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Ecological
Modelling
j ou rn a l h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l
Metapopulation
modelling
and
area-wide
pest
management
strategies
evaluation.
An
application
to
the
Pine
processionary
moth
Gianni
Gilioli
a,c,
Antonella
Bodini
b,∗,
Johann
Baumgärtner
caDepartmentofMolecularandTranslationalMedicine,UniversityofBrescia,VialeEuropa11,25123Brescia,Italy bCNR–IMATI,InstituteofAppliedMathematicsandInformationTechnology,viaE.Bassini15,20133Milano,Italy
cCASAS(CenterfortheAnalysisofSustainableAgriculturalSystems),37ArlingtonAvenue,Kensington,CA94707,UnitedStates
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received31October2012
Receivedinrevisedform18March2013 Accepted26March2013
Available online 30 April 2013
Keywords: Decisionmaking
Kullback–Leiblerdivergence Managementstrategiesranking Spatialautocorrelation
Spatiallyexplicitmetapopulationmodel Traumatocampapityocampa
a
b
s
t
r
a
c
t
Forecastingpestpopulationabundanceisatimeandresourceconsumingtask,andinparticularfor area-widepestmanagementiscomplicatedbydemographicandenvironmentalstochasticity.Thesefactors makedifficultthedevelopmentofquantitativetoolstodesignandevaluatedifferentmanagement strate-giesperformancesbytakingintoaccountvariousformofvariabilityanduncertainty.Pestmanagement couldbenefitfrommethodssupportingdecisionmakingbasedonmodelseaseofdevelopmentunder scarcedataandhighuncertainty.Hostplantsformanyagriculturalandforestpestsareoftenpatchily dis-tributed,thereforepopulationdynamicscanbesuitablydescribedintermsofmetapopulations.Despite thefactthatmetapopulationmodelswereoriginallyproposedforpests,theyremainawidelyusedtool inconservationbiologybutreceivelittleattentioninlargescalepestmanagement.
Theaimofthispaperistoproposeaframeworkallowingtherankingoftheefficacyofarea-wide pestcontrolstrategies,taking intoaccountpopulation spatialdistributionin discretepatches. The Kullback–Leiblerdivergence,wellknowninInformationTheory,ProbabilityandStatistics,isusedto measurehowfarthestateofthemetapopulationaspredictedbyaspatiallyexplicitmetapopulation modelisfromasuitablereferencestate.
Themethodisappliedtocomparetheefficacyofdifferenttypesofpredefinedcontrolstrategiesof thePineprocessionarymoth(Traumatocampapityocampa(Den.andSchiff)).Theanalysisofadataseton metapopulationdynamicsofthismothfromafragmentedMediterraneanpineforestallowstoderive somerulesofthumbfortherationalallocationofcontroleffort,intermsofspatialandtemporal distri-butionoftheinterventions.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Atthelocalandgloballevels,thespatialpatternsof agricul-turalandforestpestpopulationdynamicsmayvaryaccordingto thecharacteristicsofafragmentedlandscape(Vinatieretal.,2011). Analysesofspatialheterogeneityareeitherbasedoncorrelations that takeinto accountdetails oflandscapesand theireffect on populationprocesses(Wiens,1997;Hunter,2002),oron simula-tionmodelsbasedoncomplexassumptionsonecologicalprocesses (Sheehan,1989;SharovandColbert,1996).
Metapopulationmodelsdealwiththeoccurrenceof individ-ual populations across an ensemble of habitat fragments (or patches) suitable for the occurrence and reproduction of the species (Tscharntke and Brandl, 2004), connected by dispersal.
∗ Correspondingauthor.Tel.:+390223699524;fax:+390223699538. E-mailaddresses:gianni.gilioli@med.unibs.it(G.Gilioli),
antonella.bodini@mi.imati.cnr.it(A.Bodini).
Metapopulationmodels wereoriginally proposedfor pestsand werebasedonasimpledescriptionofthefrequencyofoccupied patches(Levins,1969;IvesandSettle,1997).Theneedtoexplicitly representpatchesinspacehasdemandedthedevelopmentof dif-ferentmetapopulationmodels.Spatiallyexplicitmetapopulation modelsareawidelyusedtoolinconservationbiology,buthave receivedlittleattentioninpestcontroldespitetheirpotentiality (Hunter,2002).Themostwidelyknownspatiallyexplicit metapop-ulationmodelfor instance,theIncidenceFunction Model(IFM) (Hanski,1994),usesareaandconnectivitytopredict metapopu-lationdynamicsandhasnotbeenappliedyettoagriculturaland forestinsect,tothebestofourknowledge.Theapparentlackof interestinapplyingtheIFManditsgeneralizationstopestis sur-prising,becauseitwouldenablemanagerstotakeintoaccountthe sizeofhabitat(Fleishmanetal.,2002),habitatquality(Moilanen and Hanski, 1998; Fleishman et al., 2002), spatialarrangement (patchstructure),patchisolation(connectivity,includingmatrix) (Fleishmanetal.,2002),andpatchoccupancy(descriptionand pos-sibilityofmodellingvariationinlocaldynamics).
0304-3800/$–seefrontmatter © 2013 Elsevier B.V. All rights reserved.
Traditionally,pestmanagementhasdisregardedtheeffectsof habitatfragmentationonpopulationdynamicsandonlyrecently, attempts have been made toovercome this and other restric-tions(Kogan et al., 1999;Elliot et al., 2008). Spatialexpansion ofpest management hasbeenmotivatedbyincreasing interest inthe effectsof landscape featureson pestpopulation dynam-ics(Hunter,2002)andfurtherstimulatedbyresearchonspecies assemblagesexhibiting differentlife strategies in a fragmented landscape.Thecarefulconsiderationofthespatialdimensionisalso importantforobtainingfurtherinsightintospeciesinteractions, includingherbivory,biologicalcontrolandpollination(Tscharntke and Brandl, 2004; Tscharntke et al., 2005). These interactions dependonspeciestraitsbutalsoonthesizeandtheconnectivity ofhabitats(TscharntkeandBrandl,2004).
