J Anim Ecol. 2017;1–14. wileyonlinelibrary.com/journal/jane  | 1© 2017 The Authors. Journal of Animal Ecology © 2017 British Ecological Society

Received:28June2016  |  Accepted:2March2017DOI: 10.1111/1365-2656.12700

A L L E E E F F E C T S I N E C O L O G Y A N D E V O L U T I O N

Differential dispersal and the Allee effect create power-law behaviour: Distribution of spot infestations during mountain pine beetle outbreaks

James A. Powell1  | Martha J. Garlick2 | Barbara J. Bentz3 | Nicholas Friedenberg4

1DepartmentsofMathematics&StatisticsandBiology,UtahStateUniversity,Logan,UT,USA2DepartmentofMathematics&ComputerScience,SouthDakotaSchoolofMinesandTechnology,RapidCity,SD,USA3USDA-FSRockyMountainResearchStation,ForestrySciencesLab,Logan,UT,USA4AppliedBiomathematics,Setauket,NY,USA

Correspondence JamesA.Powell Email:jim.powell@usu.edu

HandlingEditor:AnnaKuparinen

Abstract1. Mountainpinebeetles(MPB,Dendroctonus ponderosaeHopkins)areaggressivein-sectsattackingPinushosttrees.Pinesusedefensiveresintooverwhelmattackers,creatinganAlleeeffectrequiringbeetlestoattackenmassetosuccessfullyrepro-duce.MPBkillhosts,leavingobservable,dyingtreeswithredneedles.LandscapepatternsofinfestationdependonMPBdispersal,whichdecreaseswithhostden-sity.Awayfromcontiguouslyimpactedpatches(lowbeetledensities),infestationsarecharacterizedbyapparentlyrandomspots(of1–10trees).

2. ItremainsunclearwhetherthenewspotsarespatiallyrandomeruptionsofalocallyendemicpopulationoramodeofMPBspread,withspatialdistributiondeterminedbybeetlemotilityandtheneedtoovercometheAlleeeffect.

3. Todiscriminatebetweenthehypothesisofpopulationspreadversusindependenteruption,amodelofspotformationbydispersingbeetlesfacingalocalAlleeeffectisderived.Themodelgives rise toan inversepowerdistributionof travel timesfromexistingoutbreaks.Using landscape-levelhostdensitymaps in threestudyareas, an independently calibratedmodel of landscape resistance depending onhostdensity,andaerialdetectionsurveys,wecalculatedyearlymapsoftraveltimetopreviousbeetle impact. Isolatedbeetle spotswere sortedby travel time andcomparedwithpredictions.Randomeruptionoflocallyendemicpopulationswastestedusingartificiallyseededspots.Wealsoevaluatedtherelationshipbetweennumberofnewspotsandlengthoftheperimeterofpreviouslyinfestedareas.

4. Spotdistributionsconformedstronglytopredictedpower-lawbehaviour.Thespa-tiallyrandomeruptionhypothesiswasfoundtobehighlyimprobable.Spotnum-bersgrewconsistentlywithperimeterofpreviouslyinfestedarea,suggestingthatMPBspreadlongdistancesfrominfestationboundariesviaspotsfollowinganin-versepowerdistribution.

5. TheAlleeeffect inMPBthereforeaccelerates, rather than limits, invasion rates,contributing to recent widespread landscape-scale mortality in western NorthAmerica.

K E Y W O R D S

barkbeetle,Dendroctonus ponderosae,patchyspread,power-law,traveltime

2  |    Journal of Animal Ecology POWELL Et aL.

1  | INTRODUCTION

TheAlleeeffectistheacceleratingimpactofconspecificnumbers/densities on some aspect of fitness for small populations (Allee,1931).At the levelofpopulations, a “demographic”Alleeeffect isthepositivedensitydependenceofpopulationgrowthrateaspop-ulation size grows from zero (Stephens, Sutherland,& Freckleton,1999). Small populations experienceAllee effects through severalgoverninginteractionsthataffectindividualfitness,includinggroupforaging,defenceagainstpredatorsandmatefinding(Lande,1998).A “strong”Alleeeffectoccurswhengrowth ratesarenegative forpopulations below a critical threshold (Wang & Kot, 2001). Alleeeffectsarecommonacrosstheanimalkingdomandalsoappearfre-quentlyinotherorganisms(Taylor&Hastings,2005andreferencestherein).

Allee effects are often associatedwith “patchy” spread or in-vasion (Morozov, Petrovskii, & Li, 2006; Petrovskii, Morozov, &Venturino,2002).Patchinesscanbeanemergent,passiveresponsetoheterogeneityofspaceand/orstochasticityofdispersal.Forex-ample,geneticdiversityofSpartina alternifloraclumpsinPacifices-tuaries improves seedproduction in plants throughhybridvigour,and consequently occasional accidents of dispersal create diverseclumpswithmuchhigher reproduction rates than individuals, cre-ating anAllee effect (Taylor, Davis, Civille, Grevstad, & Hastings,2004).Patchinesscanalsodevelopduetothegrowthof instabili-tiesandsubsequentfilteringbytheAlleeeffect(Wang,Shi,&Wei,2011),amechanismwhichalsodependsonpassivedispersal.Gypsymoths(Lymantria dispar)spreadbystratifieddispersal(i.e.short-andlong-distancedispersaloccurviadifferentprocesses,withseparatedispersal kernels), establishing stable, isolated patches only afterepisodiclong-distancedispersaleventsthatsurvivethesubsequentAlleegauntlet(Sharov&Liebhold,1998).Environmentalheteroge-neitycanleadtolocalizedpopulationsexceedingtheAlleethresh-old,suchasinpatchesoffavourablehabitatwheretheAlleeeffectislocallyreduced.Inallofthesecases,patchyspreadiscreatedbypassive dispersal, and theAllee threshold is exceeded indiscrimi-nately, resulting inpatchestablishment.Asmorepatches are cre-ated they coalesce and become a source population (Liebhold &Tobin,2008).

SpeciesfacingstrongAlleeeffectswilladapttoovercomethem.Allee (1931)himself recognizedthataggregationwastheprimarymechanism by which species increase survival rates and arguedthatonlyinthesimplestorganismswouldaggregationbehappen-stance.Except in rare, small,well-mixedsystems,populationsarelikelytohaveactivedispersaladaptationsforaggregating individ-uals. Understanding the population-level expression of the Alleeeffect will require understanding the aggregation mechanisms.Example mechanisms include habitat selection in heterogeneousenvironments(Greene,2003),congregationviadensity-dependentdispersal (Turchin,1989)andpheromoneresponses inarthropods(Wertheim, van Baalen, Dicke, & Vet, 2005). Aggregations thatresult from active dispersal at lowpopulation densitieswill drawdownsurroundingpopulationsandleadtospatialtrade-offswhich

maybeexpressed independentlyfromthoseoccurringasaresultof landscapeheterogeneity.Moreover,whenthe landscapeoffersvaryingresistancetomovement,aggregationshouldbemorelikelyinpatchesthatareeasiertoaccessorcausebottleneckstoother-wise fluidmovement. Thus, patterns of patchy spread in speciesactivelyaggregating toovercomeAlleeeffectscouldbemarkedlydifferentthaninspecieswithindiscriminatepatchcreation.

