Models for Assessing the Male Annihilation of Bactrocera spp. With Methyl Eugenol Baits

16 Pages·2013·0.98 MB·English

ECOLOGYANDPOPULATIONBIOLOGY
Models for Assessing the Male Annihilation of Bactrocera spp. With
Methyl Eugenol Baits
HUGHJ.BARCLAY1,2ANDJORGEHENDRICHS3
Ann.Entomol.Soc.Am.107(1):81Ð96(2014);DOI:http://dx.doi.org/10.1603/AN13046
ABSTRACT Modelswerepresentedthatdescribetheattractionandkillingattoxicbaitsortrapsof D
o
adultmalesandfemalesofaninsectpestspecies.Thesemodelswereusedtotesttheeffectsofthe w
n
following factors on ease of control: female monogamy versus polygamy, attraction of only males lo
versusbothsexes,initiatingmatingbeforeversusafterrespondingtobaits/traps,theabilityofmales ad
e
tomatemanytimeseachdayversusonlyonceperday,andtheexistenceofatimelagofseveraldays d
before females can mate versus mating immediately after emergence. The models indicated that fro
m
matingbeforetrappingortheabilityofmalestomatemanytimeseachdaywillprobablyrenderthis
h
controlmethodineffective.Theotherfactorstested(matinghabitandspeciÞcsofattraction)hadlittle ttp
effectontheefÞciencyoftrappingmalesasacontrolmethod.Wethenincludedagestructureand s
arefractoryperiodforvirginfemalesbeforetheycanmate.ThemodelswerethenmadespeciÞcto ://a
c
Bactroceradorsalis(Hendel)(Diptera:Tephritidae)byusingparametervaluesderivedfromtheliterature. ad
Theresultsofthemodelsimplythattheattractionandkillingoflargenumbersofmalesisratherineffective em
tosuppresspopulations.However,thecombinationofattractingbothmalesandfemalescanbemore ic
.o
effectivethanattractingeithersexalone.Theincreasedattractionoffemalestomethyleugenolbaitsthat u
p
hasbeenobservedwiththedecliningpresenceofmalesduringMaleAnnihilationTechniquecampaigns .c
o
mayexplainthereportedeffectivenessagainstinvasiveBactrocerapestspecies. m
/a
e
KEYWORDS Bactrocera,maleannihilationtechnique,methyleugenol,fruitßy sa
/a
rtic
le
-a
Methyl eugenol (ME) is a powerful attractant for tance(Steineretal.1965,Ushioetal.1982,Koyama bs
males of many tropical tephritid fruit ßy species et al. 1984, Cunningham 1989b, Anonymous 1991, tra
c
(Drew1974,1989;WhiteandElsonÐHarris1992;Tan Nakamorietal.1991,Malavasietal.2000).Itisalso t/1
andNishida2012).Thisstrongattractant,alsocalled routinely used to eliminate at an early stage, the 0
7
parapheromone (i.e., a chemical that behaves like a recurringincipientoutbreaksofintroducedBactro- /1
/8
pheromonebyattractingonesex,butisnotproduced cera spp. ßies into mainland United States (Cun- 1
by the insects themselves; Cunningham 1989a), has ninghamandSuda1985)andalsoinAustralia,Japan, /77
1
beenusedsincethe1950sforpurposesofmonitoring andelsewhere(Hancocketal.2000,Seewooruthun 7
3
andcontrol(SteinerandLee1955,Cunninghamand et al. 2000, Ohno et al. 2009). In northÐeastern b
y
Suda 1986). Nevertheless, as even a few surviving Brazil,alongtheborderwithFrenchGuyana,Bac- g
multiple-matingmalescanfertilizemanyfemales,the troceracarambolae(DrewandHancock1994)con- ue
s
realdamagecausedbyovipositingfemalesisnotsig- tainmenteffortsbyMATcontinueonapermanent t o
niÞcantly reduced unless ME baits or traps are sys- basistopreventestablishmentofthispest(Godoy n
0
tematicallyappliedatalargeroperationalscale.Only 2006).Permanentcontainmenteffortsarealsoon- 5
A
suchanapproachonanarea-widebasiswilleliminate goinginIsraelattheborderwithGazaandEgyptto p
mostmalestobeabletodeprivefemalesofmatesand prevent establishment of Bactrocera zonata (Saun- ril 2
thusreducefruitinfestation(Steineretal.1970,Cun- ders)andinnorthernSouthAfrica(Manrakhanet 0
1
ningham1989b).ThisMaleAnnihilationTechnique al. 2012) near Zimbabwe to prevent the establish- 9
(MAT) has been used to suppress (Balasubrama- mentofBactrocerainvadens(Drewetal.2005).