Althougharea-widepest managementstrategies are becom-ingcommon(Kogan,1995;Faust,2008),theadvantagesofthese strategiesaredifficulttoestablishwithscientificrigour,andtheir design and evaluation may be done by modelling tools. Pest managementcouldbenefitfromquantitativemethodsbasedon modelease of developmentunder scarcedataand high uncer-tainty.Inthis respect,metapopulationmodelshaveunexplored potentiality(Hunter,2002;Grimmetal.,2003;Giliolietal.,2008; vanNouhuys,2009).However,metapopulationmodelsshouldbe embeddedinadecision-makingframeworktogivemanagersthe capabilityofrankingalternativedecisions(Westphaletal.,2003). This means that the objectives of the management should be explicitly and clearly statedin terms of metapopulation model variables(Possinghametal.,2001).Inappliedecology, metapopu-lationmodelsarefrequentlyusedinadecision-makingframework for conservation purposes mainly, to evaluate metapopulation persistence.Tothebestofourknowledge,onlyafewapplications concernpestmanagement.Forinstance,stochasticdynamic pro-gramming(SDP)hasbeenrecentlyappliedin pestmanagement relatedissues, coupledwitha spatially implicitmetapopulation model,e.g.forinvasivespeciescontroloptimization(Bogichand Shea,2008),orforbiologicalcontrolreleasestrategiesoptimization (Sheaand Possingham,2000).However,SDPis computationally complex and its applicability limited to small metapopula-tions(NicolandChadès,2011).Borrowingfromepidemiology,a susceptible-infected-susceptible(SIS)modelandafiniteMarkov decisionprocesshavebeenproposedtomanagediseases,pestor endangeredspeciesinsmall(<25nodes)networkmotifs(Chadès etal.,2011).
Inthispaper,weproposeamethodologybasedonaspatially explicitmetapopulation model and the Kullback–Leibler diver-gence(KullbackandLeibler,1951)torankarea-widesinglespecies pestmanagement strategies. Strategiesare rankedaccordingto theircapabilityinachievingatagivenpointintimeanexplicitly statedobjective,definedbymeansoftheKullback–Leibler diver-gence.Therequiredcomputationsarerelativelysimpleandfast. Themethoddealswithamosaicofdifferentspatialunits(patches) infestedby a monophagous pest and investigates theinterplay betweenpestpopulationdynamics,dispersalandarea-wide con-trolstrategies. For thetargetpestwe assumethattheproblem canbesolvedbyfocussingonthepestpopulationratherthanon thepopulationsystemcomposedofthepestanditsnatural ene-mies(Hunter,2002),andthatthreshold-basedmanagementisan appropriatecontrolstrategy.Withinthispestmetapopulation con-text,we showthatif aspatially explicitmetapopulationmodel adequately describes pest population dynamics, management strategiesexpressedintermsofvariationsinthecomponentsof suchamodelcanbecomparedandranked.Todothis,weapply themethodproposedbyGiliolietal.(2008).Weshowhowthis methodaccountsformetapopulationdynamicsandsupportsthe evaluationofshort-termstrategies(e.g.,spatialandtemporal dis-tributionofcontrolinterventions)aimingatthemanagementof
ANALYSIS PROTOCOL
PREREQUISITE
Check metapopulation models applicability
1. Select
a metapopulation model
providing a good fit to
the data.
2. Simulate
metapopulation dynamics
for each predefined
strategy according to the estimated model. Include the
do-nothing strategy
S
0.
3. For each strategy, compute the
KL
value.
4. Rank the strategies
from the higher to the lower
KL
value: good strategies have low
KL values.
5. Check the robustness
of the obtained ranking by
considering the possible sources of uncertainty, like model
parameters or sampling errors (
sensitivity analysis).
Fig.1.Schemeoftheadoptedanalysisprotocol.
speciesinhabitingagriculturalandforesthabitats,wheretheyplay theroleofpest.Shorttermstrategiesareparticularlysuitablefor peststhatdirectlyattackessentialoryieldproducingstructuresof thehostplantorpestsofsignificantpublichealthimportance,and areanessentialcomponentofIntegratedPestManagement(Kogan etal., 1999).Fig.1summarizesthecomplete analysisprotocol, describedinSection2.
Inthispaperweconsiderdatafromathreeyearssurveyofthe Pineprocessionarymoth(Traumatocampapityocampa(Den.and Schiff),PPM)populationsdynamics infragmentedforeststands ofthe NationalPark ofAspromonte,Calabria, Italy. Infestations byPPMcausedamagetopinetreesthatmayresultinyieldloss, negativelyaffectthevalueofforestsforrecreationalpurposesand diminishtheaestheticvalueofindividualtreesinurban environ-ments(MasuttiandBattisti,1990).Peststatusisfurtherincreased bytheproductionofsetaebylarvae,oftencausingserious aller-gic reactionsin humansand variousanimals(Lamy,1990).The hostplantsofthePPMareoftenpatchilydistributedinforestareas aroundtheMediterraneanBasin,thereforemetapopulationmodels canbetestedtodescribePPMpopulationdynamicsinafragmented landscape(TscharntkeandBrandl,2004;Elliotetal.,2008). 2. Materialsandmethods
2.1. Studyareaanddatacollection
TheNationalParkofAspromonteconsistsof Mediterranean-mountainforestscharacterizedbythedominanceoftheendemic Pinusnigrassp.calabrica(=Pinuslariciovar.calabrica),andvarious conifersinreforestationareas(Brulloetal.,2001).Theforestareas consideredinourstudyarepredominantlycomposedofpureP. nigracoetaneous foreststands, separated by opengrassland or deciduousforests standswhich are not inhabitedby PPM.The studyarea facestheTyrrhenian Sea and covers817ha along a 25kmtransectextendingfromSouthwesttowardsNortheast.In
FROM 2606930.04 TO 2606930.04 FROM 4230982.44 TO 4250455.94 1 2 3 4 56 7 8 9 10 111213 14 15 16 17 18 192021 22 23 24 25 26 27 2829 30 31 32
Fig.2.ThesetofPinusspp.patchessampledintheNationalParkofAspromonte Calabria(Italy).ThepatchesareinfestedbylocalpopulationsofPineprocessionary moththatmakeupthemetapopulation.Pointsrepresentthepositionofthecentroid intheplaneofnumericalXandYcoordinates(UTM33GB40)foreachforestpatch.
the preparatory phase of the study, maps were obtainedfrom differentinstitutionsincludingthenationalandregionalforest ser-vices(CorpoForestaledelloStato,AFoR)andparkauthorities(Ente Parco Nazionale d’Aspromonte). Spatial information was com-pletedwithsoildatamadeavailablebytheCorineLandCoverproject (http://www.eea.europa.eu/publications/COR0-landcover), digi-tizedaerialcolourphotographs,georeferencedforestsurveydata on the dimensions and locations of spatially separated forest fragments,andfinallywithinsectpopulationdatafromsurveys andfieldobservations(MantiandGilioli,2006).Weconsiderthe isolatedforestfragmentsaspatches.Aschematicrepresentation ofthecentroidsofthepatchesinthestudyareaisillustratedin Fig.2.