The mountain pine beetle (MPB, Dendroctonus ponderosae Hopkins) provides an excellent opportunity to study active patchformationinaspeciesfacinganAlleeeffect.MPBisaneconomicallyand ecologically important native species that has caused signifi-cantmortalityinPinusforestsacrossthewesternUnitedStatesandCanada (Meddens,Hicke,&Ferguson,2012).Dueto itseconomicimpact, there isan impressiveamountofscientific informationonMPB, and it has been established that active dispersal processesat large and small spatial scales play a central role in populationoutbreakdynamics(Logan,White,Bentz,&Powell,1998;Powell&Bentz,2014).

Unlike many phytophagous insects, successful MPB reproduc-tionusuallyresultsindeathofallorpartofthehost.Hosttreeshaveevolvedvaryingchemicalandresinresponsesthatreducevulnerabil-itytoattackbybarkbeetlesandtheirfungalandbacterialassociates(Booneetal.,2013;Kane&Kolb,2010;Raffa,Powell,&Townsend,2012).Vigorous,well-defendedtreesrequirerapidattackandcoloni-zationbyalargenumberofbeetles(i.e.amassattack)tooutpacetreeresponses(Berryman,Dennis,Raffa,&Stenseth,1985), leadingtoastrongAlleeeffectforthebeetles.Conversely,treesstressedbybioticandabioticagentshavea reducedcapacity fordefenceandcanbeovercomebylownumbersofbeetles(Raffa,Aukema,Erbilgin,Klepzig,&Wallin,2005;Safranyik&Carroll,2006).Thebetterdefended,morevigorous trees tend tobe larger andhavehighernutritional qualitythereby leading toapositive feedbackasbeetlepopulationdensityincreases (Boone,Aukema,Bohlmann,Carroll,&Raffa, 2011;Raffaetal.,2008).

EpidemicMPBpopulationdynamicsarewell-describedbyphenol-ogy,host-dependentdispersalandtheAlleeeffect (Powell&Bentz,2009,2014).Beetlesemergedailyfrompreviouslyinfestedhostsanddisperse with motility decreasing exponentially with host density.Where the dispersed population exceeds the Allee threshold newhosts are successfully colonized. Powell and Bentz (2014) showedthatthiscombinationofmechanismsdescribes85%oftheobservedspatialpatternofbeetle-killedtreesonkilometrescales.Interestingly,wherethePowellandBentz(2014)modelmissedpredictingobservedimpacts(approximately8%ofthelandscape),thetypeofimpactwassmall,isolatedspots.Thesespotsrepresentatrivialportionofagivenyear'sMPBfootprintduringanepidemic,buttheevolvingpatterninsubsequentyearsdependsstronglyonthedensityandspatiallocationofspotsacrossthelandscape.

Regardless of whether satellite spots are caused by beetlesdispersingfromthemainbodyofanoutbreak,spot initiationandgrowthdependsonMPBmovement,conditionedbyhosttreeavail-abilityandsize,MPBpopulationsize,weatherandbehaviour-mod-ifying chemicals (Mitchell & Preisler, 1991; Safranyik, Linton,

     |  3Journal of Animal EcologyPOWELL Et aL.

Silversides,&McMullen,1992).Hostkairomonesplayasignificantrolebysignallingtreesthatareunderstresscausedbybioticandabioticfactors(Chapman,Veblen,&Schoennagel,2012;Goheen&Hansen,1993).Atlowerbeetledensities,afteraweakenedtreeisfound, aggregation is facilitatedbypheromones,which throughasynergistic reactionwith host defensive compounds attract addi-tionalbeetles(Raffaetal.,2005),resultinginmassattacksonasin-gletree.Followingaggregationtooneorseveralweakenedtrees,thenumberofadultbeetlesnecessarytoovercometheAlleeeffectonmorevigorouslydefendedhosttreesbecomeavailableandat-tackingbeetlesswitchtonearbytrees,creatingaspotwithonetoseveralkilledhosts.Wehypothesizethatspotcreationisnotindis-criminate,butinsteadthatthespatialdistributionofMPBspotsaretheresultofdispersersleavingtheperimeterofpreviouslyaffectedareas,spreadingdifferentiallyduetohost-dependentmotilityandaggregating locallytomassattacktrees.Furthermore,wesuggestthat thespatialdistributionofspots reflects landscaperesistancetobeetlemovement.

MPB are not likely to disperse to distant focus trees if accept-ablehostsareencounteredfirst.PowellandBentz(2014)calibratedamodelofMPBmovementbasedonhosttreedensity-derivedbee-tlemotilitywherein resistance to beetlemovement increases expo-nentially with host tree density. Assuming that potential spot fociarerandomlydistributedpercapitaamonghosts,wearguethat thedistributionofspotsshouldfollowapower-lawdistributionintraveltime along landscape paths of least resistance frombeetle sources.Thepowerlaw,inwhichspatialprobabilityofspotoccurrenceispro-portionaltoanegativepoweroftraveltime,arisesbecauseincreasingencounter rates forweakened focal trees correlatewith decreasingmotilityashostdensitiesincrease.Moreover,becausetheperimeterofpreviouslyinfestedareaswouldbetheprimarysourceofdispersers,wehypothesizethattheyearlynumberofnewspotsshouldscalewithperimetersize.

Wetest thesehypothesesusingaerial surveysofMPB-causedtreemortality and host tree density data for three study areas inIdaho, Washington and Colorado. Annual occurrence of isolatedspotswere identified and travel time between each spot and thenearest contiguous area ofMPB-killed trees that could provide abeetlesourcewas recorded.Thedistributionofobservedspots intraveltimewasdetermined,andthenumberofspotscomparedwiththeperimetersizeofpreviouslyinfestedareas.Analternatehypoth-esis,thatspotsoccurasrandomeruptionsofanendemicbeetlepop-ulation,wasalsoexamined.Theobservednumberofspotseachyearwereseededspatiallyatrandomandtestedforpossiblepower-lawbehaviour.We found that thepower-lawhypothesiswas stronglysupportedacrossmanyyearsofobservations,while randomerup-tionwashighlyimprobable.Yearlyspotnumbersscalewitharoughmeasure of the size of the perimeter of previous infested areas,supportingthecontentionthatbeetlesdispersefromtheedgesofmajor infestations to invade new areasvia spots.Accelerating in-vasionratesareoneconsequenceofpower-lawdispersal,whichisconsistentwithobservationsofMPB'sexplosivespreadinwesternNorthAmerica.