niam et al. 1972, Qureshi et al. 1981, Vargas et al. Relevant previous modeling focused on the in-
2010)oreventoeradicateestablishedpopulations herentdensitydependenceofmaleannihilationby
of ME-responding fruit ßies of economic impor- using pheromone attractants (MAT; Barclay and
vandenDriessche1983)andmasstrapping(Barclay
1(Retired)PaciÞcForestryCentre,CanadianForestService,506 1984).Inaddition,Itoˆetal.(1989)developedaMAT
W.BurnsideRoad,Victoria,BC,CanadaV8Z1M5. modelandHorngandPlant(1993)simulatedthelek
2Correspondingauthor,currentaddress:928OldEsquimaltRd., mating system and its impact on MAT. Here we
Victoria,BC,CanadaV9A4X3(e-mail:[email protected]). investigate the area-wide implementation of MAT
3InsectPestControlSection,JointFAO/IAEADivisionofNuclear
byusingMEasthesolesuppressionmeasureforthe
TechniquesinFoodandAgriculture,InternationalAtomicEnergy
Agency,Wagramerstrasse5,P.O.Box100,A-1400Vienna,Austria. controlofahypotheticalME-respondingpestspecies.
0013-8746/14/0081Ð0096$04.00/0(cid:2)2014EntomologicalSocietyofAmerica
82 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
WethenmaketheparametervaluesspeciÞctoBac- Also,ifmalescanalwaysinseminateallfemales,then
troceradorsalis(Hendel).Thisispartoneofalarger thedistinctionbetweenmonogamyandpolygamydis-
study,withparttwo(Barclayetal.2014)studyingthe appearsinthiscontext.
simultaneousapplicationofMATandsterilereleases.
MalesLimitedtoOneMatingPerDay
TheModelsandResults
In reality, males are limited in their mating fre-
Allthemodelscontainedinthisarticlehavedaily quency and may have either an extended period of
reproductionandmostinvolvecaptureatMEbaits/ mating with one female or a refractory period after
traps.Mostalsoinvolve,andrelyon,alimitationon matingbeforetheycanmateagain.Herewepresent
howoftenmalescanmateinagivenperiod;mostof amodiÞcationoftheaforementionedmodelbyusing
the models assume that males can mate once per MEasbait.Wetreatthecasesinwhichonlymalesare
day,andwillmatewhenevertheopportunityexists. attracted,aswellaswhenbothsexesareattractedto D
the ME baits/traps. In Bactrocera spp., females are ow
MalesCanMateUnlimitedTimesEachDay attracted in reduced proportion to males (Shelly nlo
2010). We also treat the cases in which females are a
d
Amodelinwhichadultinsectsareattracteddailyby monogamous and also in which females are polyga- ed
food,otherodors,orothersourcesofattractionatbaits/ mous; although males of Bactrocera spp. are polyga- fro
trapswaspresentedbyBarclay(1987);whentheinsects mous,rematinginfemalesthathavealreadymatedis m
h
feedontoxicbaitorarecaughtintraps,theyarekilled. infrequent.Ifmalescanmateonlyonceperdaybe- ttp
Thismodelassumesthatcaptureoccursbeforemating causeoftheshortcourtshipperiodandlongmating s
and that males are capable of unlimited mating fre- timeinthesespecies,thenthereisthepossibilityof ://a
c
quency,andthatallvirginfemalesmateontheirÞrstday. restrictingmatingofvirginfemalesbyreducingmales a
d
V (cid:3)a F in sufÞcient numbers through baiting/trapping such em
t(cid:2)1 t(cid:4)k thatnotallavailablefemaleswillbecomematedona ic
Ft(cid:2)1 (cid:2) syVt (cid:3) syFt givenday.Ifbothsexesaretrapped,itismoredifÞcult .oup
Mt(cid:2)1 (cid:2) a Ft(cid:4)k (cid:3) szMt [1] ffoewremrafleemsatolesli,mbuittffeemwaelrefemmatailnegs,wbiellclaauysefetwheerreegagrse, .com
whereF,V,andM arethenumbersofadultmated sothereissomecompensation.Theparentmodelhere /a
(i.e., ferttilizted) femtales, virgin females, and males, isthatinEquation1withy(cid:3)z(cid:3)1.0. esa
respectively,attimet;sandaaredailysurvivorship Ifmalescanmateonlyonceperday,thentherewill /a
