41

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Table2-1. Modelselectionforeachexperimentaltreatment.Thenumberofcladodesisrepresentedbyn.cladodes,matrixreferstothemaximumheightofvegetationinthematrix TreatmentSexResponseModelkAICDeviance AreaBothN.cladodes*Matrix4036.1N.cladodes+Matrix31.438.8N.cladodes23.537.4Matrix21236Interceptonly112.137.3WindBothInterceptonly1033.5Wind21.233.7Matrix21.933.7Wind+Matrix33.433.8Wind*Matrix45.633.6ConspecicsBothInterceptonly1038.3Matrix21.438.6Conspecics32.738.7Conspecics+Matrix24.638.3Conspecics*Matrix39.238.1MalesConspecics3035Interceptonly11.736.1Conspecics+Matrix42.235Matrix23.836.1Conspecics*Matrix64.733.9FemalesInterceptonly1060.2Conspecics31.657.5Matrix2260.1Conspecics+Matrix43.657.3Conspecics*Matrix65.654.9ControlBothInterceptonly1054.59Matrix22.154.6ProceduralCont.BothInterceptonly1020.8Fan20.6821.78Interceptonly1020.35Mesh21.520.35 42

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Boththeobservationalandexperimentalresultsshowednosupportforprevailingorarticialwinddrivingmovementdecisions.EventhoughpreviousstudieshaveshownthatC.vittigerdirectsitsmovementstowardsprevailingwinds( SchooleyandWiens 2003 )inopen-shortgrassprairiesinColorado, Fletcheretal. ( 2013 )alsofoundlittlesupportforthiswindhypothesiswhilestudyingtheperceptualrangeofC.vittigerinourstudyarea.Theysuggestedthatwindspeedsatold-eldhabitatsinourstudyareaarerelativelylowerthanintheopenshort-grassprairiehabitatthanwhere( SchooleyandWiens 2003 )conductedtheirstudy.Nevertheless,inourexperiments,articialwindspeedsat10cmfromthegroundwerewithintherangeofthoseintheopenshort-grassprairie( SchooleyandWiens 2003 ).Thisdiscrepancymaybeexplainedbyatleasttwoalternativehypothesesrelatedtohabitatdifferencesbetweenopenshort-grassprairieinColoradoandold-eldhabitatsinFlorida.First,C.vittigerpopulationsatOSBSmayhavedevelopedalternativewaystodetectpatchesthatdonotincluderelyingonwindbecausewindisnotaprevalentfactorinthissystem.Alternatively,thetypeoramountofolfactorycuesmaydifferbetweenO.humifusaandO.polycantha(theOpuntiaspeciesin SchooleyandWiens 2003 ).WefoundnostrongsupportforthepresenceofadultconspecicsdrivingthedirectedmovementsofC.vittiger.Nevertheless,multiplestudieshavefoundthatthisspeciesusessocialinformationtoguideitsbehavior(e.g., FletcherandMiller 2008 ; Milleretal. 2012 ),suggestingthattheremaystillpotentialforother(non-tested)conspecicinteractionsdrivingdirectedmovements.Giventhatconspecicinteractionshavebeendocumentedforbothaggregativeandnon-aggregativespecies(e.g., Fletcher 2009 ; SerranoandTella 2003 ),andcanaffectcolonizationrates( HahnandSilverman 2006 ),functionalconnectivityestimates( Zeigleretal. 2011 )andmetapopulationdynamics( Alonsoetal. 2004 ; Rayetal. 1991 ),itspotentialimplicationsasadriverofasymmetricdispersalremainsanalternativehypothesistobefurthertested. 43

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Patchsizehasbeenfoundtobeanimportantdeterminantofpatchimmigrationrates( BowlerandBenton 2005 ).Bothobservationalandexperimentalresultsshowedsupportforthehypothesisthatvariationinpatchareaisadrivingmechanismofdirectedmovementsthatresultinanasymmetricpatternofmovement.Intheeldexperiments,therateofimmigrationincreasedwithincreasingpatchsize.Twopotentialmechanismsmayaccountforthispattern.First,organismsmaybedispersingbyapassive-diffusiveprocess( KindvallandPetersson 2000 ; TravisandFrench 2000 ),whichmakeslargepatches(i.e.patcheswithgreatercircumference)morepronetobedetectedbyarandomlymovingindividual.Ontheotherhand,thereisampleevidenceshowingthatanimalstendtomovetowardlargerpatches(e.g., Diffendorferetal. 1995 ),suggestingthatotherfactorsmaybecovaryingwithpatchsizeandindividualsmayhavestrongpreferenceforlargerpatches( BowlerandBenton 2005 ).OurresultssuggestthatC.vittigerpreferlargepatches.IfindividualsofC.vittigerwererandomlyndingpatchesintheareatreatment,wewouldhaveexpectedtheimmigrationratetobeproportionaltothelineardimensionofthepatch,becauseallpatcheswerelocatedatthesamedistancefromthereleasepatch( Bowmanetal. 2002 ).Giventhatinourexperimentaldesignthelineardimensionofthepatcheswasproportionaltothenumberofcladodes,wewouldhaveexpectedtheaveragemovementratetowardsthe16cladodepatchestobe4timesthatofthe4-cladodepatches.Instead,wefoundthattherateofmovementtowardsthe16cladodespatchwas11timesthatofthe4cladodespatches(Figure 2-5 c).Moreover,theaveragemovementratetowardsthe16cladodepatcheswas2.5timesthatofthe8cladodepatchessuggestinganon-linearrelationshipbetweenpatchsizeandimmigrationrate(Figure 2-5 c).C.vittigermaydirecttheirmovementstowardslargecactuspatchessimplybecauselargerpatchesprovideagreateramountoffoodresources.Alternatively,largepatchesmayprovidemorespinesforfemalestolayeggsormayincludecomplexverticalstructuresthatmayserveasrefugiafrompredators,therebydecreasingpredationrisk. 44

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Figure2-5. Movementrate(SE)atthepatchlevelintheexperimentaltreatments.Thesolidgraylinerepresentsthemeanimmigrationrateforpatchesinthecontroltreatmentandthedashedlinetheassociatedstandarderrors Acombinationofmatrixcharacteristicsandotherlandscapefeaturessuchaspatchsizeandisolationarethemostimportantdriversofinter-patchmovements( BenderandFahrig 2005 ).Moreover,variationsinmatrixpermeabilityandlandscapecharacteristicsmayresultinanasymmetricpatternofdispersalThecostofdispersalisanimportantdeterminantofwhetherdispersalleadstotnessbenets( BowlerandBenton 2005 ).Ourstudysystemcouldbeanexampleinwhich,eventhoughincreasedmatrixstructuremayprovideresistancetomovement( SchooleyandWiens 2005 ),itmayincreasesurvivalbydecreasingpredationrisk.PredationestimatesforC.vittigerarescarce,butspidershavebeenshowntopredatethemandothercactus-feedinginsects( DeVolandGoeden 1973 ; Miller 2008 ).Regardlessoftreatment,mostplotsinwhichweobservednorecaptureshadlittleornomatrixstructure.Theseindividualseitherwerepredatedorpermanentlyemigratedfromtheexperimentalplot,suggestingthattheriskoflongerdistancedispersalmaybelessthantheriskofbeingexposedmovingthroughthematrixwithoutcover.Onthecontrary, 45

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movingthroughamorecomplexmatrixmayprovidecoverinwhichindividualscouldmovemoresafelyandassesstheirsurroundingsforapatchtosettle.Asimilarpatternhasbeenfoundinothertaxa.Forinstance,predationriskforforestbirdsishigherwhiletryingtocrossanagriculturalmatrixthansecondaryforestswithrelativelymoreverticalstructureInter-patchmatrixheterogeneityandmovementbehaviorhasstrongimplicationsforpopulationdynamicsandbiodiversitypatterns( ArmsworthandRoughgarden 2005 ; Salomonetal. 2010 ).Wefoundthatpatchsizeandmatrixstructurewerethemostimportantpredictorsofinter-patchmovementsandimmigrationrateinC.vittiger,leadingtodirectedmovementsthatresultedinanasymmetricpatternofdispersal.Giventhatarelationshipbetweenpatchsizeandmatrixheterogeneityiscommoninmanytaxa,ourresultssupporttheideathatasymmetricdispersalmaybecommoninnature( GustafsonandGardner 1996 ).Variationinlandscapefeaturessuchasmatrixstructureandpatchsizemaybeimportantdriversofasymmetricdispersalandhencehavebroadimplicationsfor(meta)populationdynamics. 46

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CHAPTER3OPTIMALSITESELECTIONSTRATEGIES:PROTECTINGAGAINSTWORST-CASEDISTURBANCESCENARIOS 3.1IntroductionHuman-inducedhabitatlossandglobalclimatechangehavebeenidentiedassomeofthemostimportantlarge-scaledriversofbiodiversitydeclines( Fahrig 2003 ; LindenmayerandFischer 2006 ; Mawdsleyetal. 2009 ; Salaetal. 2000 );however,thereismuchuncertaintyabouttheirfutureecologicalconsequences( Elithetal. 2002 ; Kujalaetal. 2012 ; Moilanenetal. 2006a ; Thomasetal. 2004 ).Thereisanincreasinginterestinidentifyingconservationstrategiesthatcanamelioratesomeoftheexpectedfutureimpactsofthesedrivers,whileatthesametimeaccountingforuncertainties( Moilanenetal. 2006b ),becausefailingtoaccountfortheseuncertaintiesmayresultininadequateconservationandmanagementactions( Reganetal. 2005 ).Managingnetworksofprotectedareasisoneofthemostcommonbiodiversityprotectionstrategies( HellerandZavaleta 2009 ; KukkalaandMoilanen 2012 ; Mawdsleyetal. 2009 ).Often,theseareanetworksaredesignedunderasystematicconservationplanningframeworkthatemploysquantitativetoolsforstrategicallyselectingreserves.Thesequantitativetoolsidentifyareasfortherepresentationand/orpersistenceofbiodiversitygivensocioeconomicorbiologicalconstraints.Theseproblemscanbedescribedaslinearorintegeroptimizationprograms(thatmaximizeorminimizealinearfunctionoverasetoflinearconstraints)andbenetfromawidearrayofmodelsandalgorithmsthathavebeendevelopedintheoperationsresearcheld( CabezaandMoilanen 2001 ; KukkalaandMoilanen 2012 ; MargulesandPressey 2000 ; Moilanen 2008 ; Moilanenetal. 2009 ).Theendgoalofthesemodelsistodeterminetheoptimalsetofresourcepatchestoprotect,givenaconservationobjective.Awidearrayofobjectivefunctionshavebeenemployedintheseoptimizationmodels,suchasthemaximizationofspeciesrepresentation,themaximizationofspeciespersistenceortheminimizationofcosts( CabezaandMoilanen 2001 ).Recently,thefocusofthesemodels 47