2  | MATERIALS AND METHODS

2.1 | Study areas

2.1.1 | Sawtooth study area

TheSawtoothstudyarea incentral Idaho (Figure1) isa rectangularregion fromapproximately44◦22′N to43◦44′N (∼60km) and115◦

10′Wto114◦28′W(∼30km),comprisingover180,000ha,includingtheSawtoothNationalRecreationArea.Asinglehost,lodgepolepine,predominatesfrom1,650to2,000mandgrowsinstandswithrela-tivelyhomogeneousdemographicsatthelowestelevations.Theland-scape is characterizedby a valley and surroundingmountains,withelevationsfrom1,650to3,600m.Vegetationtypesrangefromshrubandgrasslandstoconiferousforestswithlodgepolepine,Douglasfir(Pseudotsuga menziesii (Mirb.) Franco), subalpine fir (Abies lasiocarpa [Hook.]Nutt),andwhitebarkpine(P. albicaulisEngelm.)athigheral-titudes.Densitiesofpineaverage450trees/ha,althoughthevalleyincludesmanydensestandsof1,000trees/haaswellasmeadowsandpasturelandwithnohosts.Extensivebarrenareasexistabovetree-lineatthehighestelevations.Between1995and2005,aMPBout-breakoccurredthroughouttheSawtoothstudyarea,impactingmorethanathirdofthepinehosttype(Pfeifer,Hicke,&Meddens,2011).

2.1.2 | Chelan study area

TheChelanstudyareainnorthernWashington(Figure1)encompasses446,000ha, fromapproximately47◦56′N to48◦35′Nand from119◦

F IGURE  1  Studyareasusedinthispaper.Allstudyareashadatleast10yearsofcontinuousaerialdetectionsurveyformountainpinebeetleimpact.Anoutbreakoccurredbetween1995and2005intheSawtoothandbetween2001and2010inColorado.IntheChelanstudyarea,multi-modalimpactoccurredover20years,peakingin2008

WA

ID

CO

Sawtoothstudy area

Coloradostudy area

Chelanstudy area

CA

MT

AZ

NV

NM

OR

UT

TX

WY

ND

SD

NE

KS

0 250 500 km

4  |    Journal of Animal Ecology POWELL Et aL.

52′W to120◦44′W.Elevations range from336mat LakeChelan topeaks at 2,700m. The study area is comprisedof public andprivatelands, including portions of the Methow Valley and Chelan RangerDistricts,Okanagan-WenatcheeNational Forest andNorthCascadesNational Park. TheMethowRiver drainage characterizes the easternhalf of the study area. Coniferous vegetation within the study areaincludeponderosapine(P. ponderosa),lodgepolepine,whitebarkpine,Englemannspruce(Picea englemanniiParry)andDouglasfir,withhostpinetreesaveraging500trees/ha.TheChelanstudyareaboundarywaschosentoencompasspinevegetationsusceptibletoMPB infestationandactiveMPBpatchesbasedongroundsurveys.MPBimpactbeganinthelate1990sandpeakedin2008(Crabb,Powell,&Bentz,2012).

2.1.3 | Colorado study area

The Colorado study area contains over 4,380,000ha in northernColorado, including Rocky Mountain National Park, North Park andseveralofColorado'shighestpeaks (Figure1). Elevations range from1,700mon theFrontRangewestofFortCollins toover4,300matLongsPeak.ThestudyareabeginsroughlyatInterstate70inthesouthand reaches in the north to approximately 41◦50′N (into southernWyoming),intheeastto105◦0′W,andinthewestto108◦0′W,encom-passing portions of theMedicineBow-Routt andArapaho-RooseveltNationalForests.ThesouthernboundaryalongtherouteofI-70waschosen as a natural break inMPB impacts, andwestern boundarieswere chosen to encompass impacted regions as far to the west asGlenwoodSprings.Conifersincludelodgepoleandlimberpines(P. flexi-lis),Engelmannspruce,subalpinefirandDouglasfir,andaveragepinehostdensitiesare780trees/haacrossthearea.SignificantMPBimpactbeganintheearly2000sandpeakedin2007(Crabbetal.,2012).

2.2 | Data sources

2.2.1 | Pine density data

Spatiallyexplicitdatasetsofpinedensityat30-mresolutionwerede-rivedforthestudyareasusingexistinggeospatialdatasetsofvegeta-tioncompositionandstructure.Briefly,fortheSawtoothandColoradostudyareasforestdensity(treesperhectare>2.54cmDBH)at250-mresolution,developedbytheUSDAForestServiceFIA(Blackardetal.,2008),weredownscaledto30-mresolutionusingdatafromtheinter-agency Landscape Fire and Resource Management Planning ToolsProject(LANDFIRE).DatafromtheGNNFireproject(LEMMA,2005;Pierce,Ohmann,Wimberly,Gregory,&Fried,2009)wereusedtode-rivepinedensitymatricesofpotentialpinehostsfortheChelanstudyareaat30-mresolutions.MethodsforallstudyareasaredescribedindetailbyCrabbetal.(2012).

2.2.2 | Aerial detection survey data

Geo-referenced data describing the annual number of MPB-killedtreeswereobtainedforallthreestudyareasbeginningin1991fortheSawtooth, 1980 forChelan and2001 fornorthernColorado (USDA

Forest Service, http://www.foresthealth.info/portal). The aerial de-tectionsurveys(ADS)areconductedinfixed-wingaircraftbytrainedobserverswhomanuallyrecordnumbersofkilledtreesbasedonthecolouroftreefoliage(Halsey,1998).Foliageofdead,beetle-killedtreeschangesfromgreentoredwithinasingleyear,andinsubsequentyearsthefoliageturnsgreyandneedlesarelost.ADSdatasetsincludepoly-gonshapefileswithmetadatadescribingtheestimatednumberoftreesperacreaffectedandacodeforthedamagecausalagent(s).PolygonsdepictingMPBimpactwerequeriedusingtheiruniquecode.RastersoftotalMPBimpactbyyearwerecreatedbysummingMPBimpactsacrossallpinehosttypesforeachpolygonthenconvertingto30-mrasters.Forpurposesof this study, rasterswereconverted toeitheroneorzerotoindicatewhetherornotMPBinfestationwasobservedinapixelonagivenyear.RasterswerekeptinthecoordinatesystemoftheADSshapefiles,NorthAmericanDatum1983Albers,andothergeospatialrasterdatausedinthisstudywereconvertedtothisprojec-tionat30-mresolutionusingArcGIS9.3software(ESRI,2008).