Inthreeconsecutiveyears,from2003to2005,PPMnestswere sampledinallthe32forestpatchesinthestudyarea.Duringlarval developmentthePPMaggregatesintopre-neststhatevolveinto permanent nests where thelarvae overwinter.Monthly counts of pre-nestsand nests were takenfrom October toApril, until thenumber of nestsremained constant. Thefinal nestscounts wereusedasanestimationofpopulationabundance.Thesample variableconsists ofthe number ofnests per tree.Countswere madealongaminimumnumberof2andmaximumnumberof3 transectsperpatchon25neighbouringtrees.Arandomprocedure wasusedtochoosethetransectposition.
2.2. Testingcriteriaformetapopulationdefinition
Apreliminarystepfortheapplicationoftheproposedapproach istodemonstratethataplant-pestsystemcanbeconsideredas ametapopulation(seeFig.1).Todothis,Hanski (1997) recom-mendsthat(a)hostplantsaredistributedasdiscretepatches,(b) pestpopulationswithinpatcheshaveasubstantialriskof extinc-tion,(c)emptypatchesareavailableforcolonization,and(d)pest localpopulationsdonotfluctuatesynchronously.Criterion(a)is satisfiedbecausetheoccurrenceofthePPMisrelatedtothe dis-cretedistributionofthehostplant.Forthreshold-basedpestcontrol purposes,localpestpopulationdynamicscanbedescribedas fluc-tuatingbelowandaboveagiventarget-level ofabundance,and emptypatchesarecontinuouslyavailableforcolonization.Thelast twoobservationsalsoprovecriteria(b)and(c).Asfarascriterion (d)isconcerned,anevaluationofthespatialsynchronyamonglocal populationsisconductedbymeansoftheMoran’sindexI,a mea-sureofspatialautocorrelation(Moran,1950),andoftheManteltest (Mantel,1967),ameasureofthecorrelationbetweentwosetsof
variablesbasedondistancematrices.TheindexIiscomputedfor differentdistanceclassesdas
I(d)=
n k lwkl k lwkl(xk− ¯x)(xl− ¯x) k(xk− ¯x)2 (1) wherexkisthevalueoftheobservedpestabundanceatsitek,disthedistanceclassbasedontheEuclideandistancebetweensites, wkl=1ifthepairofobservationskandlbelongstothedistance
classdandwkl=0otherwise,nisthenumberofspatialunitsand ¯x
isthemeanvalueofthexks.Azerovalueindicatesarandom
spa-tialpattern.ThesignificanceofI(d)istestedusingarandomization test(LegendreandLegendre,1998,p.20)andthemethodofHolm usedtocorrectprobabilitylevelsformultipletesting(Lehmann andRomano,2005,p.350).Thespatialrelationshipsbetweenpest abundanceobservedatdifferentdatesisassessedusingamodified Moran’sIstatistic,thebivariateMoran’sI(FortinandDale,2005, p.147)andtheMantelcorrelogram(OdenandSokal,1986).The Sturge’sruleisusedtofixthenumberofclasses,andEuclidean distancesaredistributedamongthembyconsideringbothequal distanceintervalsandequalsizedclasses(FortinandDale,2005,p. 116).
2.3. TheIncidenceFunctionModel
Toconformwiththeusualnotationsofmetapopulation mod-els,wereferto‘presence’or‘absence’(occupiedoremptypatch, respectively)meaningthatthepestlocalpopulationabundanceis above(1)orbelow(0)themanagementthreshold,respectively.
InthisworkweadopttheIFM,describingpresence/absenceofa speciesinthepatchesofahighlyfragmentedlandscapeatdiscrete timeintervals(years)astheresultofcolonizationandextinction processes,tomodelthetransitionsacrossthemanagement thresh-old(Step1inFig.1).Intermofpestmetapopulationdynamics, thetransitionofpopulationabundancefrombelowtoabovethe managementthresholdisrepresentedasacolonizationprocess. Pestspeciesareoftenr-strategiststhatbuilduphighpopulation levelssoonafterpatchcolonization,andlimitedtimeistakento reachthethreshold.Therefore,intermsofmetapopulation dynam-icsthetransitionfromemptytooccupiedpatchescanbeconsidered astheimmediateconsequenceofpatchcolonizationratherthan within-patchprocesses.Thedocumentedcontributionofthe move-ment tothelocaldynamicsin many forestLepidopteraspecies likePPM(MasuttiandBattisti,1990)furtherjustifiestheroleof connectivityand of thecolonizationprocess in metapopulation dynamics.
Thetransitionofpopulationabundancefromabovetobelowthe managementthreshold,duetoeitherthedeclineofthelocal pop-ulationortocontrolintervention,isrepresentedasanextinction process.InthecaseofthePPMinAspromonte,foreachpatchi,the meannumbermiofnestspertreeiscomparedtothethresholdlevel
of0.75nestspertree,accordingtotherecommendationprovided bytheCorpoForestaledelloStatoincompliancewiththe require-mentsoftheItalianLaw(D.M.30ottobre2007–Disposizioniperla lottaobbligatoriacontrolaprocessionariadelpino).Onlyifmi≥0.75
shouldtreatmentbeconsidered.
IntheIFM,theprocessofoccupancyofpatchiisdescribedbya first-orderMarkovchainwithtwostates,{0,1}(emptyand occu-pied,respectively).Theextinction probabilityofa populationin patchi, Ei,isconstantintime andis assumedtodecreasewith
increasingpatcharea Ei=min
A 0 Ai x ,1 (2) whereAiistheareaofthepatchi(km2),A0isthecriticalpatcharea1year,andxisaparameterreflectingtheseverityof environmen-talstochasticity(Hanski,1994).Herethepatcharearepresentsthe resourcessupportingthelocalpopulation.Thecolonization proba-bilityofpatchiattimet,Ci(t),isassumedtobeasigmoidalfunction
increasingwithconnectivity
Ci(t)= S2 i(t) S2 i(t)+y2 (3) where Si(t)=
[ıj(t)exp(−˛dij)Aj: j/=i] (4)istheconnectivityofpatchiattimet,ıj(t)=1ifpatchjisoccupied
attimetandıj(t)=0otherwise,dijistheEuclidean
centroid-to-centroiddistance(km)betweensitesiandj,˛isapositiveconstant settingthesurvivalrateofmigrantsoverthedistanceandyisa parameterdescribingthecolonizationabilityofthespecies(good colonizershavesmally).Inthispaper,duetolackofspecific infor-mationtheminimumareaA0thatmaysupportaPPMpopulation
issetequalto400m2.Thesurvivalrateoverdistance˛=2.3(10%
survivalat1km)isobtainedfromtheliteratureonthemaximum distancereachedbyaflyingadultduringitsshortlife(Manti,2006). TheuncertaintyaboutA0and˛isconsideredinSection2.6.