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haveshiftedfromthesepattern-basedobjectivestoprocess-basedobjectivesthatdriveecosystemfunctioningandspeciespersistence( Possinghametal. 2005 ).Connectivityisanecologicalprocessthatpromotesspeciespersistenceinfragmentedlandscapes,themaintenanceofbiodiversity,andotherecosystemprocesses( CrooksandSanjayan 2006 ; HellerandZavaleta 2009 ; Lindenmayeretal. 2008 ; Possinghametal. 2005 ; Tayloretal. 1993 ; TischendorfandFahrig 2000 ).Improvingconnectivityisoneofthemostfrequentrecommendationswhenmanaginglandscapesforfutureenvironmentalchangesbecausehighlyconnectedsitesareexpectedtohavegreaterpersistenceandpopulationdensities( Gillsonetal. 2013 ; Hanski 1998 ; HellerandZavaleta 2009 ).Therationalebehindthisideaisthatincreasingconnectivitywillallowindividualstorespondtofuturechangesbymovingintonewregions( Krosbyetal. 2010 ).Eventhoughconnectivityiscommonlyconsideredinavarietyofmanagementactions,recentlyitsrelevanceforsystematicconservationplanninghasbeenacknowledged( FaganandLutscher 2006 ; Moilanenetal. 2007 2009 ; Wilsonetal. 2009 )byincorporatingapenalizingfactorintheobjectivefunctiontopromotetheselectionofsitesthatareclosertoeachother(e.g., Balletal. 2009 ; Begeretal. 2010 ).Decisionsbasedoninformationfromreserveselectiontoolsare,however,subjecttodifferenttypesofuncertainties( Pressey 2004 ).Theseuncertaintiescanbegenerallyclassiedintoepistemic(relatedtolackofknowledgeordata)orlinguistic(duetomisunderstandingofconceptsorterms)( Reganetal. 2002 ).Statisticalepistemicuncertaintiesincludethosearisingfrominsufcientdata.Non-statisticalepistemicuncertaintiesarisefromplausibleassumptionsmadeintheplanningprocesssuchasstaticenvironmentalorbiologicalconditions,suchasthoserelatedtostaticspeciesdistributionsorstaticspeciesinteractions( Moilanenetal. 2006a ).Yetfutureanthropogenicactions,climatechangeornaturaldisastersmaychangebothenvironmentalandbiologicalconditionsinuncertainways( Elithetal. 2002 ; Thomas 48

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etal. 2004 ).Uncertanityanalysisisrarelyincorporatedintosystematicconservationplanning.Someuncertaintyanalysismethodssuchassensitivityanalysis,scenariobuildingandinformationgapapproacheshavebeenproposedasfeasiblewaystoaddressstatisticaluncertainties( Moilanenetal. 2006b ; Reganetal. 2005 ),butwecurrentlylackaformalapproachtoaccountfornon-statisticalepistemicuncertainty.Herewedescribeanovelapplicationofnetworkowinterdictionmodels,thatdealswithnon-statisticalepistemicuncertaintybyoptimallydeterminingwhichpatchestoprotectbasedonaworst-casedisturbancescenarioforconnectivity.Wedescribethemodelingframeworkandapplyittotwoempiricalsystemstoshowitsapplication,assumptions,sensitivity,andpotentialmodications. 3.2Methods 3.2.1NetworkFlowInterdictionModelsEcologicalspatialnetworksconsistsofnodesthatrepresentspatiallocationssuchasforestpatches,grasslands,ponds,lakes,oranimalcoloniesandlinks(arcsfromnowon)thatdescribesomeowbetweennodepairs.Inecologicalspatialnetworks,arcstypicallydescribetheowofindividuals(proportionortheprobabilityofmovement)betweennodes.Arcsandnodesmaybevulnerabletodisturbancesthatimpairowinthenetwork.Thepurposeofnetworkowinterdictionmodelsistoassessvulnerabilitiesinnetworksandplanprotectivemeasures( Brownetal. 2006 ; Smith 2010 ).Thesemodelsanalyzetheworst-casescenariofornetworkinfrastructurebyassuminganintelligentadversarythathaslimitedresources(i.e.abudgetonhowmanyarcsornodescanbedisturbed).Theadversary'sobjectiveistomaximizedamagetothenetwork.Theendgoalofnetworkowinterdictionmodelsistodeterminethevalueofprotectingagivensetnodesorarcsthatwillbestmaintainnetworkowafterthisworst-casedisturbance.Eventhoughinterdictionmodelsarewidelyappliedinmilitaryandanti-terrorismscenarios,theycanbegenerallyappliedtoanykindofscenariowhenadecisionmaker 49

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isconcernedwithdesigningarobustsystemthatwithstandsanypossiblecombinationofdisruptions.Thisviewpointisoftentakenwhenanalyzingcriticalsystems(e.g.,powersystemsandnationaldefenseinfrastructure),whereresiliencetoworst-casedisruptions,accidentalorintentional,isvital.Inecologicalspatialnetworks,large-scaledisturbances,suchaslanduseornaturaldisasters,maydecreaseconnectivity.Thisdecreasemayhaveimportantecologicalconsequences.Forinstance,thelossofanimportantpatchinametapopulationmaycauselocalextinctionsoradecreaseinmetapopulationpersistence( Hanski 1998 ).Giventheuncertaintyreleatedtofuturedisturbancesinnaturalsystems,networkowinterdicitonmodelsareanappropriatemodelingtooltoassessthevulnerabilityofanecologicalspatialnetworktoapotentialworst-casescenario.Hereweappliedadefender-attackermodel(i.e.,Stackelberggame VonStackelberg 1952 ),whichinvolvestwoplayers:aleaderandafollower.Theleaderpicksasubsetofpatchestoprotect,whichbecomeimmunetodisruption.Thefollowerthenactstooptimallydisruptasubsetofunprotectedpatcheswiththegoalofminimizingconnectivity.Weemphasizethatthefollowerneednotbeamaliciousorsentientagent,butrathersimplyarepresentationofaworst-casescenariogiventheleadersprotectionstrategy. 3.2.2ModelDescriptionInthecontextofconnectivityconservation,acommongoalistoidentifyalandprotectionstrategygivenabudget,undertheassumptionthatconnectivitymaybereducedinthefuture.Disturbancesmayreduceconnectivitybylimitingtheabilityofindividualstomovewithinthelandscapeasaresultofdecreasedmatrixpermeabilityorpatchquality( KindlmannandBurel 2008 ).Hereweconsiderthecasewhereprotectionanddisturbanceoccursatthepatch(node)level;however,themodelingapproachcanbeeasilymodiedtoincorporatedisturbancestothematrix(arcs),seeSection 3.4.1 50

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Decreasingpatch-qualityreducesconnectivitybyincreasingpatchlevelmortalityanddecreasingmovement.Themodelingapproachconsistsofthreeinterrelatedstages:assessment,disturbance,andprotection(Figure 3-1 ).Intheassessmentstageweusepatchtransitionprobabilities(includingprobabilitiesofstayinginagivenpatch)andindividualsurvivalassociatedwitheachpatch(Figure 3-2 )toquantifyconnectivity.Thesetransitionprobabilitiesandsurvivalareoftenestimatedfrommark-recapturedatainamulti-statemodelingframework( NicholsandKendall 1995 ; Whiteetal. 2006 ).WedescribethesetransitionsasareducibleMarkovchain(seeSection 3.2.2.2 )toquantifytotalindividuallifeexpectancy.Thisconnectivitymeasuredescribestheamountoftimeanindividualremainsaliveandmovingthroughthenetwork.Theusefulnessofvariousconnectivitymeasuresformanagementandconservationhasbeendebatedintheliterature( CalabreseandFagan 2004 ; KindlmannandBurel 2008 ).Generally,measuresthatincorporaterealisticestimatesofmovementincombinationwithintegrativetnesscurrencies(e.g.,survival)arepreferredovermeasuresthatmakeplausibleassumptionsaboutmovement( Belisle 2005 ).Hence,quantifyingconnectivityastotalindividuallifeexpectancyisanappropriatemeasurebecauseitincorporatesprecisebetween-patchmovementprobabilityestimates(frommark-recapture),sitedelity,andpatch-specicsurvival.Aprotectionstrategyisattainedbyoptimizingdisturbanceandprotectionactions.First,aworst-casescenarioforconnectivityisassessedbydetermininganoptimaldisturbanceaction(asetofpatchestodisturb)thatwillminimizeconnectivity(totalindividuallifeexpectancy).Thesedisturbanceactionsmayincludefactorsthatdecreasepatchqualitysuchasdeforestation,re,hurricanes,oodingorchangesinabioticfactors.Second,wesimulateanoptimalprotectionaction(anoptimalsetofpatchestoprotect)thatwillmitigatetheworst-casescenariobymaximizingtheamountofconnectivitylossprevented(i.e.maxminbilevelproblem).Theseprotectionmeasures 51

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includeactionsthatpreventpatchdeteriorationsuchasprotectingagainstfuturedevelopment,reducehumanuse(e.g.,reduceshingorhunting),managingforreorcontrolpredatorpopulations.Inthiscontext,weareuncertainabouthowfuturedisturbanceswillaffectconnectivity,hence,weprotectagainsttheworst-possiblescenario(Figure 3-1 ). Figure3-1. Schematicdiagramofthethreestagesofthemodelingprocess:assessment,disturbanceandprotection.Theprotectionanddisturbancemodelsinteractrecursivelytoassesstheoptimalsetofpatchestoprotectgivenaworst-casescenarioforconnectivity. 3.2.2.1CasestudiesWeappliedthenetworkowinterdictionmodelingframeworktotwopublisheddatasets:adultRoseateTerns(Sternadougallii)livinginnortheasternUSA( Spendelowetal. 1995 )andSkipjackTuna(Euthynnuspelamis)livingthewesternPacic( Hilborn 1990 ).Thesedatasetsincludedsurvival,sitedelityandbetween-patchmovementestimates. 52

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RoseateTerncolonieswerelocatedinBirdIsland(BI)inMarionMassachusetts,GreatGullIslandinNewYork(GG),FalknerIslandinConnecticut(FI)andCedarBeach(CB)inLongIsland,NewYork(Figure 3-2 a).Individualadultswerecapturedyearlyfrom1988inthesesitesusingtreadletrapsandindividuallymarkedwithcolorbands.See Spendelowetal. ( 1995 )formoredetailsonpotentialbiasduetocolorbandlossandfading,andhowtheycontrolledforit.Weusedthearithmeticmeanfor1988oftheirsurvival,transitionandcolonysizeestimates(seebelow).SkipjackTunaweretaggedandrecovered(>150,000individualstagged),monthlyfrom1977,insevensheriessitesthatincludedwatersnearPalau(PAL),Yap(YAP),Truk(TRK),Ponape(PON),PapuaNewGuinea(PNG),SolomonIslands(SOL)andtheinternationalwatersbetweenYAP,TRK,PONandIndonesia(Figure 3-2 b).ThedataanalyzedwerecollectedaspartoftheSkipjackTunasurveyandassessmentprogramoftheSouthPacicCommission.Fishtaggedand/orrecoveredfromFijiwereexcludedfromtheiranalyses.Formoredetailsonthedataortheestimatessee Hilborn ( 1990 ).Bothdatasetsvariedinthenumberofpatches,scaleofmovement,survival,andlevelofphylopatry.SurvivalwasmorevariablefortheSkipjackTuna(0.664.99)thantheRoseateTerns(0.74).PhylopatrywasgenerallylowerintheSkipjackTuna(0.60.87)thanintheRoseateTerns(0.88.98).Bothspecieshadsimilarrangesofbetween-patchmovementrates(SkipjackTuna:0.11;RoseateTerns:0.09).SurvivalandtransitionprobabilitiesforbothspecieswereestimatedusingvariationsoftheArnason-Schwarzmulti-statemodel.Inthismulti-statemodelingframework,movementismodeledasarst-orderMarkovchain,whichmodelsthestochasticmovementpatternsofindividualsamongpatches,accountingforthepossibilitythatanindividualdiesateachtimestep.Thismulti-stateapproachallowsthedecompositionoftransitionprobabilitiesintosurvival(Si)andmovement(qij).Notethatinthiscasemortalityhappensbeforeanindividualmoves.See Spendelowetal. ( 1995 )for 53