2.2.3 | Diagnosing isolated spots

We defined a spot as a single 30-m pixel withMPB impact, sur-roundedbypixelswithno impact.LetADSn

i,j indicatethepresence

(1)orabsence(0)ofMPBimpactinapixelinrowi, column joftheADSrasterforyearn.Todeterminethelocationofapotentialiso-latedspot,weappliedadiscretesecondderivativetest,recordingallpositionsbelow50%ofthemaximumsecondderivativefortheyear,

whereΔx is thepixelwidth (30m).Theminussign isusedbecausespotsarelocalmaximawithlargenegativeconcavity.Eachpotentialspotwasthenscreenedtotestwhethertheeightsurroundingpixelswere impact-free (guaranteeing it to be separated from contiguousregionsof impact).The listofverifiedspot locations inayearwererecordedforfurtheranalysis.

2.3 | Determining travel times to spots from previously infested areas

2.3.1 | Resistance to movement, motility and pixel residence time

Theecologicaldiffusionmodel(Turchin,1998)describesthepopula-tion-level distribution,P(x,y,t), that emerges from individual randomwalkswithmovementprobabilitiesbasedonlocalhabitatinformation:

(Okubo&Levin,2001;Patlak,1953).Theindividualmovementprob-abilityat anypoint in space isproportional to the “motility” at thatpoint,μ(x,y),resultinginvariablepatchresidencetimeswhicharein-versely proportional to μ. In a homogeneous environment, motility

ΔADSn

i,j

def=

1

Δx2

(ADS

n

i+1,j+ ADS

n

i−1,j+ ADS

n

i,j+1+ ADS

n

i,j−1− 4ADS

n

i,j

)

≤−1

2maxi,j

|||ΔADS

n

i,j

|||,

∂P

∂t=

(∂2

∂x2+

∂2

∂y2

)[μ(x, y)P

]

     |  5Journal of Animal EcologyPOWELL Et aL.

isthesameasthediffusionconstantandhasunitsofareapertime.In variable environments, ecological diffusion is verydifferent fromstandard(“Fickian”)diffusion,inwhichthediffusionconstantisinter-mingledwithderivatives

(e.g. ∂

∂x(D(x) ∂

P

∂x

) inonedimension). Ineco-

logicaldiffusion,allspatialderivativesapplytotheproductofmotilityandpopulationdensity(μ(x, y)P),supporting“weak”solutionswithdis-continuitieswherehabitattypeschangeandlong-termsolutionswithdensities inverselyproportional tomotility leadingtoaggregation infavourable(highresidencetime)habitat.Intuitively,themathematicaljustificationforecologicaldiffusion is that thediffusionprocessap-pliesonlytothoseindividualschoosingtoleaveapatch(thenumberofwhichisproportionaltoμP,theproductofmovementprobabilityandthedensityavailabletodepart).Thus,theLaplacian, ∂

2

∂x2+ ∂

2

∂y2,ap-

pliesonly to themovingpopulation,μP.Moremathematical detailsaboutthedifferencesbetweenecologicalandFickiandiffusion,andtheconsequencesforlarge-scalepopulationmovement,canbefoundinGarlick,Powell,Hooten,andMcFarlane(2011).

Motilityinapatchisinverselyrelatedtomeanresidencetimesforindividuals in thepatch (Turchin,1998); inapixelwithareaΔx2 theexpectedresidencetimeofindividualsis

ForMPB,PowellandBentz(2014)showedthatbeetlepopulationshavemotilityfollowinganegativeexponentialwithhostdensity,

where Si,j is the density of hosts in pixel i,j, scaled in thousands ofhostsperhectare,μ0 is themaximummotility (3.79km

2/day) in theabsenceofhosts,andμ1 = −10.9 is therelativerateofmotilityde-cline with host density. This model for motility reflects the timespent by beetles searching an increasingly complex environmentfor chemical plumes and potentially susceptible hosts. The param-eter μ̂1 =

μ1+ln (μ0)

1000= 1.3472×10−3 is introduced for convenience.

Converting tominutesduringa10-hr flightday, thismodel gives ameanresidencetimeof64minina30-mpixelcontainingadensityof500hosts/ha.Residencetimeinapixelwithnotreesis8.5s(corre-spondingtoanaveragespeedof3.5m/sforMPBcrossingunforestedpixels).Thisvariability inresidencetimescausesbeetlestodisperserapidly through areas with few hosts and aggregate in areas withhigherhostdensity.

2.3.2 | Minimum travel time for attacking MPB

If a beetle follows a path passing throughK pixels,{(ik, jk)}Kk=1, theexpectedtraveltimeis

Pathsfollowedbybeetleswhenparticipatinginaspotattackcannotbeknownapriori.However,beetlesaremorelikelytobesuccessfulatovercominghostdefencesinlocationswheretraveltimestobeetle

sourcesare shorter.We thereforehypothesize thatobservedspotswillbestructuredaccordingtominimaltraveltimesfromthenearestbeetlesources.

Theminimumtraveltimetoapointinalandscape,T,satisfiestheeikonalequation,

whichconnectsminimumtraveltimes(T)andresidencetimesthroughthe gradient vector, (Tx,Ty) = ∇T (subscripts indicate partial deriva-tives).InEquation(1),thetemporalcostofmovementfromonepixeltothenext (i.e. therateofchangeoftravel time) isproportional tohowlongbeetlesspendinthespacebetween(i.e.theresidencetime).Theeikonalequation isdifficult tosolveanalytically,butcanbeef-ficiently solved numerically using the fast sweepingmethod (Zhao,2004). This approach iterates to a solution in a pixel by examiningneighbourstodeterminewhichhasthelowesttraveltime,thenup-datingprojected travel timebyadding thecurrentpixel's residencetimetotheminimumamongnearestneighbours.

BecausebeetlescomefromtreesinfestedinthepreviousyearweuseADSdatainyearn−1tosetTi,j = 0 inpixelswithsourcepopu-lationsofbeetles.ThefastsweepingmethodthengeneratesminimaltraveltimesfromtheperimeterofobservedADSimpact inthepre-viousyeartoallotherpointsonthelandscape,conditionedbyinter-veninghostdensitywhichaltersmotilityinpixelsbetweenspotsandthenearestsources(1).Ineveryyear,thetraveltimemapisdifferentbecausethelocationofsourcepopulationschanges.Figure2depictsprojected travel times for a portionof the Sawtooth study area for2001.