Parame-tersxandyareestimatedbymaximizationofthepseudo-likelihood (Moilanen,1999)correspondingtotheinitialdistributiongivenby thefirstobservedmetapopulationstate(year2003,seeTable1).
Table1
ThestudyofaPineprocessionarymothmetapopulationinforeststandsofthe NationalParkofAspromonte,Calabria,Italy.Foreachforestpatch,thesize(inm2)
andtheaverageannualnestabundancearereported.Theaveragenestabundance perpatchisdefinedintermsofnumberoffinalpermanentnestspertree.Thelast rowshowsthemeanvaluesofpatchareaandnestabundance.
Patch Patcharea(m2) Nestabundance
2003 2004 2005 1 76,952.12 1.52 1.08 2.16 2 118,040.78 1.13 0.97 1.47 3 75,634.44 0.68 2.20 1.80 4 46,699.49 0.72 1.04 1.72 5 9626.39 1.12 0.72 1.52 6 32,839.29 0.40 0.40 0.40 7 156,918.33 1.10 1.10 1.10 8 80,640.96 0.30 0.30 0.30 9 489,647.84 0.89 1.83 1.91 10 22,778.76 1.80 1.80 1.80 11 124,423.97 1.60 1.60 1.60 12 41,644.60 0.64 2.04 1.34 13 37,503.63 0.92 2.00 1.28 14 368,181.27 0.56 1.10 1.66 15 426,183.47 0.86 1.50 1.77 16 652,758.32 0.48 2.13 1.97 17 407,366.32 0.44 0.84 2.00 18 1,175,910.81 0.90 1.23 1.86 19 100,794.14 1.72 2.28 2.60 20 51,305.24 1.44 1.20 3.40 21 273,488.23 0.80 0.80 0.80 22 69,012.79 0.52 0.84 2.00 23 63,327.84 0.32 0.96 1.32 24 475,172.94 0.84 1.52 1.88 25 369,247.02 0.84 1.88 1.86 26 82,197.46 0.00 0.92 1.68 27 249,482.81 0.01 0.95 1.68 28 41,686.47 0.36 2.04 1.20 29 134,546.93 1.80 1.80 1.80 30 190,317.93 0.44 2.09 1.53 31 1,626,188.30 0.59 1.62 1.66 32 98,395.17 2.60 2.60 2.60 Mean 255,278.56 0.89 1.42 1.68 Table2
Pineprocessionarymothmetapopulationmodel:comparisonofthemodel param-etervaluestothevaluesestimatedunderthedifferenthypothesesconsideredfor sensitivityanalysisinSection2.6.Boldtypedenotestheparameterwithrespectto whichsensitivityisassessed.A0isthecriticalpatchareaforwhichthelocal
popula-tionhasaunitprobabilityofextinctionin1year,˛isthesurvivalrateoverdistance, xreflectstheseverityofenvironmentalstochasticityandydescribesthe coloniza-tionabilityofthespecies.Thelastrowreferstothehypothesisthatthetruenest densityisoverestimatedbyafactorof1.25.
Fixedbiological parameters x y ˛=2.3 A0=400m2 0.77724 0.00314 ˛=0.7 A0=400m2 0.77724 0.09608 ˛=3.9 A0=400m2 0.77724 0.00014 ˛=2.3 A0=40m2 0.52239 0.00314 ˛=2.3 A0=4000m2 1.65224 0.00314 ˛=2.3 A0=400m2 Overestimation 0.74035 0.00331 2.4. Strategydefinition
Theproposedevaluationmethodcomparesdifferentstrategies ofpestcontroldefinedintermsofspatialandtemporalallocationof treatmentsirrespectiveoftheadoptedcontroltechnique.Theonly requirementsarethatthepatchareaistheminimumspatialunitof interventionandcontroloperationstargettheentirepest popula-tionsinapatch.Weconsiderathreshold-basedpestmanagement, andthereforelocalpopulationabundanceiskeptundera thresh-oldwhateverdefined(e.g.,actionthreshold)asaconsequenceof aneffectiveintervention.Then,fromthemodellingpointofview, eachstrategyconsidersasetofoccupiedpatchesthatareforced tobeunoccupied(belowthethreshold,1→0)atselectedtimes.In otherwords,here‘strategy’meansagroupofpatchestobetreated (ifoccupied,i.e.abovethethreshold)andthestrategiesdifferin whichpatchesaretreatedandinthetimeofthetreatments.First, weevaluatetheeffectofdifferentallocationofthetreatmentsin spaceandtimeseparately,thenweshowbyanexamplehowto combinespaceandtime.
Forthespatialallocation,threekindsofstrategiesarecompared: (a)scatteredsites,(b)closesitesand,duetothegeometryofthe PPMmetapopulation(seeFig.2),(c)inlinesites.Threelevelsof interventionareconsidered,15%,30%and50%ofthetotalarea(low, mediumandhighinterventionlevel).Byfixingtheintervention areaweareimplicitlyconsideringabudgetconstrainedproblem. Table3liststheselectedpatchesfortheidentifiedstrategies.In eachinterventionlevel,strategiesaredenotedbyacapitalletter indicatingthelevel(L,MandH)andincreasinglynumbered.We supposethatasingletreatmentisappliedin2006(thefirstyear fol-lowingthesamplingperiod)andstrategiesoutcomesareevaluated in2008,forthesakeofillustration.
Forthetemporalallocation,wecomparethetreatment applica-tionin2006and2007tothetreatmentapplicationin2006and 2008.Strategiesoutcomesareevaluated in2010.Weapplythe treatmenttothetwolargestgroupsconsideredinthepreviously describedcaseofhighinterventionlevel(H7andH8,seeTable3). Finally,toevaluatetheroleofsynchronousandasynchronous spa-tialtreatments(Elliotetal.,2008)wecombinespecificpatternof spatialhighinterventionlevel andtemporal allocationof treat-mentfrom2006to2009.Inparticular,treatmentsare assigned totwosubgroupsofthepatchesinthestrategyalternatelyinthe fouryears.WeconsideredthesubgroupsofH7andH8indicatedin Table5.
Table3
ValuesoftheindexKLfordifferentpurelyspatialcontrolstrategiesofthePineprocessionarymothmetapopulationinthenationalparkofAspromonte(Calabria,Italy)under threelevelsofintervention:treatedareaof15%(low,L),30%(medium,M)and50%(high,H).Inmodelsimulationsthestrategiesareappliedin2006andevaluatedatyear T=2008.Boldtypedenotesthelowestvalue(thebeststrategy)foreachlevelofintervention.Parameters:A0=400m2,˛=2.3,x=0.77724andy=0.00314.