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comparisonwithalternatemodelingapproachestoestimatesurvivalandtransitionprobabilities. Figure3-2. SpatialnetworkrepresentationofRoseateTerncoloniesinnortheastUSAandSkipjackTunaintheWesternPacic.Thecolorofthearcscorrespondtotheprobabilityofmovementfrompatchitoj.Notethatselfarcshadthelargestprobabilitiesshowinghighsitedelity. 3.2.2.2ConnectivityassessmentWecalculatefunctionalconnectivityasthetotalindividuallifeexpectancy,whichdescribesthenumberoftime-stepsanindividualremainsaliveinthenetwork.Toaddmorebiologicalrealitytothismeasure,weweighttotallifeexpectancybypatchabundancetoaccountforthenumberofindividualssubjecttoconditionsinpatchi.Ateachtime-stepanindividualhassomeprobabilityofmovingtoanotherpatch,stayinginthesamepatch,ordying.Theunitsofthismeasurearethesameastheinputdata.Forexample,iftheestimatedtransitionprobabilitiescorrespondtoaperiodoftwoweeks,thenthelifeexpectancydescribehowmanyperiodsoftwoweeksanindividualwillsurvive.WecalculateindividuallifeexpectancybymodelingtransitionprobabilitiesusingareducibleMarkovchain.Accordingly,weconsideralandscapecomposedofdiscretepatchesdenotedbyP=f1,...,jPjg.TheMarkovchaincontainsjPj+1states,wheretheadditionalstateisanabsorbingstatedthatcorrespondstomortality(Figure 3-2 ). 54

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TheMarkovchainstatesaregivenbyM=PSfdg.WeinitiallyrepresentthisMarkovchainasamatrixQ,comprisedofelementsqijfori2Pandj2P,andqdi=0,qid=1)]TJ /F5 11.955 Tf 12.42 0 Td[(Siforeachi2P,andqdd=1.EachcolumnofQrepresentsapatchinthenetwork,exceptforthelastcolumn,whichrepresentsmortality.Byassumptionwehavethatqij0foralli,j2MandPj2Mqij=1froalli2M.Intheabsenceofanydisturbances,thetotalindividuallifeexpectancy(z)canbecalculatedasfollows.LetpbeajPj-dimensionalcolumnvectorwhoseithelement,pi,describestheabundanceinpatchi2P,andletAbethesubmatrixofQobtainedbyremovingthelastrowandcolumnofQ( Ross 2006 ).SeeAppendix B foradditionalmathematicaldetailsoftheconnectivityassessment. z=pT(I)]TJ /F7 11.955 Tf 13.05 2.66 Td[(A))]TJ /F3 7.97 Tf 6.59 0 Td[(11, (3) whereIisthejPjjPjidentitiymatrixand1isthejPj-dimensionalcolumnvectorofallones.Intheunweightedcasep=1,andhencelifeexpectancyisbasedsolelyonthetransitionprobabilities. 3.2.2.3Disturbance:simulatingaworst-casescenarioThedisturbancemodelidentiesaworst-casescenarioforconnectivitybyselectingalimitednumberofunprotectedpatchestodisturbthatwillminimizetotallifeexpectancy.Herewedenedisturbanceasanyactionthatmaydecreasepatchquality.Todescribethedisturbancedecisionweusethebinaryvariableyi,whichequals1ifpatchi2Pisdisturbed,andequals0,otherwise.DisturbancetoanygivenpatchwillaffecttheindividuallifeexpectancybychangingthetransitionprobabilitiesintheMarkovchain.Wedeneqijastheupdatedtransitionprobabilityofgoingfromstatei2Mtostatej2M.Themagnitudeofthischangedependsonthedisturbanceregime,whichisdescribedbytheparametersij,ij,andij.Biologically,theseparametersdescribehowadisturbanceactioninapatch(decreasinghabitatquality)affectspatch-specic 55

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survivalandthebetween-patchmovementsassociatedwiththedisturbedpatch.Weassumethatdisturbanceactionswilldecreaseconnectivity;hence,ijandijarebothnonnegative.Forinstance,ifpatchjisdisturbed(yj=1),thentheprobabilityofmovingfrompatchitopatchj(qij)decreasesbyij.Similarly,ifpatchiisdisturbed(yi=1),qijdecreasesbyij.Theparameterijisusedasacorrectiontermwhenbothyi=1,andyj=1.Alsowesetii=ii=0,meaningthatifpatchiisdisturbedtheupdatedprobabilityofstayinginthesamepatch(qii)isaffectedonlybyii(seeAppendix C formoredetails). qij=8>>>>>><>>>>>>:qij)]TJ /F9 11.955 Tf 11.95 0 Td[(ijyj)]TJ /F9 11.955 Tf 11.95 0 Td[(ijyi+ijyiyjifi,j2Pqid+Xk2P(ikyk+ikyi)]TJ /F9 11.955 Tf 11.96 0 Td[(ikyiyk)ifi2Pandj=dqijifi=dandj2M (3) wherePdescribesthesetofallpatchesinthenetworkandMthesetofallstatesintheMarkovchain.Weestimateij,ij,andijasapercentageoftheinitialtransitionprobabilityqij.Tothisend,wedeneij=ijqij,ij=ijqij,andij=!ijqij,whereij,ij2[0,1]and!ij2[ij+ij)]TJ /F8 11.955 Tf 11.95 0 Td[(1,ij+ij],8i,j2P.Weselecttheoptimalsetofpatches(H)todisturbbymeansofthemixed-integerlinearprogramdescribedinAppendix C 3.2.2.4ProtectionTheendgoaloftheprotectionmodelistodeterminetheoptimalsubsetofpatchestoprotect.Wedenebinaryvariableswi,8i2P,whichequal1ifandonlyifpatchiisprotected.Thewvariablesaffectthey-variablesinthedisturbanceproblembecauseyi=0wheneverwi=1.Theprotectionmodelmaximizesthepotentiallossinthetotallifeexpectancythatcanbepreventedbyprotectingaparticularsetofpatches.Weusetheintegerlinear 56

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programdescribedinAppendix D todetermineanoptimalsetofpatchestoprotect,givenasetofpossibledisturbanceactions. 3.2.2.5GeneralmodelassumptionsWeassumethatindividualsfollowarandomwalkbetweenpatches.Thatis,anindividualattempttomovefromonepatchtoanotherisindependentofthesequenceofpatchespreviouslyvisited.Thisassumptioniscommontootherconnectivitymodelingapproaches( McRaeetal. 2008 ).Wealsoassumethatqid>0,foralli=1,...,jPj,whichmeansthatnomatterthepatchwheretheindividualiscurrentlyliving,thereisalwaysapositiveprobabilityofdeath.Weconsiderthesurvivalrates(Si)estimatedfrommark-recaptureastruesurvival;however,thesesurvivalestimatesmaybenegativelybiasedduetothedifcultyofdiscerningbetweenmortalityandpermanentemigration( Gilroyetal. 2012 ).Weconsiderthenetworktobeaclosedsystem,hence,theonlyavailablepatchestomoveareincludedinP.Weassumeindividualhomogeneityinthefactorsthataffectmovementandsurvival.Wealsoassumethatmovementdecisionsandmortalityaredensityindependent. 3.2.3ScenariosWeillustratethemodelingframeworkbybuildingdifferentprotectionscenariosforeachspecies.Wedevelopedscenariosinwhichtheprotectionbudget(u)wasroughlyhalfthelandscape(RoseateTern:u=2,b=2;SkipjackTuna:u=3,b=4).WeoptimizedboththeweightedandunweightedtotallifeexpectancyfortheRoseateTern,becausethedataincludedpatchabundances.WeoptimizedonlytheunweightedtotallifeexpectancyfortheSkipjackTuna,becausepatchabundanceswerenotincludedinthedataset.Wealsoshowhowtotalindividuallifeexpectancyvariesforallcombinationsofprotectionbudgetsandnumberofpatchesallowedtobedisturbed.Becausetherewasnodataavailabletoestimateijandij,wegeneratedij=ijqijandij=ijqijasdescribedinSection 3.2.2.3 .Parametersijandijwererandomlygeneratedonthecontinuousuniformdistributionswithbounds0and0.35, 57

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whichrepresentsdisturbanceactionsoflowtomoderatemagnitude.Furthermore,wegenerateparameters!ijonthecontinuousuniforminterval[ij+ij)]TJ /F8 11.955 Tf 12.02 0 Td[(1,ij+ij].AllthecodewaswritteninC++,usingCPLEX12.3optimizationsoftware. 3.2.4SensitivityAnalysisWeperformedtwotypesofsensitivityanalysis.First,weassesstherobustnessofthesolution(setofoptimalpatchestoprotect)givenchangesintheinputdata(Q).Second,westudyhowtheoptimalindividuallifeexpectancychangeswithincreasingprotectionbudget(amountofsitesallowedtobeprotected)ornumberofpatchesallowedtobedisturbed.Weobservedhighsitedelityinbothspecies(highprobabilityofindividualsstayinginthesamepatch).Wewantedtoknowhowmuchchangeintheseprobabilitieswilltaketochangetheoptimalsolution.Recallthat!iidescribestheproportionaldecreaseintheprobabilityofstayinginpatchigiventhatthepatchwasdisturbed.Wevary!ii(1+),where2[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,0],toassesshowmuchvariationisrequiredtochangetheoptimalsolution.FromSection 3.2.2.3 wehavethatqii=qii(1+!iiyi)8i2P,showingthatasourceofchangeinqiiis!ii.Wedeneastherateofchangein!ii,sothatqii=qii(1+(1+)!iiyi)8i2P.Sinceqii2[0,qii],weknowthat2[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,0].Inaddition,wehavethatqii=qii8i2T,soweonlyallowtoaffectqii8i=2T.Wedenez()astheoptimallifeexpectancyobtainedaftersolvingtheprotectionproblemusing(1+)!ii(insteadof!ii)8i2Tamongtheinputdata.Dene^z()asthelifeexpectancycalculatedbyassumingthatsetofpatchesTisprotected,asinputdata.Theratio()=z()=^z()measuresthechangeintheoptimaltotallifeexpectancyasafunctionof.If()=1,thenTisstilloptimalaftertheperturbationgivenby,whereasif()<1thenTprovidesasuboptimalsolutionwithagapfromtheoptimallifeexpectancygivenby1)]TJ /F9 11.955 Tf 11.96 0 Td[(().GiventhatabundancedatawasnotavailablefortheSkipjackTuna,forconsistencyweperformthisanalysisontheunweightedindividualtotallifeexpectancy. 58

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3.3Results 3.3.1RoseateTerns:Step-by-StepTotalindividuallifeexpectancy(unweighted),inthecasethatnopatchisdisturbedwasz=20.43.Ifweallowedatleasthalfofthenetworktobedisturbed(b=2patches),withaprotectionbudgetoftwopatches(u=2).Theworst-caseforconnectivityhappenswhenthesetofpatchesH=f2,4garedisturbed,andacorrespondingtotallifeexpectancyofz=15.59(Figure 3-3 b).ThecorrespondingoptimalsetofpatchestoprotectthatwillpreventthemostlossoftotallifeexpectancyisT=f1,4gwithanassociatedlifeexpectancyofz=15.97(Figure 3-3 j).WhentheweightedtotallifeexpectancyisoptimizedusingpatchabundancespT=[3773,2467,350,172]( Spendelowetal. 1995 ),thetotallifeexpectancywhennopatchisdisturbedwasz=32368.9.Theworst-casescenarioforconnectivityhappenswhenpatchesH=f1,2garedisturbedwithanassociatedlifeexpectancyofz=20545.7.ThecorrespondingoptimalsetofpatchestoprotectisT=f1,2gwithanassociatedlifeexpectancyofz=30638.9(Appendix E ). 3.3.2SkipjackTunaThetotalindividuallifeexpectancyinthecasethatnopatchisdisturbedwasz=46.07.Ifweallowedatmostfourpatchestobedisturbed(i.e.b=4),andhaveaprotectionbudgetofatmostthreepatches(i.e.u=3),theoptimalstrategyistoprotectthesetofpatchesT=f2,3,7g,withanassociatedoptimaltotalindividuallifeexpectancyofz=35.99. 3.3.3SensitivityAnalysisWefoundthat,inbothspecies,astheprotectionbudget(u)decreases,totallifeexpectancydecreasesandasthenumberofpatchesallowedtobedisturbedincreases,totallifeexpectancydecreasesasymptotically(Figure 3-4 ).Whenoptimizingtheunweightedtotalindividuallifeexpectancy,theoptimalsolutionfortheSkipjackTunawasmorerobustthantheoptimalsolutionforRoseateTerns(Figure 3-5 ).Theoptimal 59