Observed travel times at spot locations were recorded yearly,using that year's spatial pattern of travel times resulting from hostdensitiesandADSimpactfromthepreviousyear.Acrossallyearsineachstudyarea(8intheSawtooth,10inColoradoand20inChelan),ahistogramofobservedtraveltimetospotswascreated.Thenumberofbinswaschosensothatthebincorrespondingtothelargesttraveltimesheldat leastonepercentofthetotalnumberofobservations.Thebincontainingzerotraveltimewasignored(asapeculiarityoftheADSdataisthatisolatedspotsaresometimesobservedinsidepoly-gons indicatingcontiguous impact inthepreviousyear,seeFigure2foranexample).Ineachcase,apower-lawcurvewasfittothebinnedobservations using nonlinearmaximum likelihood on the arithmeticscaleandassumingnormaldistributionoferrors.

2.4 | Predicted distribution of spots

We assume that spotswhere beetles have overcome theAllee ef-fect are most likely at foci located with minimal travel time fromsourcepopulations.ConsiderabeetlepathpassingthroughKpixels,{(ik, jk)}

K

k=1.Ifpotentialfociarerandomlydistributedamonghosts,the

probabilityofpassingthroughpixelk and notencounteringaweak-enedfocustreeisexp [−αΔx2Sik ,jk ],whereαisthepercapitaencoun-terrateforweakenedtrees.TheprobabilityofpassingpixelKonthepathistherefore

ΔT =Δx2

μ.

(1)μ = μ0 exp

[

−(μ1+ ln (μ0)

) Si,j

1,000

]

= μ0 exp[−μ̂1Si,j

],

(2)K∑

k=1

Δx2

μ0eμ̂1Sik , jk .

(3)‖∇T‖ =

�T2x+T2

y=

Δx

μ0eμ̂1S,

6  |    Journal of Animal Ecology POWELL Et aL.

Minimizingtraveltimetothefinalpixelrequiresthatthepathencoun-terasfewhostsaspossibletoreducetime in interveningpixels,soSik ,jk ≪SiK ,jKonpathsthatendinisolatedspots.Theprobabilityofpass-ingpixelKbecomes

becauseexp (−αΔx2Sik ,jk )≈1 for theearlier, low-densitypixelsalongthepath.Actualtraveltimealongthepathwillbedominatedbythecontributionofthefinal,mostdenselystockedpixel,givinganapprox-imatetraveltimetopixelKof

Wecannowfind thecumulativedensity function (CDF) forT≤TK, using(4),

Equation(5)canberearrangedtoexpressdensityinthefinalpixelasafunctionofTK,

andnowtheCDF,(6),canbewrittenintermsofTK alone,

Theprobabilitydensityfunction,p(T), fortraveltimestospots isproportionaltothederivativeof(7),

givingapowerlawinminimaltraveltime.

2.5 | Relating spot numbers and perimeter of previously impacted areas

IfactivespotformationisamodeofdispersalthatallowsMPBtoinvadenewareas,onewouldexpectthenumberofnewspotsformedeachyeartoscaleroughlywiththeperimeteroftheinfestedareainthepreviousyear,astheperimeteristheprimarysourceofdispersers.However,theshapesof contiguously impacted regions are spatially complex, making directmeasurementofthetotalperimeterlengthuntenable.Instead,weadopttheapproachofShigesadaandKawasaki(1997),whousedsquarerootofimpactedareaasasurrogateforperimeter.Ineachyear,thetotalimpactedareawascalculatedbysummingallpixelswithADSimpactandsubtractingthenumberofnewspotsforthatyear(aseachspothasbeenfilteredtoimpactonlyasinglepixel).Thenumberofspotsinyearnwasthenfittothesquarerootofimpactedareainyearn−1usinglinearregression.

2.6 | An alternative: Random spot formation

Totestanalternativehypothesisthatspotsformrandomlyinspace,wegeneratedartificialdatasetsofisolatedspots.Foreachyearandineachstudyarea,randomindiceswerechosenfromdiscreteuniformdistributionswiththeonlyrestrictionbeingthata“spot”locationmustappearinanareawithhostcovertype.Randomlocationsweregener-ateduntilthenumberof“spots”wasequaltothenumberofobservedspots inthatyearforthatstudyarea.Therandomspotdistributionwasfittoapower-lawusingnonlinearregression,andthepredictedcumulativedistributioncalculateddirectlybyanalyticintegrationfromthesmallesttraveltime:

where1+α̂ is the (negative) fitted exponent and the coefficientin front of the integral normalizes the distribution. To test the

P(k>K)=

K∏

k=1

e−αΔx2Sik ,jk

(4)P(k>K)=

K∏

k=1

e−αΔx2Sik ,jk ≈e

−αΔx2SiK,jK

(5)TK≈Δx2

μ0eμ̂1Si

K,jK

(6)P(T≤TK

)=1−P(k>K)≈1−e

−αΔx2SiK,jK

eSiK,jK ≈

[μ0

Δx2TK

] 1

μ̂1

(7)P(T≤TK

)≈1−

[μ0

Δx2TK

]− αΔx2

μ̂1

(8)p(T)∝T−

(

1+αΔx2

μ̂1

)

,

F(T)=α̂Tmin∫

T

Tmin

t−(1+α̂) dt=1−Tα̂min

Tα̂,

F IGURE  2  Traveltime(inminutes,seecolourbartoright)frombeetlesources(boundariesindicatedbysolidcontours)tootherlocationsina10×15kmportionoftheSawtoothstudyarea,2001.NewspotsidentifiedintheannualADSappearascircles.Occasionallyspotsareobservedinpreviouslyimpactedarea,asindicatedbycirclesinsideofsolidcontours

     |  7Journal of Animal EcologyPOWELL Et aL.

hypothesisthatthedatawereactuallygeneratedbyapower-lawdis-tribution“spots”werebinnedaccordingtotraveltimefrompreviousyear's impactedareaandgoodnessof fit testedusingCramér-vonMises’ A2 (as recommended by Choulakian, Lockhart, & Stephens,1994). The statistic tests the correspondence between observedandpredicted cumulative distributions, and is defined for discreteobservations:

whereNisthenumberofspots,kisthenumberofbins,andifej and ojarethenumberofexpectedandobservedspotsinbinj,thenpj=

ej

N

and

(Choulakianetal.,1994).CalculatedA2werecomparedwithtabulatedcritical values,A2

crit, usingdegreesof freedomcorresponding to the

numberofbins (k) +number fittedparameters (2)−1 todeterminewhetherthenullhypothesis(randomspotsfollowapower-lawdistri-bution)canberejectedwith90%confidence.