KL(S0)=144.3
Strategy Treatedsites Shortdescription Treatedarea(km2) KL
Lowlevel(∼15%ofthetotalarea)
L1 1,7,16,21,28 Scatteredsites 1.20 142.5 L2 1,9,15,30 Scatteredsites 1.18 143.5 L3 4,18 Scatteredsites 1.22 143.4 L4 3,7,21,22,23,27,28,29,30 Scatteredsites 1.25 131.4 L5 15,17,19,21 Closesites 1.21 143.5 L6 16,17,19,20 Closesites 1.21 142.4 L7 7,10,11,12,13,14,15,20 Closesites 1.23 142.9
L8 1,2,3,4,5,6,7,8,9,10,11,12,13 Inlineclosesites 1.31 119.5
L9 23,25,26,27,28,29,30,32 Inlineclosesites 1.23 110.3
Mediumlevel(∼30%ofthetotalarea)
M1 1,2,3,7,9,11,16,23,24,27 Scatteredsites 2.48 130.8 M2 9,12,19,24,26,31,32 Scatteredsites 2.91 132.7 M3 3,9,14,15,17,25,26,27 Scatteredsites 2.47 133.6 M4 3,4,5,14,25,27,28,31 Scatteredsites 2.79 128.1 M5 11,14,16,18,19 Closesites 2.42 140.6 M6 14,15,17,18,20,21 Closesites 2.70 133.0 M7 7,10,11,12,13,14,15,18 Closesites 2.35 134.6
M8 15,16,17,22,23,24,25 Inlineclosesites 2.46 123.1
M9 17,22,23,26,27,28,29,31 Inlineclosesites 2.67 120.4
Highlevel(∼50%ofthetotalarea)
H1 1,4,9,14,16,22,24,30,31,32 Scatteredsites 4.09 131.7 H2 1,4,9,11,15,16,18,21,24,27,30,32 Scatteredsites 4.28 133.7 H3 9,16,18,25,31 Scatteredsites 4.31 136.2 H4 2,5,9,11,18,21,22,25,31 Scatteredsites 4.22 137.3 H5 7,9,10,11,12,13,14,15,16,17,18,19,20,21 Closesites 4.33 81.3 H6 7,8,9,10,11,12,13,14,15,17,18,19,20,21 Closesites 3.75 77.2
H7 17,19,20,21,22,23,24,25,26,27,28,29,30,31,32 Inline,closesites 4.23 96.1 H8 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18 Inlinesites(upperedge) 4.34 82.7 H9 16,22,23,24,25,26,27,28,29,30,31,32 Inlinesites(loweredge) 4.05 94.2
ThestrategyS0={}meaningnotreatmentforanypatchhas
beenconsideredasareferencelevelaswell.
2.5. Kullback–Leiblercriterion
Pestmanagementstrategiesareorderedaccordingtoanindex measuringhowfarthemetapopulationstateresultingafterthe interventionisfromthetheoreticalreferencestateofpopulations inallpatchesbeingbelowthethreshold.AttimeT,weevaluatethe ‘global’metapopulationstatusbycomparingtheprobability distri-butionPoftherandomvectorofpresence/absenceı(T)=(ı1(T),...,
ın(T))predictedbyanystochasticmetapopulationmodel,withthe
probabilitydistributionP0indicatingthatpopulationsinallpatches
arecertainlybelowthethreshold:P0(ı1(T)=0,...,ın(T)=0)=1.The
distanceofPfromP0ismeasuredbytheKullback–Leibler
diver-gence,definedasKL(P)=−logP(ı1(T)=0,...,ın(T)=0).
InthecaseofthemultivariateIFMconsideredhere,KL(P)can notbeexplicitlyobtained,butitcanbeeasilyapproximatedby simulations.Thechainrule:
P(ı1(t+1),...,ın(t+1))
=P(ı1(t+1),...,ın(t+1)|ı1(t),...,ın(t))P(ı1(t),...,ın(t))
=P(ı1(t),...,ın(t))
×
iP(ıi(t+1)|ı1(t),...,ıi−1(t),ıi+1(t),...,ın(t)) (5)
allowstheapproximationofKL(P)byiteratingformula(5)uptoan initialprobabilityP(ı1(1),...,ın(1))andbyusingformulas(3)and
(4)forthetransitionprobabilities
P(ıi(t+1)|ı1(t),...,ıi−1(t),ıi+1(t),...,ın(t)) =
⎡
⎢
⎣
Ci(t) if ıi(t+1)=1 & ıi(t)=0 1−Ci(t) if ıi(t+1)=0 & ıi(t)=0 Ei if ıi(t+1)=0 & ıi(t)=1 1−Ei if ıi(t+1)=1 & ıi(t)=1⎤
⎥
⎦
.For theestimated parameters set,start thesimulation froman initialvector(ı1(1), ..., ın(1))ofstates (either0or 1)ofthen
localpopulations(thechoiceoftheinitialdistributionwillbecome clearerlateroninthetext).Foreachtimet=1,...,T−1,calculate Si(t)andCi(t),andgeneratearandomnumberUaccordingtoa
uni-formdistributionon(0,1).Finally,determineıi(t+1)bycomparing
EiandCi(t)toU: ıi(t+1)=
⎡
⎢
⎣
1 if ıi(t)=0 & U≤Ci(t) 0 if ıi(t)=0 & U>Ci(t) 0 if ıi(t)=1 & U≤Ei 1 if ıi(t)=1 & U>Ei⎤
⎥
⎦
.Repeatthisproceduremanytimes(here,100,000)andcalculatethe fractionofsimulateddynamicsendinginpattern(0,...,0)toobtain P(ı1(T)=0,...,ın(T)=0).Asthisprobabilityisusuallyvanishingly
small,abetterapproximationcanbeobtainedbysimulatingthe metapopulationforT−1yearstoestimateP(ı1(T−1),...,ın(T−1))
inaccordancewiththepreviouslydescribedprocedure,andthen multiplyingthis estimatedvaluebytheexact probabilityofthe finaltransition(eitherEior1−Ci(T−1)).Forfurtherdetailssee
Moilanen(1999).
TheframeworkdescribedinSection2.4.allowsthedescription ofeachstrategyintermsofthestateofthelocalpopulations.The
patchesoccupancyissimulated,andatthetreatmenttimesthe stateofthetreatingpatchesisforcedtobe0iftheyareoccupied (Step2inFig.1).Weassumeasinitialvectorinallthesimulations thelastobservedmetapopulationstate(year2005,seeTable1). Finally,foreachstrategythecorrespondingvalueKL(P)iscomputed (Step3inFig.1).Amongagivensetofstrategies,theoneswith lowerKLvaluesarethosemakingthemetapopulationstatecloser tothereferencestate(Step4inFig.1).