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Figure3-3. Exampleoftheiterativeprocessbetweenthedisturbanceandprotectionmodelsforaprotectionbudgetofu=2andallowingb=2patchestobedisturbed.Thegureshowsthe(a)totalindividuallifeexpectancywithoutanyprotectionordisturbance,the(b)worst-casescenarioforconnectivityand(j)theoptimalprotectionagainstthisworst-casescenario. 60

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solutionfortheSkipjackTunaremainedoptimalfor2[)]TJ /F8 11.955 Tf 9.29 0 Td[(0.7,0],whichimpliesthatsitedelityestimatescandecreasedbyatmost70%andthesolutionwillremainoptimal.Onthecontrary,theoptimalsolutionfortheRoseateTernnetworkremainedoptimalfor2[)]TJ /F8 11.955 Tf 9.3 0 Td[(0.2,0],whichimpliesthatsidedelityestimatescandecreaseatmost20%forthesolutiontoremainoptimal.Whenconsideringtheweightedtotalindividuallifeexpectancy,thesolutionremainedoptimalregardlessoftheamountofvariation(Figure 3-5 ). Figure3-4. Sensitivityoftheoptimallifeexpectancyintheprotectionstagegivenchanges()inprobabilityofstayinginthesamepatch(qii). 3.4DiscussionSystematicconservationplanningtoolsprovideguidanceforthepropermanagementofprotectedareanetworks.Uncertaintyispresentinmanystagesofthisplanningprocess,andoftentimesnotacknowledged,whichmayresultininadequatemanagementactions.Recently,methodshavebeendevelopedthattesttherobustnessofsiteselectionmodelsgivenuncertaintyintheinputdata( Moilanenetal. 2006a b );however,evenwhenuncertaintyinthedataisaccountedfor,thereisstilluncertaintyaboutfuturechangesinthesystemduetohumandisturbances,ornaturaldisasters.Herewepresentedanovelapplicationofnetworkinterdictionmodelsthatdealswiththese 61

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Figure3-5. Optimallifeexpectancy(z)givenallpossibleprotectionbudgets(u)andnumberofpatchesallowedtobedisturbed(b).Lifeexpectancydecreaseswithdecreasingprotectionbudgetandwithincreasingnumberofpatchesallowedtobedisturbed unaccounteduncertaintiesbyproposingaprotectionplangivenaworst-casescenarioforconnectivity.Wegroupthedatarequirementstoparameterizedthemodelintotwo:movementdataandscenariobuildingvariables.Movementdataincludestransitionestimates( ),patchsurvival(S)andpatchabundances.Italsoincludestheestimatesonhowdisturbanceinapatchwillaffectthetransitionprobabilities(,and).Scenariobuildingvariablesincludethemaximumnumberofpatchesallowedtobedisturbed(b) 62

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andtheprotectionbudget(themaximumnumberofpatchestoprotectu).Thedatarequirementstoparameterizetheinterdictionmodelmaybeanimportantlimitationofthisapproach.Eventhoughbetween-patchtransitionandsurvivalestimatesareoneofthebestdataavailabletoquantifylandscapeconnectivity( Belisle 2005 ; KindlmannandBurel 2008 ),thesedatasetsarescarceintheliteraturebecauseitrequiresmucheldeffortandtheyarecomplextotinamulti-statemark-recapturecontext( CalabreseandFagan 2004 ).Nevertheless,thesedatasetsarebecomingmorecommoninconservationefforts.Parameterestimatesthatdescribehowmovementchangeswhenpatchesaredisturbedcanbetakenfromtheliteraturewhenavailable(e.g., Bechetetal. 2003 ; Bowneetal. 2006 ).Iftheseestimatesarenotreadilyavailableforaparticularspecies,themodelcanbeparameterizedassumingdifferentscenariosfortheseparameters.Thesensitivityoftheoptimalsolutiontovariationsinthedatawilldependonthestructureofthenetwork.WefoundthemodeltobeveryrobusttochangesinsitedelityestimatesfortheSkipjackTuna,butlessrobustfortheRoseateTerns.TheRoseateTernnetworkconsistedofonly4nodeswithlittlevariationinsitedelityandmortalityprobabilities(Figure 3-2 ),whichmeansthatevensmallchangesinsomeofthemovementestimatesmaysignicantlychangetheoptimalsolution.Incontrast,therobustnessofthesolutionfortheSkipjackTunanetworkmaybeduetoincreasevariationinsitedelityandmortalityprobabilities.TheoptimalprotectionforRoseateTernnetwork,whenconsideringtheweightedtotalindividuallifeexpectancy,wasveryrobust.Thisisbecausetheabundancesintheoptimalpatchestobeprotectedwasanorderofmagnitudegreaterthantheotherpatches.Scenariobuildingvariablesareusefultounderstandvariousscenariosforconnectivity,dependingonhowmanypatchesareexpectedtobedisturbedandtheprotectionbudget.Generally,lifeexpectancyisexpectedtodecreaseswithincreasingnumberofpatchesallowedtobedisturbedandincreasewithincreasingnumberof 63

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patchesallowedtobeprotected(Figure 3-5 ).Theshapeoftheserelationshipswillbespecicforeachapplicationandwillbeausefultoolformanagersmakingprioritizingdecisions.Insomeinstances,increasingtheprotectionbudgetbyonepatchmayresultinasignicantincreaseinlifeexpectancy,suggestingthatincreasingtheprotectionbudgetmaybeuseful.Inotherinstances,decreasingtheprotectionbudgetmaynotnecessarilyresultinasignicantlifeexpectancydecreasesuggestingthatareducedprotectionbudgetmaybeappropriate. 3.4.1ModicationsThenetworkinterdictionmodelpresentedhereisaexibleframeworkthatcanbeappliedtoawidearrayofinstancesinconservationandmanagement.Thisapproachcanbeappliedtoorganisms,bothactiveandpassivedispersers,forwhichbetween-patchmovementandsurvivaldataareavailable.Eventhoughwedescribeamodelinwhichthegoalistoassesvulnerabilitiestopatchdisturbances,themodelcanbemodiedtoassessvulnerabilitiesinthematrix.Inthiscasethedisturbanceandprotectionmodelswilloptimizelifeexpectancygiventheprotectionordisturbanceofarcsinthenetworkinsteadofnodes.Thismodicationwillallowtheidenticationofplacesinthematrixtoperformrestorationactivitiesorwheretobuildcorridorstoincreaseconnectivity.Thecostofacquiringorprotectingapatchiscommonlyincorporatedinmostprioritizationmodels( CabezaandMoilanen 2001 ; KukkalaandMoilanen 2012 ; MargulesandPressey 2000 ; Moilanen 2008 ; Sarkaretal. 2006 ).Inthismodelingapproachtheprotectionmodelisgivenabudgetormaximumnumberofpatchestoprotect.Thisinherentlyassumesthatthecostofprotectingeachpatchwillbethesame.Variationsincostcanbeincorporatedasconstraintsinbothdisturbanceandprotectionmodels.Prioritizingdisturbanceactionstominimizeconnectivityforinvasivespeciesisapotentialalternateapplicationofnetworkinterdictionmodels.Inthiscasetheproblem 64

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maybesimplerbecauseitonlyrequiresthedisturbancemodel,inwhichweidentifytheworst-casescenarioforconnectivity.Hence,theoptimalsetofpatchesthatminimizesconnectivitywillbethebestpatchestodisturbedtocontroltheinvasivespecies. 3.4.2ConservationPlanningforaWorst-caseScenarioforConnectivityThefewprioritizationmodelsthataccountforconnectivityfollowthearea/isolationparadigmfrommetapopulationtheorythatstatesthattheprobabilityofrecolonizationincreaseswithdecreasingisolation( Hanski 1998 )andattempttoincreasestructuralconnectivitybyeitherminimizingtheperimeterlengthofthenetworkorthedistancebetweenpatches( Balletal. 2009 ; Begeretal. 2010 ; Nalleetal. 2002 ; Williamsetal. 2004 ).Here,whenprioritizingareasfortheRoseateTerns,patches1and4wereselectedastheoptimalpatchestobeprotected,evenwhenthedistancebetweenthesetwopatchesisthelongestwhencomparedtootherbetween-patchdistancesinthisnetwork(Figure 3-2 a).Similarly,theoptimalsolutionfortheSkipjackTunaincludesthenodethatrepresentsinternationalwaters,whichisthemostisolatedpatchintermsofdistance(Figure 3-2 b).Thispatchwaspartoftheoptimalprotectionsolutionbecauseithadhighsitedelityandverylowmortality.Theseresultssuggeststhatminimizingbetween-patchdistancemaynotbealwaysanappropriatecriteriaforprioritization.Thepracticalityofincreasingstructuralconnectivitybetweenpatcheshasbeendebated.Decreasingdistancebetweenpatchesmaydecreasetheprobabilityoflocalextinctionandreducepotentialinbreeding,butitmayincreasethespreadofdiseasesorthepersistenceofexoticspecies( Williamsetal. 2005 ).Hence,ithasbeenarguedthatincreasingstructuralconnectivity(i.e.distancesbetweensiteswithoutinformationonhoworganismsactuallymoveamongsites)maynotbeanoptimalmanagementstrategy,whenprioritizingareasforanuncertainfuture( Hodgsonetal. 2011 2009 ).Measuresofstructuralconnectivitydonotincorporateinformationaboutthebehaviorofthespecies,orhowmovingthroughlandscapeaffectsindividualtness.Here,theoptimalprotectionisassessedoptimizingaconnectivity 65

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measurebasedonprecisemovementdata(i.e.movementprobabilitiesarequantiedthroughrigorousmark-recapturemethods)andpatch-specicsurvivalestimates,whichamelioratessignicantlysomeoftheconcernsthathavebeenbroughtaboutconnectivityconservationforfuturechanges( Doerretal. 2011 ).Thecurrentparadigmofmetapopulationecologyalsodeterminespatchimportancebyconsideringpatchesindividually( Hanski 1998 ).Highlyconnectedpatchesareconsideredmoreimportanttomaintainecologicalprocessinaspatialnetwork( Bunnetal. 2000 ; MinorandUrban 2008 ).Moreover,theresilienceofanetworkisoftenmeasuredbyquantifyinghowmanyindividualnodescanberemovedwithoutalteringnetworkconnectivity( BodinandSaura 2010 ; Bunnetal. 2000 ; Pascual-HortalandSaura 2006 ; UrbanandKeitt 2001 ; Watsonetal. 2011b ).Herewepresentedanalternativemethodbyconsideringtheimportanceofaparticularsubsetofnodesinsteadnodesindividually.Consideringacombinationofpatchesinsteadofthesumcontributionofindividualpatches,oftenresultsinnon-intuitivesolutions.Thisisacommonresultintheoperationsresearchliterature( Brownetal. 2006 ).Solutionsfromthenetworkowinterdictionmodels,likeanyotheroptimizationmodel,aresimplyatoolfordecisionsupport.Itspurposeistoguidethemanagementprocessanditssolutionsshouldnotbenecessarilyconsideredinisolation( Williamsetal. 2004 ).Instead,itshouldbeconsideredinconjunctionwithothermodelswithdifferentobjectivesaspartoftheconservationplanningprocess. PresseyandBottrill ( 2009 )summarizedtheconservationplanningprocessin11stages.Thesestages,includetheidenticationof(1)thescope,(2)thestakeholders,(3)thespatialextentandthe(4)goalsoftheconservationprogram,(5)thecollectionofsocio-economicvariablesandthreats,(6)collectdataonbiodiversity,(7)setconservationtargets,(8)evaluationoftheexistingareanetwork,(9)identicationofimportantareas,(10)applyingtheconservationactionsand(11)monitoring.Themodelingframeworkpresentedhereencompassesstages6to9ofthisprocess.Thedatacollection(stage6) 66