3  | RESULTS

3.1 | Spot distributions and relation to perimeter size of previous year infestations

In the Sawtooth study area, nonlinear regression gave an expo-nentof−2.398andr2 = .990(Figure3).Thepredictedexponentis1+

αΔx2

μ̂1,indicatingαΔx2 = 0.0102,or10.2potentialfocustreesper

1,000trees.InChelan,thepredictedexponentwaslower,−1.215,fittedwithr2 = .998(Figure4).Thislowerexponentindicatesthattheper-capitarateoffocustreeencounterwasαΔx2 = 0.00158, or 1.58focustreesper1,000trees.ResultsfromColoradowereclosetotheChelanresults,withαΔx2 = 0.00201 and r2 = .985(Figure5).Viewed in terms of focus trees per hectare, using average hostdensitiestoconvertencounterratestopotentialfocustreedensi-ties, rates varied from 0.79 trees/ha in Chelan to 1.56 trees/hainColoradoandupto4.59trees/ha in theSawtoothstudyarea.Thesespot initiationratescomparefavourablywiththeobserva-tionsofCarroll etal. (2006) inBritishColumbia.Onsevenstudyplots,theseauthorsobservedisolatedmassattacksofMPBcorre-spondingtoincipientepidemicsin2000–2004andreportedspotdensities varying from a low of .403 trees/ha to a high of 4.04trees/hawithameanof1.78trees/ha.

Thenumberofspotsformedinayearisexpectedtoincreasewiththe perimeter of area impacted in the previousyear, and perimeterscaleswiththesquarerootofimpactedarea,providedtheshapeoftheimpactedareasisnottoocomplex.FollowingShigesadaandKawasaki(1997),wefitthenumberofspotsobservedinyearn, Nn,tothesquareroot of the previous year's impacted area, An−1 = Σi,jADS

n−1

i,j, using

simple regression. Regression coefficientswere consistent, althoughgoodnessoffitvaried:

A2=1

N

k−1∑

j=1

Z2jpj

Hj(1−Hj),

Sj=

j∑

i=1

oj, Tj=

j∑

i=1

ej, Zj=Sj−Tj, andHj=1

NTj

F IGURE  3  Distributionofpredictedtraveltimes(minutes)toobservedspotsintheSawtoothstudyareabetween1997and2003.Traveltimestoobservedspotswereplacedin20bins;thebincontainingzerowasnotincluded.Bestmaximumlikelihoodfitofapowerlawtothehistogramisdepictedinasasolidline;linearityofthedataandthefittedcurvearedepictedonlog–logscaleintheinset.Thefittedcurvedescribes99.0%ofthevariabilityinthedataaccordingtor2 20 40 60 80 100 120 140 160 180 200

0

500

1000

1500

2000

2500

T (min)

Num

ber o

f spo

ts

Sawtooth spots 1997−2003

ObservedT−2.398

3 3.5 4 4.5 5 5.53

4

5

6

7

8

log(T)

log(

N)

15.56 − 2.398 log(T)

Study area Fitted modelCoeff. determination Graph

Sawtooth Nn = 82.7√An−1

r2 = .939 Figure6

Chelan Nn = 78.9√An−1

r2 = .327 Figure7

Colorado Nn = 75.9√An−1

r2 = .784 Figure8

8  |    Journal of Animal Ecology POWELL Et aL.

ThefittedcurvesaredisplayedintermsofimpactedareaandyearofimpactinFigures6–8.Asexpected,thenumberofspotsincreaseswith the sizeof themain infestation and in particularwith a roughmeasureofinfestationperimeter,withbetween75and83spotsgen-eratedperkilometreofperimeter.Thissupportstheideathatbeetlescreatingisolatedspotstravelfromtheperimeterofmajorinfestationsinthesurroundinglandscape.

3.2 | Random seeding of spots

To illustrate the differences between random formation of spotsand the power-law distribution of spots,we seeded forested areaswithartificial“spots”whoselocationswereselectedfromauniform

distributionover areaswithhost cover type. In each study area, ineach year as many random “spots” were generated as were actu-ally observed in that year, and the spot numbers binned accordingtotheyear'straveltimemapinthesamewayastheobservedspots(Figure9).TheCramér-vonMisesA2statisticwascalculatedforeachstudyarea.

F IGURE  4  Distributionofpredictedtraveltimes(minutes)toobservedspotsintheChelanstudyareabetween1990and2009.Traveltimestoobservedspotswereplacedin22bins;thebincontainingzerowasnotincluded.Bestmaximumlikelihoodfitofapowerlawtothehistogramisdepictedasasolidline;linearityofthedataandthefittedcurvearedepictedonlog–logscaleintheinset.Thefittedcurvedescribes99.8%ofthevariabilityinthedataaccordingtor2 50 100 150 200 250 300 350 400 450

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

T (min)

Num

ber o

f spo

tsChelan spots 1990−2009

ObservedT−1.215

2 3 4 5 6 74

5

6

7

8

9

10

log(T)

log(

N)

12.34 − 1.215 log(T)

F IGURE  5  Distributionofpredictedtraveltimes(minutes)toobservedspotsintheColoradostudyareabetween2002and2010.Traveltimestoobservedspotswereplacedin25bins;thebincontainingzerowasnotincluded.Bestmaximumlikelihoodfitofapowerlawtothehistogramisdepictedasasolidline;linearityofthedataandthefittedcurvearedepictedonlog–logscaleintheinset.Thefittedcurvedescribes98.5%ofthevariabilityinthedataaccordingtor2 0 50 100 150 200 250 300 350 400 450

0

0.5

1

1.5

2

2.5

x 104

T (min)

Num

ber o

f spo

ts

Colorado spots 2001−2010

ObservedT−1.291

3 3.5 4 4.5 5 5.5 6 6.56

7

8

9

10

11

log(T)

log(

N)

14.15 − 1.291 log(T)

Study area A2 df A2

critResult

Sawtooth 63.95 20 1.834 Reject(p<.0001)

Chelan 1.842 22 1.835 Reject?(p<.1)

Colorado 73.12 25 1.837 Reject(p<.0001)

     |  9Journal of Animal EcologyPOWELL Et aL.