2.6. Sensitivityanalysis
We firstassess therobustness of theobtainedrankingwith respecttoindependentvariationsintheestimatesofthecritical patchareaA0andofthesurvivalrateofmigrants˛.Two
alterna-tivevaluesforA0are40m2and4000m2,correspondingto0.1and
10timestheadoptedvalueforA0respectively.Regarding˛,two
alternativevaluesare3.9and0.7,correspondingtoa2%and50% survivalat1kmdistance.
Wealsoanalysetheeffectofsamplingerrorsonstrategy rank-ing,byconsideringpossibleunder/overestimationofthetruevalue. Let mi bethe sampledvalueat patchi andm∗i thetrue value:
wesuppose mi=0.75m∗i in thecase of underestimation, while
mi=1.25m∗i inthecaseofoverestimation.
Summingup,sixmorecompleteanalysesarecarriedout(Step 5 in Fig. 1)by considering the original data and four alterna-tiveparameter sets:(A0=400m2,˛=3.9), (A0=400m2, ˛=0.7),
(A0=4000m2,˛=2.3)and(A0=40m2,˛=2.3),andthetwonew
datasetswiththeoriginalparametersset(A0=400m2,˛=2.3).
3. Results
3.1. Testingcriteriaformetapopulationdefinition
Localpopulationabundancehasa verylow correlationwith patcharea,varyingfrom−0.16in2003to0.09in2004and2005. Randomizationtestsbasedonatleast4999randomizationsare usedtotest thesignificanceoftheuni-andbivariateMoran’sI andoftheMantelcorrelograms(LegendreandLegendre, 1998). TheMantelcorrelogramiscomputedfordifferentdistance matri-ces.Allthecorrelogramsareinterpretedfordistances≤15,000m. Noneofthetestshighlightsanysignificantautocorrelation (proba-bilitylevelequalto0.01)ofandbetweenPPMpestabundance.The obtainedresultspartiallyagree withSamalensand Rossi(2011) who,unlike us,considered a spatially homogeneouslandscape. Fig.3showsthecorrelogramsbasedonequaldistanceclasses. Anal-ogousresultsareobtainedbyconsideringequalsizedclasses.Thus, thecriterion(d)ofasynchronousfluctuationsoflocalpopulations (seeSection2.2)canbeconsideredassatisfiedandtheprerequisite fortheanalysis(seeFig.1)aswell.
3.2. IFMparameterization
Theestimatedvaluesforparametersxandyare0.77724and 0.00314, respectively. The 95% confidence intervals, computed bypseudo-likelihood(see Moilanen,1999), are(0.46,1.55) and (0.0011,0.0783)forxandy,respectively.Thegoodnessoffit(see Step1inFig.1)ischeckedbystarting100,000simulationsfrom thefirstyearofdata(2003,seeTable1)andsimulatingthe pres-ence/absenceforthetwoconsecutiveyears(2004and2005).From thesesimulations,wecomputethe95%confidenceintervalforthe fractionofoccupiedsitesin2004and2005.Fig.4showsthatin 2004theobservedmeanoccupancyisclosetotheupperboundof theconfidenceinterval,whilein2005italmostcoincideswiththe intervalcentre.
Table2comparestheobtainedparametersvaluestothevalues estimatedunderthedifferenthypothesesconsideredforsensitivity
Fig.3. UnivariateMoran’sIcorrelogramforthepestdensityofthePine proces-sionarymothonthethreeyearsofdata(top).Mantel’scorrelogrambasedonthe EuclideandistanceforthepestdensitiesofthePineprocessionarymoth(down). Spatialautocorrelationisneversignificant,accordingtothepermutationstest(4999 permutations)andtheHolm’scorrectionmethod.
analysisinSection2.6.Sensitivityisassessed independentlyfor each parameter. Changes in thevalue of ˛ affect parameter y, becauseconnectivityonlyismodified(seeformula(4)).Changesin thevalueofA0affectparameterx,becauseextinctiononlyis
modi-fied(seeformula(2)).Sensitivityanalysiswithrespecttosampling errorsis carriedoutby considering˛=2.3andA0=400m2.The
hypothesis of underestimation of the truenest abundancecan notbeanalysedbecausethishypothesisproducestheabsenceof turnover(i.e.,extinctionisnotobserved)and,therefore,prevents theestimationofx.Underthehypothesisofoverestimationofthe truenestabundanceweobtainx=0.74035andy=0.00331,which areclosetotheestimatedvaluesfromtheobserveddata.
2003 2004 2005 0.2 0.4 0.6 0.8 1
Fig.4.ThegoodnessoffitestimatedIFMforthePineprocessionaymothinthe Aspromonte(Calabria,Italy)ischeckedbycomparingthefractionofoccupiedsites atyears2004and2005(blackdiamonds)totheir95%confidenceintervals(gray verticalbars).The95%confidenceintervalswereobtainedfrom100,000simulations ofoccupancy,startingfromthefirstyearofdata,2003.
Table4
Sensitivityanalysis.ValuesoftheindexKLforthecontrolstrategiesofthePineprocessionarymothmetapopulationinthenationalparkofAspromonte(Calabria,Italy) describedinTable3forthedifferentparametersetsdefinedinSection2.6.Thelastcolumnreferstothehypothesisthatthetruenestdensityisoverestimatedbyafactorof 1.25,accordingtoSection2.6.BoldtypedenotesthelowestKLvalue(thebeststrategy).
Strategy KL ˛=2.3 A0=400m2 KL ˛=2.3 A0=40m2 KL ˛=2.3 A0=4000m2 KL ˛=3.9 A0=400m2 KL ˛=0.7 A0=400m2 KL Overest. ˛=2.3 A0=400m2 S0 144.3 135.5 183.7 148.2 141.8 135.9
Lowlevel(15%ofthetotalarea)
L1 142.4 131.7 181.3 143.2 139.6 135.3 L2 142.5 133.8 182.3 143.1 136.7 134.4 L3 141.4 133.4 183.2 143.1 141.7 135.6 L4 129.9 117.6 171.9 131.1 126.6 120.7 L5 139.6 133.7 181.2 142.8 140.6 135.3 L6 140.9 132.6 178.7 140.4 140.9 134.1 L7 142.9 134.5 183.5 144.1 141.5 135.9 L8 119.5 110.4 156.5 121.4 122.9 114.5 L9 118.6 105.7 153.1 111.9 111.9 113.0
Mediumlevel(30%ofthetotalarea)
M1 130.8 120.8 166.2 134.9 125.9 123.2 M2 132.7 125.2 166.4 134.6 128.7 125.9 M3 133.6 126.5 169.8 132.2 130.1 127.6 M4 128.1 124.2 167.3 136.0 123.6 125.6 M5 140.6 133.7 179.4 142.4 140.0 135.3 M6 133.0 115.9 164.8 131.0 133.2 119.9 M7 134.6 135.0 180.7 147.3 130.0 134.1 M8 123.1 116.4 154.3 124.5 126.6 117.2 M9 120.4 111.5 152.0 120.1 122.4 113.5
Highlevel(50%ofthetotalarea)
H1 131.8 123.2 165.5 132.7 122.8 124.6 H2 133.2 123.3 168.0 132.3 124.6 125.8 H3 135.0 129.3 172.3 138.1 133.2 129.7 H4 136.8 128.0 174.9 137.6 130.0 130.4 H5 80.3 76.4 100.1 85.4 73.5 77.9 H6 68.9 74.2 100.6 79.0 76.8 75.1 H7 96.5 91.5 119.7 97.4 97.4 91.9 H8 82.7 77.6 112.3 87.5 78.1 74.1 H9 94.3 89.6 116.3 126.5 95.4 90.0
Thecomputationtimetocarryout100,000simulationsofa 1-yearmetapopulationoccupancybyMathematica5.0is123s,Intel®
CoreTMi7-2720QMprocessor.