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requiresbetween-patchtransitionprobabilitiesandpatchspecicsurvivals.Thenetworkinterdictionframeworkalsoallowstosetthespecicconservationobjectives(stage7)intermsofthenumberofpatchestobeprotected(andthenumberofpatchestobeallowedtobedisturbed).Theconnectivityassessmentstage(Section 3.2.2.2 ;stage8)whichcalculatestheexpectedlifeexpectancygiventhecurrentconditions.Theresultofthemodelisthesetofpatchesthatwillminimizefuturedecreasestothelifeexpectancyofindividualsgiventhescenariosbuiltininthemodel(stage9). 3.4.3UncertaintyandWorst-caseScenarioPlanningUncertaintymaybepresentinmultiplestagesoftheconservationplanningprocess,butrarelyacknowledged.Informationgapdecisiontheoryhasbeenproposedasanefcientframeworktoincorporatestatisticaluncertaintiesinconservationplanning( Moilanenetal. 2006a b ).Theaimofinformationgapdecisionmodelsistoidentifysolutionsthatarerobusttothepresenceofuncertainty.Evenwhenstatisticaluncertaintiesareincorporatedintheplanningprocessbytheuseofinformationgap,orothersensitivityanalysis,othernon-statisticaluncertaintiesmaystillaffectthesysteminthefutureandresultininadequatemanagementactions.Thenetworkowinterdictionmodelacknowledgesthatthereareadditionaluncertaintieswhicharedifculttoquantify,butusestheavailabledatatoprotectagainsttheworstpossibledisturbance.Hence,evenwhenclimatechangeorregional-scalechangesinenvironmentalconditionsmayaffectthenetworkinanunpredictablemanner,thesystemhasalreadybeenprotectedagainsttheworstpossiblescenario. 67

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APPENDIXAMODELSUMMARIESANDADDITIONALFIGURES TableA-1. Parameterestimatesandstandarderrors(SE)forthemulti-seasonoccupancymodelthatresultedinthebestt; (.),(SasymAPh),(SasymAPh),P(APh). ColonizationExtinctionDetectionEstimateSEEstimateSEEstimateSE INTERCEPT-4.8970.198-2.7240.2852.1420.113Sasym0.3300.1740.5650.200A0.0080.001-0.0160.0050.0040.001Ph(TREES)0.8000.3830.5750.415-0.4100.184Sasym:A0.0020.001-0.0160.005Sasym:Ph(TREES)0.4100.255-0.4140.394A:Ph(TREES)-0.0030.006-0.0050.0100.0010.002Sasym:A:Ph(TREES)-0.0150.0050.0040.012 68

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TableA-2. Parameterestimatesandstandarderrors(SE)forthemodelthatincludedasymmetricdispersalkernel(Ssym)asasitecovariate; (.),(SsymAPh),(SsymAPh),P(APh). ColonizationExtinctionDetectionEstimateSEEstimateSEEstimateSE INTERCEPT-4.8670.195-2.600.2672.1340.112Ssym0.2310.1920.5580.219A0.0070.001-0.0210.0060.0040.001Ph(TREES)0.8000.4090.4000.400-0.4130.184Ssym:A0.0010.001-0.0180.005Ssym:Ph(TREES)0.2890.284-0.6670.380A:Ph(TREES)-0.0020.007-0.0010.0100.0010.002Ssym:A:Ph(TREES)-0.0080.0050.0200.009 FigureA-1. Spatialrepresentationofpatches(bouldersandtreetrunks)sampledforLepanthesrupestrisoccurrenceintheLuquilloExperimentalForest. 69

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FigureA-2. Theoreticaldistributionsoftheparameterij.Arandomspatialarrangement(a)resultsinarelativelyuniformdistribution(b).Ifpatchesarealignedalmostperfectlyinthesameangleofwinddirection(c),itresultsinabimodaldistributionhighlyskewedtothelimitsofthedistribution(d). 70

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FigureA-3. Predictedpartialrelationshipsbetweencolonization,extinctionandpatchareaforbothphorophytesforthebest-supportedmodel 71

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FigureA-4. Partialrelationshipbetweenpatchareaanddetectabilityfor(a)rockand(b)treephorophyteforthebest-supportedmodel 72

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APPENDIXBCONNECTIVITYASSESSMENT:MATHEMATICALDETAILSWecalculatetotalindividuallifeexpectancy(gi)asthenumberoftransitionsthatanindividuallivinginpatchisurvivesbeforedying(goingtostated)( Ross 2006 ).Thiscalculationisderivedbasedonthefollowinglinearequation.Theexpectedlifetime,gi,startingfrompatchi2P,isgivenbyoneplus zeroiftheindividualdiesatthenextperiod(withprobability1)]TJ /F5 11.955 Tf 11.96 0 Td[(Si) giiftheindividualsuccessfullyreachespatchj2P(withprobabilityqij).Therelationshipcanbewrittenas gi=1+Xj2Pqijgij,(B)orinmatrixfrom: Ig=Ag+1.(B)If(I)]TJ /F7 11.955 Tf 13.05 2.66 Td[(A)isinvertible(whichweassumetobethecase,byaaslightperturbationofAifnecessary),then g=(I)]TJ /F7 11.955 Tf 13.06 2.66 Td[(A))]TJ /F3 7.97 Tf 6.59 0 Td[(11. (B) Theweightedtotallifeexpectancy,z,iscalculatedas: z=pT(I)]TJ /F7 11.955 Tf 13.05 2.66 Td[(A))]TJ /F3 7.97 Tf 6.58 0 Td[(11=pTg. (B) 73

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APPENDIXCDISTURBANCEOPTIMIZATIONThedisturbanceoptimizationmodelincludestheinitialtransitionprobabilitiesgivenbyqij(8i,j2M)andtheestimatedimpactsonthetransitionprobabilitiescausedbytheinterdictor'sdisturbances,whicharegivenbyij,ij,andij(8i,j2M).Aformalstatementthatdescribesthecalculationofqij,8i,j2MisgivenbyEquation 3 .Toillustratetheuseofij,considerthecaseinwhichdisturbingpatchesiandjtogetherreducesqijbylessthanthesumoftheindividualdisturbancesij+ij.Inthiscase,ijisapositivecorrectionterm.Ontheotherhand,ifthereisasynergisticeffectattributabletojointlydisturbingpatchesiandj,suchthatdisturbingiandjtogetherreducesqijbymorethantheindividualdisturbances,thenijisasanegativecorrectionterm.Weassumethatij2[ij+ij)]TJ /F8 11.955 Tf 12.86 0 Td[(qij,ij+ij],sothatincaseofdisturbingbothpatchesiandj,wehaveqij2[0,qij].Inthecasethatonlyonepatchisdisturbed,werequiretheconditionqij)]TJ /F8 11.955 Tf 12.2 0 Td[(maxfij,ijg0sothatqij2[0,qij].Also,wesetii=ii=0withoutlossofgenerality,meaningthatifpatchiisdisturbed,thenqiiisupdatedonlyaccordingtoii2[)]TJ /F8 11.955 Tf 9.54 0 Td[(qii,0].Wedenetheparameterb2f1,,jPj)]TJ /F8 11.955 Tf 31.12 0 Td[(1gasthemaximumnumberofpatchesthatcanbedisturbed.Ourmodelalsousestheparametergi(8i2P),givenbyEquation B .Weknowthatgiisanupperboundonthenumberoftransitionsspentintransientstates,becausethetransitionprobabilitiesbetweentransientstatesareeitherreducedwhensomepatchesaredisturbedorremainunchangedwhennopatchisdisturbed.Themodelidentiesthosepatchesthatshouldbedisturbedtominimizetheindividualtotallifeexpectancy.Thedisturbanceoptimizationmodelusesvariablesyi,8i2P,andupdatedtransitionprobabilityvariablesqij,8i,j2PasdenedbytherstcaseofEquation 3 .Thedisturbanceproblemobjectiveisgivenby minz=pTg, (C) 74

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andgisagaincalculatedas g=(I)]TJ /F7 11.955 Tf 11.96 0 Td[(A))]TJ /F3 7.97 Tf 6.59 0 Td[(11, (C) whereAisobtainedbydeletingthelastrowandcolumnfromQ.BysubstitutingqijusingEquation 3 ,andwritingEquation C infunctionalform,wecanrewriteEquation C as gi)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj2Pqijgj+Xj2Pijyjgj+Xj2Pijyigj)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj2Pijyiyjgj=1,8i2P. (C) NotethateachequationinEquation C isnonlinearintheyandgvariables.TolinearizetheseequationsinEquation C ,weintroducetheadditionalvariableswij(=yigj)andxij(=yiyjgj).Theresultingsystemisgivenby gi)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj2Pqijgj+Xj2Pijwjj+Xj2Pijwij)]TJ /F12 11.955 Tf 11.95 11.36 Td[(Xj2Pijxij=1,8i2P. (C) ThesystemshowninEquation C istherstsetofconstraintsinourmodel.Thissetdescribesnotonlythewaytoupdatethetransitionprobabilitiesgiventheinterdictor'sactions(qvariables),butalsothecalculationoftheexpectedtimespentintransientstates(gvariables).Toguaranteethatwij=yigj,weintroducethesetoflinearconstraintsshowninEquations C C .Notethatifyi=1,thenwij=gij.Ifyi=0,thenwij=0,asdesired. wij0,8i,j2P (C) wijgj,8i,j2P (C) wijgj)]TJ /F8 11.955 Tf 12.2 0 Td[(gj(1)]TJ /F5 11.955 Tf 11.96 0 Td[(yi),8i,j2P (C) wijgjyi,8i,j2P (C) Similarly,wedenethesetofconstraintsshowninEquations C C toguaranteethatxij=yiyjgj.Sincethetermyiyjgjisalsononlinear,weintroducethenew 75

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variablevijandthecorrespondingsetofconstraintsshowninEquations C C toguaranteethatvij=yiyj.Notethatifyi=0,thenvij=0andxij=0.Moreover,ifyi=yj=1,thenvij=1andxij=gj,asdesired. xij0,8i,j2P (C) xijgj,8i,j2P (C) xijgj)]TJ /F8 11.955 Tf 12.19 0 Td[(gj(1)]TJ /F5 11.955 Tf 11.96 0 Td[(vij),8i,j2P (C) xijgjvij,8i,j2P (C) vij0,8i,j2P (C) vijyi,8i,j2P (C) vijyj,8i,j2P (C) vijyi+yj)]TJ /F8 11.955 Tf 11.96 0 Td[(1,8i,j2P. (C) ThemaximumnumberofpatchesthatcanbedisturbedisconsideredinEquation C ,whiletheconstraintshowninEquation C denesthebinarynatureofthey-variables. Xi2Pyib (C) yi2f0,1g,8i2P (C) Themixed-integerlinearproblemconsistsinminimizingtheobjectivefunctionshowninEquation C subjecttoconstraintsinEquations C C 76