F IGURE  6  ObservedspotsintheSawtoothstudyareabetween1996and2004asafunctionofinfestedareaperimeter.In(a)observedspots(*)arefittedto

√Impacted Area;fithasr2 = .939.In(b)bothnumberofobservedspots(*)andpredictednumberofspots(solidline)are

plottedasafunctionofyear.Thequalityofthefitindicatesthatthenumberofspotsispredictedbytheperimeteroftheinfestedarea,aswouldbeexpectedifbeetlesfromtheedgeoftheinfestedareaovercometheAlleeeffectbyactivedispersaltospots

1996 1998 2000 2002 20040

2000

4000

6000

8000

10000

12000

14000

16000

18000

Year

Sawtooth spots

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

12000

14000

16000

18000

ADS area, (ha)

Num

ber o

f spo

tsSawtooth spots

Observed

N = 82.70 ADS1/2

(a) (b)

F IGURE  7  ObservedspotsintheChelanstudyareabetween1990and2010asafunctionofinfestedareaperimeter.In(a)observedspots(*)arefittedto

√Impacted Area;fithasr2 = .327.In(b)bothnumberofobservedspots(*)andpredictednumberofspots(solidline)

areplottedasafunctionofyear.TheChelanseriesofinfestationsoccurredinseveralgeographicallyseparatedareas,andtheyearswithverylowspotformationcorrespondtothecollapseofasub-infestationbecauseavailablehostswereexhausted

1990 1995 2000 2005 20100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 104 Chelan spots

Year0 50 100 150 200 250 300

0

0.5

1

1.5

2

2.5x 104

ADS area, (ha)

Num

ber o

f spo

ts

Chelan spots

ObservedN = 78.89 ADS1/2

(a) (b)

10  |    Journal of Animal Ecology POWELL Et aL.

In the cases of the Sawtooth and Colorado study areas,the hypothesis that the randomly generated “spots” followeda power-law distribution in travel time from previously im-pacted areas could be rejected with a high degree of con-fidence. In the Chelan study area, the pattern of spots wasmore random.While it is not clearwhy the Chelan data did notmore closely adhere to the power-law prediction, we note thatthe area had substantially lower spot densities and stronger spatialstructuretoitshostpopulation.

4  | DISCUSSION

WehaveshownthatastrongAlleeeffect,requiringbeetleaggrega-tiontoovercomehostpinedefences,incombinationwithlandscaperesistance, inwhichmotilitydecreasesexponentiallywithhostden-sity,leadstodispersivespreadviaspotsunderapower-lawdistribu-tionoftraveltimesfromsourcepopulations.Lowdensitiesofbeetles,dispersing fromtheperimeterofprevious infestedareas,aggregateatweakened focus treeswhichnucleate isolated spots.The impactoftheAlleeeffect is thataggregationawayfrommajor infestationsdrawsdownthedispersingbeetlepopulationsothatmorespotsarepossibleatlocationswithlowertraveltime,withalgebraicallyfewerspotsinregionswithhighertraveltime.AerialsurveysofannualMPBinfestationinIdaho,WashingtonandColoradostudyareaswereana-lysedandfoundtoconformverystronglytothepower-lawprediction(r2 ≥ .985).Thealternatehypothesisthatspotsarisespontaneously,

wasnotsupportedinIdahoandColorado,however,spontaneousspotcreationcouldnotberuledoutintheWashington(Chelan)studyarea.

Ourresultsalsosuggestthatdispersingbeetles leavetheperim-eterofsource infestations (asmeasuredbysquarerootof impactedarea),althoughthiswas lessstronglysupported(.327 ≤ r2≤ .939).Alowcorrelationwaspotentiallyduetothepoorrelationshipbetweenthe actual and estimated (square root of impacted area) perimetersize,inadditiontothefactthatwedidnotaccountfortemperature- dependentdifferentialMPBproductivityamongyears,whichisknowntohaveasubstantial impactonpopulationgrowth (Powell&Bentz,2009).Nevertheless,thenumberofspotscreatedhadaconsistentre-lationshiptopreviousyearinfestationperimeter,andtherelationshipwasstrongestinthesmallest(Sawtooth)studyarea,wherethespatialstructureofMPB-causedtreemortalitywasgeographicallysimplest.In Colorado, the largest study area, hosts had a relatively homoge-neousdistributionbutmountainoustopographybrokeuplargeareasofinfestation,weakeningtherelationshipbetweenperimeterandspotnumbers.IntheChelanarea,wheretherelationshipwasonlymargin-allysignificant,theMPBoutbreakwasdissectedintothreeareassepa-ratedbydeeprivervalleys,causingasynchronousandseparatedMPBactivity.Clear outlierswith lownumbersof spotsoccurred inyearswhentheintensityofMPBactivityshiftedbetweentheareas.Largebodiesofwater,deepvalleyswithnohostsandhigh-altituderidgesmayhaveobscuredthepower-lawprocessofspotformationthroughdisruptionofdispersalandstrongspatialstructuringofpotentialhosts.

Inheterogeneousenvironments,thepower-lawtheorypredictsthat MPB spread preferentially along corridors of relatively high

F IGURE  8  ObservedspotsintheColoradostudyareabetween2002and2010asafunctionofinfestedareaperimeter.In(a)observedspots(*)arefittedto

√Impacted Area;fithasr2 = .784.In(b)bothnumberofobservedspots(*)andpredictednumberofspots(solidline)

areplottedasafunctionofyear.TheColoradooutbreakspreadthroughamuchlargerarea,butnumberofspotscorrelatesstronglywithperimeter,indicatingthatbeetlesfromtheedgesareovercomingAlleeeffectsbyactivedispersaltospots

2002 2004 2006 2008 20100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 105 Colorado spots

Year0 1000 2000 3000 4000 5000 6000 7000

0

0.5

1

1.5

2

2.5x 105

ADS area, (ha)

Num

ber o

f spo

tsColorado spots

ObservedN = 75.93 ADS1/2

(a) (b)

     |  11Journal of Animal EcologyPOWELL Et aL.

motilityhabitat,whichreducestraveltime,withimpactoccurringattheboundaries.ForMPB,thisincludesforestedgesandboundariesof areaswith lowhost density.Thinning is a control strategypre-scribedtoreducetheoverallnumberofhostsandencounterratesforweakenedhosts,inadditiontoincreasinghostvigour(Fettig,Gibson,Munson,&Negrón, 2014;Waring&Pitman, 1985). Inconsistencyin the spatial pattern of thinning, however, could produce move-mentcorridorswithfewhoststherebyresultinginacceleratedMPBspreadacrossalandscape.Ontheotherhand,directlymanipulatingvariation in host densitymay be a strategy for control ofMPB. If

patchesofhighhostdensityaresmallenough,theymayactaseco-logicaltraps(Gilroy&Sutherland,2007)fordispersingbeetles.Forexample, in the related species southern pine beetle (D. frontalis),spot extinction risk increases dramatically as the number of hostsinvolveddecreases(Friedenberg,Powell,&Ayres,2007;Hedden&Billings,1979).