3.3. Strategyranking
TheKLvaluescalculatedin2008forthethreekindsofstrategies definedbyallocatinginspaceoneonlytreatmentin2006(see Sec-tion2.4)arepresentedinTable3.Resultsaregroupedaccordingto theinterventionlevel,fromlowtohigh.StrategyS0hasthe
high-estKullback–Leiblervalue(144.3),indicatingthatallthestrategies turnouttobeeffectiveinpestreduction. Allbutonescattered strategieshavehigherKLvaluesinallthethreeintervention lev-els.Forthelowinterventionlevel,thesevaluesarenotsignificantly differentfromKL(S0).Forlowandmediuminterventionlevels,the
lowestKLvaluesarereachedbythein-linestrategies.Bothclose andin-linestrategiesshowlowKLvaluesatthehighintervention level.Withinthislatterlevel,thebeststrategyistheclose-type strategyH6forminglikeaclump,clearlysplittingtheremaining patchesintwoparts.Thefactthatastheinterventionlevel(i.e.,the treatedarea)increases,closeandin-linestrategiesprovidecloseKL valuesisanevidenteffectofboththespatialnetworkshapeandthe limitednumberofpatches(seeFig.2).Therankingdoesnotshow anyspecificroleofthenumberoftreatedsitesandofthearea(see Table3).Thiscanbeclearlyseenwithineachlevelandacrossthe levelsaswell.
Theobtainedresultsindicatethatthemosteffectivestrategies aretheonesreducingpatchconnectivity.Thismightalsobedue tothespecificfeatureoftheIFM,a modelessentiallybased on
patchconnectivity.Thisinterpretationisconfirmedbytheresults ofthesensitivityanalysis,showninTable4.Thefactthatforthe highinterventionlevelbothstrategiesH5andH6canbeoptimal, accordingto theparameterset,is dueto thefactthat thetwo strategiesareverysimilaranddifferbyasmallareaonly.
Theroleofthetemporalpatternoftheinterventionis evalu-atedunderthehypothesisofafixedbutdifferentlydistributedon timecontroleffort.Efficacyofthestrategiesisassessedatafixed timeintervalafterthelastobservedyear(2005,seeTable1).The resultsshown inTable5 revealtheimportanceof thetimelag betweenthelasttreatmentandthetimeofevaluation.Forthetwo assessedstrategies,H7andH8,themosteffectivetemporalpattern istheonewithtreatmentin2006and2008.Asgeneralinsight,it mightbeconsideredthatpestcontrolinterventionin metapopu-lationsystemsliketheoneunderstudydoesnothavelonglasting effect.
Finally,anexampleofhowtocombinespatialandtemporal pat-ternsofinterventionisprovided.ThelowestvalueofKLisobtained forH8c(seeTable5),astrategycombiningtheeffectof concen-tratingthemajorcontroleffort(i.e.thetreatedarea)closetothe assessmenttimewiththeadvantagesofferedbyaclumpstrategy. Analogousresultsareobtainedbycarryingoutthesensitivity analysisoftemporal and spatio-temporalstrategies (notshown heretosavespace).
4. Discussion
For area-wide pest management monitoring population dynamics and realistic models able to forecast variation of
Table5
ValuesoftheindexKLfordifferentthetemporalandspatio-temporalcontrolstrategiesofthePineprocessionarymothmetapopulationinthenationalparkofAspromonte (Calabria,Italy).Inmodelsimulationsthestrategiesareappliedatdifferentpointintimesfrom2006to2009andevaluatedatyearT=2010.Foreachyear,thetreatedarea isindicatedbyusingthesamenotationsofTable3Boldtypedenotesthelowestvalue(thebeststrategy)forthetwotypesofintervention.Parameters:A0=400m2,˛=2.3,
x=0.77724andy=0.00314.
Shortdescription 2006 2007 2008 2009 KL
Temporaltreatmentallocation
H7 H7 – – 99.4
H7 – H7 – 96.1
H8 H8 – – 91.5
H8 – H8 – 83.0
Spatio-temporaltreatmentallocation
H7a 17,19,20,21,22,23,24,25,26,27 28,29,30,31,32 17,19,20,21,22,23,24,25,26,27 28,29,30,31,32 97.2 H7b 28,29,30,31,32 17,19,20,21,22,23,24,25,26,27 28,29,30,31,32 17,19,20,21,22,23,24,25,26,27 91.5 H7c 19,20,21,26,28,31 17,22,23,24,25,27,29,30,32 19,20,21,26,28,31 17,22,23,24,25,27,29,30,32 98.2 H7d 17,22,23,24,25,27,29,30,32 19,20,21,26,28,31 17,22,23,24,25,27,29,30,32 19,20,21,26,28,31 102.1 H8a 7,11,17,18 1,2,3,4,5,6,8,9,10,12,13,14,15 7,11,17,18 1,2,3,4,5,6,8,9,10,12,13,14,15 112.4 H8b 1,2,3,4,5,6,8,9,10,12,13,14,15 7,11,17,18 1,2,3,4,5,6,8,9,10,12,13,14,15 7,11,17,18 117.3 H8c 1,2,3,4,5,6 7,8,9,10,11,12,13,14,15,17,18 1,2,3,4,5,6 7,8,9,10,11,12,13,14,15,17,18 69.5 H8d 7,8,9,10,11,12,13,14,15,17,18 1,2,3,4,5,6 7,8,9,10,11,12,13,14,15,17,18 1,2,3,4,5,6 82.7
populationabundancebytakingintoaccounthabitat fragmenta-tion(TscharntkeandBrandl,2004)arerequired.Spatiallyexplicit metapopulationmodelscanincorporatecomplexdetailson land-scapestructure andconnectivity,providinginsightintotherole of habitat fragmentation on pest populationdynamics. In par-ticular,the Incidence FunctionModel takesadvantage of using only a few parameters and some plausible assumptions on populationprocessesthataresatisfiedformostpestpopulations. Thedisadvantage of possiblesimplifications introduced by this metapopulation model (Baguette, 2004) is outweighed by the advantageof enablingto dealwithcomplex systemsin a cost-efficientmannerwithsimplemodellingtools(Bodinietal.,2008). Metapopulationmodelparameterscanbeestimatedfrom snap-shotdataorlimitedtimeseriesdataacrossthemetapopulation. Subsequently,theparameters canbeused toiteratethemodel usingdifferentscenariosofthespatialandtemporaldistributionof treatmentsforagivensetofpatches.Themodellingapproachwe usedisparticularlyrelevantifothermodellingapproachescannot beappliedduetoscarcedataonthepestbiology(e.g.,the biode-mographicratefunctionsandbetween-patchesmovement),asin thecase of thePineprocessionary moth(Nethererand Schopf, 2009).