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APPENDIXDPROTECTIONOPTIMIZATIONTheprotectionoptimizationmodelincludestheparameteru2f1,,jPj)]TJ /F8 11.955 Tf 31.43 0 Td[(1gtodenotetheprotectionbudget(themaximumnumberofpatchesthatcanbeprotected).Weletthebinarydecisionvariablewiequal1ifandonlyifpatchi2Pisprotected.Firstobservethatallfeasibleprotectionsolutionsobeytheconstraints: Xi2Pwiu (D) wi2f0,1g,8i2P. (D) Equation D enforcesthemaximumnumberofpatchesthatcanbeprotected,whileconstraintsinEquation D denethebinarynatureofthedecisionvariables.Theobjectivefortheprotectionproblemislessstraightforwardtomodel.Conceptually,letHbethesetofallcardinalitybsubsetsofP,i.e.,thesetofallpossibledisturbancesolutions.ForeveryH2H,letzHbetheindividuallifetimegivendisturbancesetH.WecandenebinaryvariablesH,8H2H,whichequal1ifandonlyiftheprotectionactiondisablesdisturbancesetH(i.e.,ifthereexistsani2Psuchthati2Handwi=1).Thenwecanstateconstraints: Xi2HwiH,8H2H.(D)Theprotectionproblemobjectiveisdenotedby maxminH2H:H=0zH.(D)However,suchaproblemisnonlinearandrequiresexponentiallymanyconstraintsandvariables.AsanalternativeconsiderthefeasibilityproblemgivenbyF(H):ConstraintsinEquations D and D 77

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Xi2Hwi18H2H,(D)whereHHisasubsetofthesetH.NotethatF(H)isfeasibleifandonlyifallpossibledisturbancesinthesetHcanbedisabledbyaprotectionaction.Therefore,considerthefollowingalgorithm,inwhichanupperboundvalueisinitiallysettoUB=z(computedbyEquation B ),Hisinitiallyempty,andwhichtheprotectionsolutionisgivenbyw=0. Step1.SolveF(H),andiffeasiblesolution^wexists,gotothesecondstep;otherwise,gothethirdstep. Step2.Solvetheoptimaldisturbanceproblemaugmentedwiththeconstraintsthatyi^wi,8i2P,whichenforcetheconditionthatnoprotectedpatchisdisturbed.Let^Hbeanoptimalsolutiontothedisturbanceproblem,given^w,notingthatconstraintsyi1)]TJ /F8 11.955 Tf 14.57 0 Td[(^wi,8i2P,areaddedtothedisturbanceproblem.Add^HtoH.IfUB>z^H,thensetUB=z^Handw=^w.Thisstepreturnstostep1. Step3.TerminatewithanoptimalsolutiongivenbywhavinganobjectiveUB.Twonotesarenecessaryregardingthisalgorithm.Therstregardstheconvergenceandcorrectnessofthealgorithm.EachtimeF(H)issolved,onemoresubsetisaddedtoH.AsecondissueisthatsolvingF(H)inearlyiterationsofthealgorithmyieldsprotectionstrategiesthatareblindtomanypossibledisturbancesolutionsthatarenotyetinH,butarelikelytobeaddedtoHinfutureiterations.Accordingly,wecanapplyalower-boundaryobjectivefunctiontoF(H).Toestimatethelowestpossiblelifeexpectancywedene~gasavectorwhoseentries~girepresentthemeannumberoftransitionsspentinthetransientstates,giventhattheMarkovchainstartsatstatei.Toguaranteethat~gisalower-boundonlifeexpectancy,weassumethatthetransitionprobabilitiesbetweentransientstatesaretheminimumpossible,whichisequivalenttoassumethatthetransitionprobabilitiesfromanytransientstatetothedeathstatearemaximum.Tothisendwedenethetransition 78

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matrix~Q,whoseentries~qijaregivenby ~qij=(1)]TJ /F8 11.955 Tf 11.96 0 Td[(maxfij,ij,ij+ij)]TJ /F9 11.955 Tf 11.95 0 Td[(ijg)qij. (D) ByourconstructionofthetransitionprobabilitiesinEquation D ,nodisturbancecanproduceatotallifeexpectancyoflessthanpT~g,andsopT~gisavalidlowerboundontotallifeexpectancy.TheobjectivefunctionforF(H)inthiscasecanbewrittenas f(w)=Xi2Ppi(gi)]TJ /F8 11.955 Tf 12.2 0 Td[(~gi)wi, (D) wherepi(gi)]TJ /F8 11.955 Tf 12.47 0 Td[(~gi)isanestimateofthepotentiallossinthetotallifeexpectancythatcanbepreventedifpatchiisprotected. 79

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APPENDIXEMODELITERATIONSFORWEIGHTEDLIFEEXPECTANCYOPTIMIZATION TableE-1. ModeliterationsshowingtheinteractionbetweendisturbanceandprotectionstageswhenpT=[3773,2467,350,172] IterationStageOptimalsolutionzz 1ProtectionwT=[0000]20545.7)]TJ /F8 11.955 Tf 9.29 0 Td[(10093.2DisturbanceyT=[1100]2ProtectionwT=[1100]30638.90DisturbanceyT=[0011]3ProtectionwT=[1010]27084.8)]TJ /F8 11.955 Tf 9.3 0 Td[(3554.1DisturbanceyT=[0101]4ProtectionwT=[1001]26465.4)]TJ /F8 11.955 Tf 9.3 0 Td[(4173.5DisturbanceyT=[0110]5ProtectionwT=[0110]24499.1)]TJ /F8 11.955 Tf 9.3 0 Td[(6139.8DisturbanceyT=[1001]6ProtectionwT=[0101]23846.9)]TJ /F8 11.955 Tf 9.29 0 Td[(6792DisturbanceyT=[1010] 80

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REFERENCES Ackerman,J.,1995.AnorchidoraofPuertoRicoandtheVirginIslands.MemoirsoftheNewYorkBotanicalGarden73:1. Ackerman,J.,A.Sabat,andJ.Zimmerman,1996.Seedlingestablishmentinanepiphyticorchid:anexperimentalstudyofseedlimitation.Oecologia106:192. Alonso,J.C.,C.A.Martin,J.A.Alonso,C.Palacin,M.Magana,andS.J.Lane,2004.DistributiondynamicsofaGreatBustardmetapopulationthroughoutadecade:inuenceofconspecicattractionandrecruitment.BiodiversityandConservation13:1659. Armsworth,P.andL.Bode,1999.Theconsequencesofnon-passiveadvectionanddirectedmotionforpopulationdynamics.ProceedingsoftheRoyalSocietyofLondon.SeriesA:Mathematical,PhysicalandEngineeringSciences455:4045. Armsworth,P.andJ.Roughgarden,2005.Theimpactofdirectedversusrandommovementsonpopulationdynamicsandbiodiversitypatterns.TheAmericanNaturalist165:449. Ball,I.,H.Possingham,andM.Watts,2009.Marxanandrelatives:softwareforspatialconservationprioritisation.InA.Moilanen,K.A.Wilson,andH.P.Possingham,editors,SpatialConservationPrioritization:QuantitativeMethodsandComputationalTools,pages185,Oxford,UnitedKingdom.OxfordUniversityPress. Bechet,A.,J.Giroux,G.Gauthier,J.Nichols,andJ.Hines,2003.SpringhuntingchangestheregionalmovementsofmigratingGreaterSnowGeese.JournalofAppliedEcology40:553. Beger,M.,S.Linke,M.Watts,E.Game,E.Treml,I.Ball,andH.Possingham,2010.Incorporatingasymmetricconnectivityintospatialdecisionmakingforconservation.ConservationLetters3:359. Belisle,M.,2005.Measuringlandscapeconnectivity:thechallengeofbehaviorallandscapeecology.Ecology86:1988. Bender,D.andL.Fahrig,2005.Matrixstructureobscurestherelationshipbetweeninterpatchmovementandpatchsizeandisolation.Ecology86:1023. Bode,M.,K.Burrage,andH.Possingham,2008.Usingcomplexnetworkmetricstopredictthepersistenceofmetapopulationswithasymmetricconnectivitypatterns.EcologicalModelling214:201. Bodin,O.andS.Saura,2010.Rankingindividualhabitatpatchesasconnectivityproviders:integratingnetworkanalysisandpatchremovalexperiments.EcologicalModelling221:2393. 81

PAGE 82

Bowler,D.andT.Benton,2005.Causesandconsequencesofanimaldispersalstrategies:relatingindividualbehaviourtospatialdynamics.BiologicalReviews80:205. Bowman,J.,N.Cappuccino,andL.Fahrig,2002.Patchsizeandpopulationdensity:theeffectofimmigrationbehavior.ConservationEcology6:9. Bowne,D.,M.Bowers,andJ.Hines,2006.Connectivityinanagriculturallandscapeasreectedbyinterpondmovementsofafreshwaterturtle.ConservationBiology20:780. Brown,G.,M.Carlyle,J.Salmeron,andK.Wood,2006.Defendingcriticalinfrastructure.Interfaces36:530. Brown,J.andA.Kodric-Brown,1977.Turnoverratesininsularbiogeography:effectofimmigrationonextinction.Ecology58:445. Bunn,A.,D.Urban,andT.Keitt,2000.Landscapeconnectivity:aconservationapplicationofgraphtheory.JournalofEnvironmentalManagement59:265. Burnham,K.andD.Anderson,2002.Modelselectionandmulti-modelinference:apracticalinformation-theoreticapproach.Springer,NewYork,USA. Cabeza,M.andA.Moilanen,2001.Designofreservenetworksandthepersistenceofbiodiversity.TrendsinEcology&Evolution16:242. Calabrese,J.M.andW.F.Fagan,2004.Acomparison-shopper'sguidetoconnectivitymetrics.FrontiersinEcologyandtheEnvironment2:529. Castellon,T.D.andK.E.Sieving,2006.Anexperimentaltestofmatrixpermeabilityandcorridorusebyanendemicunderstorybird.ConservationBiology20:135. Chardon,J.,F.Adriaensen,andE.Matthysen,2003.Incorporatinglandscapeelementsintoaconnectivitymeasure:acasestudyfortheSpeckledWoodButtery(ParargeaegeriaL.).LandscapeEcology18:561. Colbert,J.,E.Danchin,A.Dhondt,andJ.Nichols,2001.Dispersal.OxfordUniversityPress,NewYork,USA. Compton,S.,2002.Sailingwiththewind:dispersalbysmallyinginsects.InJ.Bullock,R.E.Kenward,andR.Hails,editors,DispersalEcology,pages113,Oxford,UK.BlackwellScience. Crooks,K.andM.Sanjayan,2006.Connectivityconservation,volume14.CambridgeUniversityPress,NY,USA. DeVol,J.andR.Goeden,1973.BiologyofChelinideavittigerwithnotesonitshost-plantrelationshipsandvalueinbiologicalweedcontrol.EnvironmentalEntomol-ogy2:231. 82