IndirectcontrolmeasurestoreduceMPBpopulationsize(e.g.in-secticides,semiochemicalsandremovalofinfestedhosts)couldben-efit by considering thatMPB spreadmay be fastest through areasof low host density including along meadows and other clearings.

F IGURE  9  Comparisonoftraveltimehistogramsforobserved(solidbars)andrandomlyseeded(openbars)spotsinthethreestudyareas.Thesamenumberofrandomspotsweregeneratedyearlyaswereobserved,withrandomspotssampledfromauniformdistributioninlocationswithhostcovertype.IntheSawtooth(a)andColorado(c)studyareas,thedistributionsofrandomlyselectspotsaremulti-modalandclearlynotofpower-lawtype.InChelan(b)thedistributionisunimodal,andcouldnotberejectedasapotentialpower-lawdistribution.SpotdensitiesweresubstantiallylowerinChelanandmanybarrierstobeetlemovementwerepresent(deepvalleys,broadbodiesofwaterandhigh-altituderidges),potentiallyobscuringthepowerlawrelationship

50 100 150 200 2500

1000

2000

3000

4000

5000

Sawtooth study area

50 100 150 200 250 3000

1000

2000

3000

4000

5000

Chelan study area

Num

ber o

f spo

ts Observed

Randomly seeded

50 100 150 200 250 300 350 4000

0.5

1

1.5

2

x 104 Colorado study area

T (min)

(a)

(c)

(b)

12  |    Journal of Animal Ecology POWELL Et aL.

Consequently, controlmeasures directed along boundaries ofmini-mumtraveltimecorridorsare likelytohavedisproportionately largerewards.Usingthepower-lawmodelandestimatesofhosttreeden-sity,traveltimemapsforMPBspreadacrossalandscapecanbecalcu-lated,helpingtoprioritizetreatmentapplication.

5  | CONCLUSION

The relationshipbetweenactiveaggregation toovercometheAlleeeffectandthepatchyspreadofpopulationshasnotbeenpreviouslyconsidered.Themechanismsevaluatedhere,differentialdispersalre-flectinglandscaperesistancetomovementandactiveaggregationtoovercomeastrongAlleeeffectat lowpopulationdensities,arerea-sonablygeneral.Exponentialrepresentationsofhabitatinfluenceonresidencetimesarethemostfrequentmodelforlandscaperesistancetomovement(e.g.Hanks&Hooten,2013).AnyPoissonprocessforencountering spot nucleation conditions will generate exponentialfailureprobabilities for stopping inpixels.Asdiscussed inNewman(2005),thecombinationofsuchexponentialeffectsisacommonwayforpowerlawstoariseinnature.WehaveshownthatforMPBtheseexponential mechanisms do, in fact, combine to create power-lawdistributionsofobservablepatchyspread.

ThereisalonghistoryofconsideringtheconsequencesofAlleeeffects on the passive dispersal and spread of organisms (see re-viewsbyLiebhold&Tobin,2008;Taylor&Hastings,2005).Generallyspeaking, theAlleeeffect isexpectedtoslowdown invasions (Kot,Lewis,&denDriessche, 1996).The inertia of a strongAllee effectdiminishesspreadratesbecausesmallpopulations(belowthreshold)cannotestablishawayfromtheperimeterand“pull”thewaveofinva-sion;sourcepopulationsbehindtheperimeterofthewaveofinvasionmustgrowsufficientlyto“push”outenoughdisperserstoovercometheAlleeeffect.Putmoremathematically,theAlleeeffecttruncatesthepassivedispersal kernel so that even fat-tailed (includingpow-er-law) kernels end up with finite moments, making the effective meandispersaldistancemuchsmallerthanthekernel'smeandisper-saldistance.Thus,withpassivedispersalAlleeeffectssloworstopthe spread of invasives. Invasions thatwould otherwise acceleratebecomeconstantspeedinvasionsinthepresenceoftheAlleeeffect(Wang,Kot,&Neubert,2002); inheterogeneousenvironments,theAlleeeffectcanstopinvasionsthrough“rangepinning”(Keitt,Lewis,&Holt,2001).

Wehaveshownthatactivedispersalandspot formationduetotheAllee effect result in power-law dispersal of propagules (spots)spreading fromtheperimeterof invadedareas.This isanalogous toclassicexamplesofspeciesinvasionswithouttheAlleeeffect(Andow,Kareiva,Levin,&Okubo,1990;Shigesada&Kawasaki,1997;Skellam,1951),andwethereforeproposethatthedispersedspotsofimpact“pull”thewaveofinvasion.Aslow-exponentpower-lawkernelsmayhaveonlyonemoment,spreadratesarelimitedonlybythenumberoftimesthedispersalpatternissampled(i.e.75–83timesperkilometreofever-expandingperimeter),leadingtoacceleratinginvasions(Clark,Lewis, & Horvath, 2001) in spite of an obviousAllee effect. These

mechanismscouldhavecontributedtotherecentrapidspreadofMPBacrosswesternCanada(delaGiroday,Carroll,&Aukema,2012).

ACKNOWLEDGEMENTS

TheauthorsthankTomEdwardsandEthanWhitefor formativedis-cussionsandfeedback,aswellasthreeanonymousreviewerswhoof-feredmanyexcellentsuggestions.TheUSDAForestServiceWesternWildlandsEnvironmentalThreatAssessmentCenterprovidedsupportthrough a cooperative agreement with USU. This project was alsosupported in part by the Small Business Innovation Research (SBIR)programmeof theUSDANational Institute forFoodandAgriculture(NIFA).

AUTHORS’ CONTRIBUTIONS

B.B.andJ.P.procureddatausedinthispaper;N.F.,B.B.andJ.P.de-signedanalyses;whileM.G.andJ.P.organizedandimplementedcom-putational approaches used in the analyses;B.B. andN.F. providedecologicalbackgroundonmountainpinebeetle;andJ.P.ledthewrit-ingofthemanuscript.Allauthorscontributedcriticallytothedraftsandgavefinalapprovalforpublication.

DATA ACCESSIBILITY

DatausedinthispaperarearchivedinUtahStateUniversity'sDigitalCommons, http://digitalcommons.usu.edu/all_datasets/24/ (Powell,2017;https://doi.org/10.15142/T31C73).

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How to cite this article:PowellJA,GarlickMJ,BentzBJ,FriedenbergN.DifferentialdispersalandtheAlleeeffectcreatepower-lawbehaviour:Distributionofspotinfestationsduringmountainpinebeetleoutbreaks.J Anim Ecol. 2017;00:1–14.https://doi.org/10.1111/1365-2656.12700