Theoutcomesofmetapopulationmodellingcansuggestgeneral managementrecommendations.However,conservationliterature warnsusthattohelpmanagersinachievingthebestresultswith limitedresources,populationecologyshouldbemergedwith deci-sion theory (Possingham et al., 2001). Our work shows the usefulness of linking metapopulation models and the Kullback–Leiblerdivergencetorepresenttheregionaldynamics ofpestpopulationsandevaluatetheconsequencesofshort-term controltechniquesapplicationattheregionallevel.Theproposed methodrequiresrathersimpleandfastcomputations,andallows toefficiently use data onpest populationdynamics to predict theoutcomesofcontrolinterventionsunderconditionoflimited knowledge.The methodhasbeenapplied topestmanagement strategyevaluationforthePineprocessionarymothpopulations intheCalabrianforestsinhabitingspatiallyseparatedpatches.We preliminarilyprovedbyusingtoolsfromspatialstatistics(Liehbold andGurevitch, 2002)that Pineprocessionarymothpopulations undergo asynchronous fluctuations. This enables to apply a metapopulationapproach (Hanski, 1997). Themethod allows a comparativeassessment of theefficacy of different predefined typesofstrategydefinedintermsofspatialandtemporalpatternof
allocationoftheinterventions.Withineachtypeofcontrol strate-giesthemethodallowsaclearidentificationofthebestone.The resultsweobtainedforthePineprocessionarymothpopulationsin theCalabrianforestsindicatethatthebeststrategiesaretheones reducingmetapopulationconnectivity.Theimportanceofregular applicationof theinterventionand thedifficultyin achievinga long lasting controleffect emerged as a generalresult as well. Thereare alsoindications that anasynchronous distributionof thetreatmentcouldcontributetothecontrolofthemoth.Atleast inthecaseextensively studiedhereofpurely spatialtreatment allocation,sensitivityanalysisindicatesthatthebeststrategyin each intervention level is invariant withrespectto parameters variations.ConsistentresultslikethoseobtainedforthePPMin Aspromontemean thatthe methodis ableto providea robust indicationtomanagers.Nevertheless,managementprogrammes areoftenvulnerabletonaivelyusingmathematicalandsimulation models inappropriately (Walsh et al., 2012; Peck and Bouyer, 2012).Testingthemetapopulationhypothesis andverifyingthe goodnessoffitoftheselectedmodel(theIFM)arefundamental stepsofthedecisionprocessweproposehere.Moreover,aswe analysedarealsystemwithalimitednumberofsitesandaspecific patches geometry along a transect, the possibility of deriving completelygeneralindicationsformanagersonthecomparative efficacyofthetestedpattersislimited.Theperformancesofthe KLdivergenceonmore generalnetworkswillbepartof future researches, together with improvements of the approximation methodforKLcomputation.
Thoughfocusingonthresholdbasedcontrol,themoregeneral assumptionsconsideredinthispaperallowtoextendthemethod toamorecomprehensivesystemofcompatiblecontroltechniques asusedinIntegratedPestManagementprogrammes(Koganetal., 1999)fortheevaluationofmetapopulationmanagement scenar-ios. The IFM is suitable for representing global dynamics with respecttoathresholddensityratherthanwithin-patchdynamics. Consequently,theKLvaluesobtainedfromtheevaluationofIFM scenariosareusefulforquantifyingthebenefitsresultingfromthe applicationofmanagementstrategiesattheglobalratherthanthe locallevel.
Inpestmanagement,decisionmakersarefacedwithdifferent typeofuncertaintiesaboutbiologicalandenvironmentalprocesses, andobservationaluncertainty(Sheaetal.,2002).Infact,sampling methodologyandtechniquesproducebothrandomand system-atic errors in the estimates of populationdensity (e.g., Schaub
etal.,1988)anddifferenttypesoferrorscanoccurin metapop-ulation data, affecting parameter estimation (Moilanen, 2002). Thisproducesuncertaintiesintheoutcomeofinterventionsand exposesthedecisionmakertorisk(e.g.,RouxandBaumgärtner, 1988).Thesensitivityanalysisappliedinourstudyshowshowto accountforvariabilityofbothmodelparametersandpopulation abundanceestimatesintheevaluationofpestcontrolstrategies (Drechsler,2004).Sensitivityanalysisisalsoimportantbecauseit mayshowpossiblerestrictionstotheapplicabilityoftheadopted model,andindicatedifferentrankingofcontrolstrategies.
Aformalcost-benefitanalysiscertainlyprovidesabetter sup-porttomanagementstrategyevaluationwithrespecttomultiple criteria.Apreliminaryextensionoftheproposedmethodologyto includeacostfunctionrelatedtothetreatedareahasbeen pro-posedinBodiniandGilioli(2009).Themethodcanbeextended to consider the optimal selection of a sequence of strategies underconstraintslikecosts,maximuminterveningareaand pro-tected(i.e.,pest-free)areascreation,forinstance.Thisalsoallows toconsider long-termcontrol strategies that assume particular importanceinarea-widepestmanagement.However,this exten-sionisnotstraightforwardastheminimizingfunctionisnotlinear andthereforeadhocalgorithmsshouldbedeveloped.
Acknowledgments
WearegratefultoMarkHunterforhisencouragingcomments andusefulsuggestions.Theauthorswouldliketothankthe anony-mousreviewersfortheirvaluablecommentsandsuggestionsto improvethequalityofthepaper.
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