PAGE 83

Dias,P.,1996.Sourcesandsinksinpopulationbiology.TrendsinEcology&Evolution11:326. Diffendorfer,J.,M.Gaines,andR.Holt,1995.Habitatfragmentationandmovementsofthreesmallmammals(Sigmodon,Microtus,andPeromyscus).Ecology76:827. Doerr,V.,T.Barrett,andE.Doerr,2011.Connectivity,dispersalbehaviourandconservationunderclimatechange:aresponsetoHodgsonetal.JournalofAp-pliedEcology48:143. Dressler,R.,1993.Phylogenyandclassicationoftheorchidfamily.CambridgeUniversityPress,NewYork,USA. Elith,J.,M.A.Burgman,andH.M.Regan,2002.Mappingepistemicuncertaintiesandvagueconceptsinpredictionsofspeciesdistribution.EcologicalModelling157:313. Erwin,R.,J.Nichols,T.Eyler,D.Stotts,andB.Truitt,1998.Modelingcolony-sitedynamics:acasestudyofGull-billedTerns(Sternanilotica)incoastalvirginia.TheAuk115:970. Fagan,W.andF.Lutscher,2006.Averagedispersalsuccess:linkinghomerange,dispersal,andmetapopulationdynamicstoreservedesign.EcologicalApplications16:820. Fahrig,L.,2003.Effectsofhabitatfragmentationonbiodiversity.AnnualReviewofEcology,Evolution,andSystematics34:487. Fedrowitz,K.,M.Kuusinen,andT.Snall,2012.MetapopulationdynamicsandfuturepersistenceofepiphyticcyanolichensinaEuropeanborealforestecosystem.JournalofAppliedEcology49:493. Ferreras,P.,2001.Landscapestructureandasymmetricalinter-patchconnectivityinametapopulationoftheendangeredIberianLynx.BiologicalConservation100:125. Fiske,I.andR.Chandler,2011.unmarked:AnRpackageforttinghierarchicalmodelsofwildlifeoccurrenceandabundance.JournalofStatisticalSoftware43:1. Fletcher,R.,M.Acevedo,B.Reichert,K.Pias,andW.Kitchens,2011.Socialnetworkmodelspredictmovementandconnectivityinecologicallandscapes.ProceedingsoftheNationalAcademyofSciencesUSA108:19282. Fletcher,R.,C.Maxwell,J.Andrews,andW.Helmey-Hartman,2013.Signaldetectiontheoryclariestheconceptofperceptualrangeanditsrelevancetolandscapeconnectivity.LandscapeEcology28:57. Fletcher,R.andC.Miller,2008.Thetypeandtimingofsocialinformationaltersoffspringproduction.Biologyletters4:482. 83

PAGE 84

Fletcher,R.J.,2009.Doesattractiontoconspecicsexplainthepatch-sizeeffect?Anexperimentaltest.Oikos118:1139. Gillson,L.,T.P.Dawson,S.Jack,andM.A.McGeoch,2013.Accommodatingclimatechangecontingenciesinconservationstrategy.TrendsinEcology&Evolution28:135. Gilpin,M.andJ.Diamond,1976.Calculationofimmigrationandextinctioncurvesfromthespecies-area-distancerelation.ProceedingsoftheNationalAcademyofSciencesUSA73:4130. Gilroy,J.J.,T.Virzi,R.L.Boulton,andJ.L.Lockwood,2012.Anewapproachtotheapparentsurvivalproblem:estimatingtruesurvivalratesfrommark-recapturestudies.Ecology93:1509. Gustafson,E.andR.Gardner,1996.Theeffectoflandscapeheterogeneityontheprobabilityofpatchcolonization.Ecology77:94. Gustafsson,L.,A.Fiskesjo,T.Ingelog,B.Petterssonj,andG.Thor,1992.Factorsofimportancetosomelichenspeciesofdeciduousbroad-leavedwoodsinsouthernSweden.TheLichenologist24:255. Hahn,B.A.andE.D.Silverman,2006.Socialcuesfacilitatehabitatselection:Americanredstartsestablishbreedingterritoriesinresponsetosong.BiologyLetters2:337. Hamrick,J.,G.M.J.W.,andS.Sherman-Broyles,1995.Geneowamongplantpopulations:evidencefromgeneticmarkers.InP.HockandA.Stephenson,editors,Experimentalandmolecularapproachestoplantbiosystematics,pages215,St.Louis,USA.MissouriBotanicalGardenPress. Hanski,I.,1994.Apracticalmodelofmetapopulationdynamics.JournalofAnimalEcology63:151. Hanski,I.,1998.Metapopulationdynamics.Nature396:41. Hanski,I.andI.Hanski,1999.Metapopulationecology.OxfordUniversityPressOxford,UK. Hazell,P.,O.Kellner,H.Rydin,andL.Gustafsson,1998.Presenceandabundanceoffourepiphyticbryophytesinrelationtodensityofaspen(Populustremula)andotherstandcharacteristics.ForestEcologyandManagement107:147. Heller,N.andE.Zavaleta,2009.Biodiversitymanagementinthefaceofclimatechange:areviewof22yearsofrecommendations.BiologicalConservation142:14. 84

PAGE 85

Hilborn,R.,1990.Determinationofshmovementpatternsfromtagrecoveriesusingmaximumlikelihoodestimators.CanadianJournalofFisheriesandAquaticSciences47:635. Hodgson,J.,A.Moilanen,B.Wintle,andC.Thomas,2011.Habitatarea,qualityandconnectivity:strikingthebalanceforefcientconservation.JournalofAppliedEcology48:148. Hodgson,J.,C.Thomas,B.Wintle,andA.Moilanen,2009.Climatechange,connectivityandconservationdecisionmaking:backtobasics.JournalofAppliedEcology46:964. Holt,R.,1985.Populationdynamicsintwo-patchenvironments:someanomalousconsequencesofanoptimalhabitatdistribution.TheoreticalPopulationBiology28:181. Holt,R.,1996.Adaptiveevolutioninsource-sinkenvironments:directandindirecteffectsofdensity-dependenceonnicheevolution.Oikospages182. Hovestadt,T.,B.Binzenhofer,P.Nowicki,andJ.Settele,2011.Doallinter-patchmovementsrepresentdispersal?Amixedkernelstudyofbutterymobilityinfragmentedlandscapes.JournalofAnimalEcology80:1070. Johnson,S.,A.Covich,T.Crowl,A.Estrada-Pinto,J.Bithorn,andW.Wurtsbaugh,1998.Doseasonalityanddisturbanceinuencereproductioninfreshwateratynidshrimpinheadwatersstreams,PuertoRico?InternationaleVereinigungfuerTheo-retischeundAngewandteLimnologieVerhandlungen26:2076. Kawecki,T.andR.Holt,2002.Evolutionaryconsequencesofasymmetricdispersalrates.TheAmericanNaturalist160:333. Keddy,P.,1981.Experimentaldemographyofthesand-duneannual,Cakileedentula,growingalonganenvironmentalgradientinNovaScotia.TheJournalofEcologypages615. Kery,M.andK.Gregg,2003.Effectsoflife-stateondetectabilityinademographicstudyoftheterrestrialorchidCleistesbifaria.JournalofEcology91:265. Kery,M.,J.Spillmann,C.Truong,andR.Holderegger,2006.Howbiasedareestimatesofextinctionprobabilityinrevisitationstudies?JournalofEcology94:980. Kindlmann,P.andF.Burel,2008.Connectivitymeasures:areview.LandscapeEcology23:879. Kindvall,O.andA.Petersson,2000.Consequencesofmodellinginterpatchmigrationasafunctionofpatchgeometrywhenpredictingmetapopulationextinctionrisk.EcologicalModelling129:101. 85

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Krosby,M.,J.Tewksbury,N.Haddad,andJ.Hoekstra,2010.Ecologicalconnectivityforachangingclimate.ConservationBiology24:1686. Kujala,H.,M.Burgman,andA.Moilanen,2012.Treatmentofuncertaintyinconservationunderclimatechange.ConservationLetters0:1. Kukkala,A.andA.Moilanen,2012.Coreconceptsofspatialprioritisationinsystematicconservationplanning.BiologicalReviews. Liebhold,A.,W.D.Koenig,andO.N.Bjrnstad,2004.Spatialsynchronyinpopulationdynamics.AnnualReviewofEcology,Evolution,andSystematics35:467. Lindenmayer,D.andJ.Fischer,2006.Habitatfragmentationandlandscapechange:anecologicalandconservationsynthesis.IslandPress. Lindenmayer,D.,R.Hobbs,R.Montague-Drake,J.Alexandra,A.Bennett,M.Burgman,P.Cale,A.Calhoun,V.Cramer,P.Cullen,etal.,2008.Achecklistforecologicalmanagementoflandscapesforconservation.EcologyLetters11:78. Lomolino,M.,1990.Thetargetareahypothesis:theinuenceofislandareaonimmigrationratesofnon-volantmammals.Oikos57:297. Lutscher,F.,E.McCauley,andM.Lewis,2007.Spatialpatternsandcoexistencemechanismsinsystemswithunidirectionalow.Theoreticalpopulationbiology71:267. MacKenzie,D.,J.Nichols,J.Hines,M.Knutson,andA.Franklin,2003.Estimatingsiteoccupancy,colonization,andlocalextinctionwhenaspeciesisdetectedimperfectly.Ecology84:2200. Margules,C.andR.Pressey,2000.Systematicconservationplanning.Nature405:243. Mawdsley,J.,R.O'Malley,andD.Ojima,2009.Areviewofclimate-changeadaptationstrategiesforwildlifemanagementandbiodiversityconservation.ConservationBiology23:1080. McRae,B.H.,B.G.Dickson,T.H.Keitt,andV.B.Shah,2008.Usingcircuittheorytomodelconnectivityinecology,evolution,andconservation.Ecology89:2712. Miller,C.,R.Fletcher,B.Anderson,andL.Nguyen,2012.Natalsocialenvironmentinuenceshabitatselectionlaterinlife.AnimalBehaviour83:473. Miller,T.,2008.Bottom-up,top-down,andwithin-trophiclevelpressuresonacactus-feedinginsect.EcologicalEntomology33:261. Minor,E.andD.Urban,2008.Agraph-theorytrameworkforevaluatinglandscapeconnectivityandconservationplanning.ConservationBiology22:297. 86

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Moilanen,A.,2008.Twopathstoasuboptimalsolutiononcemoreaboutoptimalityinreserveselection.BiologicalConservation141:1919. Moilanen,A.andI.Hanski,1998.Metapopulationdynamics:effectsofhabitatqualityandlandscapestructure.Ecology79:2503. Moilanen,A.,J.Leathwick,andJ.Elith,2007.Amethodforspatialfreshwaterconservationprioritization.FreshwaterBiology53:577. Moilanen,A.andM.Nieminen,2002.Simpleconnectivitymeasuresinspatialecology.Ecology83:1131. Moilanen,A.,M.Runge,J.Elith,A.Tyre,Y.Carmel,E.Fegraus,B.A.Wintle,M.Burgman,andY.Ben-Haim,2006a.Planningforrobustreservenetworksusinguncertaintyanalysis.EcologicalModelling199:115. Moilanen,A.,K.Wilson,andH.Possingham,2009.Spatialconservationprioritization:quantitativemethodsandcomputationaltools.OxfordUniversityPress. Moilanen,A.,B.Wintle,J.Elith,andM.Burgman,2006b.Uncertaintyanalysisforregional-scalereserveselection.ConservationBiology20:1688. Moser,D.,T.Drapela,J.Zaller,andT.Frank,2009.Interactingeffectsofwinddirectionandresourcedistributiononinsectpestdensities.BasicandAppliedEcology10:208. Nalle,D.,J.Arthur,andJ.Sessions,2002.Designingcompactandcontiguousreservenetworkswithahybridheuristicalgorithm.ForestScience48:59. Nichols,J.D.andW.L.Kendall,1995.Theuseofmulti-statecapture-recapturemodelstoaddressquestionsinevolutionaryecology.JournalofAppliedStatistics22:835. Olaya-Arenas,P.,E.Melendez-Ackerman,M.Perez,andR.Tremblay,2011.Demographicresponsebyasmallepiphyticorchid.Americanjournalofbotany98:2040. Pascual-Hortal,L.andS.Saura,2006.Comparisonanddevelopmentofnewgraph-basedlandscapeconnectivityindices:towardsthepriorizationofhabitatpatchesandcorridorsforconservation.LandscapeEcology21:959. Pellet,J.,E.Fleishman,D.Dobkin,A.Gander,andD.Murphy,2007.Anempiricalevaluationoftheareaandisolationparadigmofmetapopulationdynamics.BiologicalConservation136:483. Pollock,K.,1982.Acapture-recapturedesignrobusttounequalprobabilityofcapture.TheJournalofWildlifeManagement46:752. 87

PAGE 88

Possingham,H.,J.Franklin,K.Wilson,andT.Regan,2005.Therolesofspatialheterogeneityandecologicalprocessesinconservationplanning.InG.M.Lovett,C.G.Jones,M.G.Turner,andK.C.Weathers,editors,EcosystemFunctioninHeterogeneousLandscapes.Springer. Pressey,R.,2004.Conservationplanningandbiodiversity:assemblingthebestdataforthejob.ConservationBiology18:1677. Pressey,R.andM.Bottrill,2009.Approachestolandscape-andseascape-scaleconservationplanning:convergence,contrastsandchallenges.Oryx43:464. Prevedello,J.andM.Vieira,2010.Doesthetypeofmatrixmatter?Aquantitativereviewoftheevidence.BiodiversityandConservation19:1205. Prokopy,R.J.andE.D.Owens,1983.Visualdetectionofplantsbyherbivorousinsects.AnnualReviewofEntomology28:337. Pulliam,H.,1988.Sources,sinks,andpopulationregulation.AmericanNaturalist132:652. Ramrez,A.andE.Melendez-Colom,2003.MeteorologicalsummaryforElVerdeFieldStation:1975-2003.InstituteforTropicalEcosystemStudies,UniversityofPuertoRico. Rasmussen,H.andD.Whigham,1993.Seedecologyofdustseedsinsitu:anewstudytechniqueanditsapplicationinterrestrialorchids.AmericanJournalofBotany80:1374. Ray,C.,M.Gilpin,andA.T.Smith,1991.Theeffectofconspecicattractiononmetapopulationdynamics.BiologicaljournaloftheLinneanSociety42:123. Regan,H.M.,Y.Ben-Haim,B.Langford,W.Wilson,P.Lundberg,S.Andelman,andM.Burgman,2005.Robustdecision-makingundersevereuncertaintyforconservationmanagement.EcologicalApplications15:1471. Regan,H.M.,M.Colyvan,andM.A.Burgman,2002.Ataxonomyandtreatmentofuncertaintyforecologyandconservationbiology.EcologicalApplications12:618. Revilla,E.,T.Wiegand,F.Palomares,P.Ferreras,andM.Delibes,2004.Effectsofmatrixheterogeneityonanimaldispersal:Fromindividualbehaviortometapopulation-levelparameters.TheAmericanNaturalist164:E130E153. RiveraGomez,N.,R.Tremblay,andE.Melendez-Ackerman,2006.Distributionoflifecyclestagesinalithophyticandepiphyticorchid.FoliaGeobotanica41:107. Ross,S.M.,2006.IntroductiontoProbabilityModels,NinthEdition.AcademicPress,Inc.,Florida,USA. 88

PAGE 89

Rota,C.,R.Fletcher,R.Dorazio,andM.Betts,2009.Occupancyestimationandtheclosureassumption.JournalofAppliedEcology46:1173. Sala,O.,F.Chapin,J.Armesto,E.Berlow,J.Bloomeld,R.Dirzo,E.Huber-Sanwald,L.Huenneke,R.Jackson,A.Kinzig,etal.,2000.Globalbiodiversityscenariosfortheyear2100.Science287:1770. Salomon,Y.,S.Connolly,andL.Bode,2010.Effectsofasymmetricdispersalonthecoexistenceofcompetingspecies.EcologyLetters13:432. Sarkar,S.,R.Pressey,D.Faith,C.Margules,T.Fuller,D.Stoms,A.Moffett,K.Wilson,K.Williams,P.Williams,etal.,2006.Biodiversityconservationplanningtools:presentstatusandchallengesforthefuture.AnnualReviewofEnvironmentandResources31:123. Sawyer,S.,C.Epps,andJ.Brashares,2011.Placinglinkagesamongfragmentedhabitats:doleast-costmodelsreecthowanimalsuselandscapes?JournalofAppliedEcology48:668. Schick,R.andS.Lindley,2007.Directedconnectivityamongshpopulationsinariverinenetwork.JournalofAppliedEcology44:1116. Schooley,R.andJ.Wiens,2003.Findinghabitatpatchesanddirectionalconnectivity.Oikos102:559. Schooley,R.andJ.Wiens,2004.Movementsofcactusbugs:patchtransfers,matrixresistance,andedgepermeability.LandscapeEcology19:801. Schooley,R.andJ.Wiens,2005.Spatialecologyofcactusbugs:areaconstraintsandpatchconnectivity.Ecology86:1627. Serrano,D.andJ.Tella,2003.Dispersalwithinaspatiallystructuredpopulationoflesserkestrels:theroleofspatialisolationandconspecicattraction.JournalofAnimalEcology72:400. Serrano,D.,J.Tella,M.Forero,andJ.Donazar,2001.Factorsaffectingbreedingdispersalinthefacultativelycoloniallesserkestrel:individualexperiencevs.conspeciccues.JournalofAnimalEcology70:568. Smith,A.andM.Peacock,1990.Conspecicattractionandthedeterminationofmetapopulationcolonizationrates.ConservationBiology4:320. Smith,J.,2010.Basicinterdictionmodels.WileyEncyclopediaofOperationsResearchandManagementScience. Snall,T.,J.Pennanen,L.Kivisto,andI.Hanski,2005.Modellingepiphytemetapopulationdynamicsinadynamicforestlandscape.Oikos109:209. 89

PAGE 90

Snall,T.,P.RibeiroJr,andH.Rydin,2003.Spatialoccurrenceandcolonisationsinpatch-trackingmetapopulations:localconditionsversusdispersal.Oikos103:566. Spendelow,J.,J.Nichols,I.Nisbet,H.Hays,andG.Cormons,1995.EstimatingannualsurvivalandmovementratesofadultswithinametapopulationofRoseateTerns.Ecology76:2415. Taylor,P.,L.Fahrig,K.Henein,andG.Merriam,1993.Connectivityisavitalelementoflandscapestructure.Oikos68:571. Team,R.etal.,2011.R:Alanguageandenvironmentforstatisticalcomputing.RFoundationStatisticalComputing. Thomas,C.D.,A.Cameron,R.E.Green,M.Bakkenes,L.J.Beaumont,Y.C.Collingham,B.F.Erasmus,M.F.DeSiqueira,A.Grainger,H.Hannah,Lee,L.Hughes,B.Huntley,A.vanJaarsveld,G.Midgley,L.Miles,M.A.Ortega-Huerta,A.T.Peterson,O.L.Phillips,andS.E.Williams,2004.Extinctionriskfromclimatechange.Nature427:145. Tilman,D.,R.May,C.Lehman,andM.Nowak,1994.Habitatdestructionandtheextinctiondebt.Nature371:65. Tischendorf,L.andL.Fahrig,2000.Ontheusageandmeasurementoflandscapeconnectivity.Oikos90:7. Travis,J.andD.French,2000.Dispersalfunctionsandspatialmodels:expandingourdispersaltoolbox.EcologyLetters3:163. Tremblay,R.,1997.DistributionanddispersionpatternsofindividualsinninespeciesofLepanthes(orchidaceae).Biotropica29:38. Tremblay,R.,2008.Ecologicalcorrelatesandshort-termeffectsofrelocationofarareepiphyticorchidafterHurricaneGeorges.EndageredSpeciesResearch5:83. Tremblay,R.andJ.Castro,2009.Circulardistributionofanepiphyticherbontreesinasubtropicalrainforest.TropicalEcology50:211. Tremblay,R.,E.Melendez-Ackerman,andD.Kapan,2006.Doepiphyticorchidsbehaveasmetapopulations?evidencefromcolonization,extinctionratesandasynchronouspopulationdynamics.BiologicalConservation129:70. Treml,E.,P.Halpin,D.Urban,andL.Pratson,2008.Modelingpopulationconnectivitybyoceancurrents,agraph-theoreticapproachformarineconservation.LandscapeEcology23:19. Urban,D.andT.Keitt,2001.Landscapeconnectivity:agraph-theoreticperspective.Ecology82:1205. 90

PAGE 91

VonStackelberg,H.,1952.Thetheoryofthemarketeconomy.OxfordUniversityPress,London,UK. Vuilleumier,S.,B.Bolker,andO.Leveque,2010.Effectsofcolonizationasymmetriesonmetapopulationpersistence.Theoreticalpopulationbiology78:225. Vuilleumier,S.andH.Possingham,2006.Doescolonizationasymmetrymatterinmetapopulations?ProceedingsoftheRoyalSocietyB:BiologicalSciences273:1637. Watson,J.,C.Hays,P.Raimondi,S.Mitarai,C.Dong,J.McWilliams,C.Blanchette,J.Caselle,andD.Siegel,2011a.Currentsconnectingcommunities:nearshorecommunitysimilarityandoceancirculation.Ecology92:1193. Watson,J.,D.Siegel,B.Kendall,S.Mitarai,A.Rassweiller,andS.Gaines,2011b.Identifyingcriticalregionsinsmall-worldmarinemetapopulations.ProceedingsoftheNationalAcademyofSciences108:E907E913. Whigham,D.,J.ONeill,H.Rasmussen,B.Caldwell,andM.McCormick,2006.Seedlongevityinterrestrialorchidspotentialforpersistentinsituseedbanks.BiologicalConservation129:24. White,G.C.,W.L.Kendall,andR.J.Barker,2006.MultistatesurvivalmodelsandtheirextensionsinProgramMARK.JournalofWildlifeManagement70:1521. Wiegand,T.,K.Moloney,J.Naves,andF.Knauer,1999.Findingthemissinglinkbetweenlandscapestructureandpopulationdynamics:aspatiallyexplicitperspective.TheAmericanNaturalist154:605. Williams,I.,D.Frearson,H.Barari,andA.McCartney,2007.Firsteldevidencethatparasitoidsuseupwindanemotaxisforhost-habitatlocation.EntomologiaExperimentalisetApplicata123:299. Williams,J.,C.ReVelle,andS.Levin,2004.Usingmathematicaloptimizationmodelstodesignnaturereserves.FrontiersinEcologyandtheEnvironment2:98. Williams,J.,C.ReVelle,andS.Levin,2005.Spatialattributesandreservedesignmodels:areview.EnvironmentalModelingandAssessment10:163. Wilson,K.,M.Cabeza,andC.Klein,2009.Fundamentalconceptsofspatialconservationprioritization.InA.Moilanen,K.Wilson,andH.Possingham,editors,Spatialconservationprioritization:QuantitativeMethodsandComputationalTools,pages16,London,UK.OxfordUniversityPress. Zeigler,S.L.,M.C.Neel,L.Oliveira,B.E.Raboy,andW.F.Fagan,2011.Conspecicandheterospecicattractioninassessmentsoffunctionalconnectivity.BiodiversityandConservation20:2779. 91

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BIOGRAPHICALSKETCH MiguelA.AcevedowasborninSanJuan,PuertoRicoin1980.HegraduatedfromtheUniversityofPuertoRicowithaB.S.inbiology2002.HelatergraduatedwithaM.S.intheTropicalBiologyProgramfromtheUniversityofPuertoRico-RoPiedrasin2006.MiguelreceivedhisPh.D.fromtheUniversityofFloridainthespringof2013where,aspartoftheQuantitativeSpatialEcology,EvolutionandEnvironment(QSE3IGERT),developedaninterdisciplinarydissertationresearchthatincorporatedspatialecology,networktheoryandconceptsfromindustrialandsystemsengineering.HehopestoreturnsomedaytoPuertoRicoandjointheirprofessorateconductingquantitativeecologicalresearch,andteaching. 92