andfertility,respectively,andkisthedevelopmental be several sets of equations, depending on whether rtic
le
periodindaysfromovipositiontoadultemergence. females are monogamous or polygamous, whether -a
Further,thefertility,a,isassumedtoincludepreadult only males or both sexes are attracted to the baits/ bs
survivorship.Ageclassesarenotexplicitlymodeled, traps,whethertrappingoccursbeforeoraftermating, tra
c
butbothFandMareassumedtocontainpotentially and whether males outnumber receptive females at t/1
aninÞnitenumberofadultageclasses,withallbutthe mating time or whether they are outnumbered by 07
Þrstfewbeingclosetozero.Inaddition,yandzare receptivefemales.Whethermalesoutnumberrecep- /1
/8
proportions representing the daily survivorships of tivefemaleswilldependontherelativesusceptibilities 1
matedandvirginfemales(y),andofmales(z),after to capture at baits/traps of the different population /77
1
someofthemvisitthebaitsortraps;therefore,yand componentsaswellasthetimingofvisitingthebaits/ 7
3
zaretheproportionsofthosepopulationcomponents traps.Ifmalesvisitandarekilledatbaits/trapsbefore b
y
thatdonotvisitanddieatthebaits/trapseachday. mating,thenitispossiblethatreceptivefemalesout- g
ThismodelhasonepositiveequilibriumconÞguration numbermalesatmatingtime(maledeÞcit).Ifmales ue
s
ofsizesofthethreecomponents(F,V,andM),andit mate Þrst and then visit the baits/traps, males will t o
is obtained by dropping time subscripts and solving probablyoutnumberreceptivevirginfemalesatmat- n 0
theresultingthreesimultaneousequations;theequi- ingtime(maleexcess),astheratesofrecruitmentof 5
A
librium is neutrally stable. The value of y that will bothareidenticalunlessthesexratioatadultemer- p
achievethisequilibrium(calledthecriticalvalueofy, genceisdifferentfromone-to-one.Hereweassume ril 2
y*)isgivenby: that virgin females and mated females are equally 0
1
attractedtobaits/traps,atadailyrateof1(cid:4)y.Wewill 9
y*(cid:3)1/s(a(cid:2)1) [2]
treat the two conditions in which attraction to the
Notethatzdoesnotenterintothiscondition,mean- baits/trapsoccursbeforematingsothatreceptivefe-
ingthatzhasnoeffectonthefemaleequilibrium,but malesoutnumbermales,andalsoinwhichattraction
onlyonthatofthemales.Ify(cid:5)y*,thenthepopulation occursaftermating,sothatatmating,malesoutnum-
collapses,whereasify(cid:6)y*,thepopulationgrowsin berreceptivefemales.Inaddition,wewillassumethat
anunlimitedmanner.Ifmalescanalwaysinseminate allvirginswillmateassoonastheyhavetheoppor-
allthevirginfemales,regardlessoftheirrelativenum- tunity (day 1 if males are in excess of receptive fe-
bers, then z has no effect, except on the size of the males).
malepopulationrelativetothefemaleswhenatequi- FemaleMonogamy.Hereweassumethatonlyvir-
librium,andthepopulationcannotbecontrolledby gin females mate and that mated females will not
deploymentofbaits/trapsifonlymalesareattracted. remate.
January2014 BARCLAYANDHENDRICHS:MODELOFMATFORCONTROLLINGBactrocera 83
Table1. Equationsforthemodelsinvolvinglimitedmalematingfrequency
Trappingbeforemating Trappingaftermating
Onlymalesattractedtotraps:Femalemonogamy
Vt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)sVt(cid:7)1(cid:4)zMt/Vt(cid:8)(cid:3)aFt(cid:4)k(cid:2)sVt(cid:4)szMt
Ft(cid:2)1(cid:3)sVt(cid:7)zMt/Vt(cid:8)(cid:2)sFt(cid:3)szMt(cid:2)sFt Nocontrolpossiblehere
Mt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)szMt
Onlymalesattractedtotraps:femalepolygamy
Vt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)sVt(cid:7)1(cid:4)zMt/(Vt(cid:2)Ft)(cid:8) Vt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)sVt(cid:7)1(cid:4)Mt/(Vt(cid:2)Ft)(cid:8)
Ft(cid:2)1(cid:3)sVt(cid:7)zMt/(Vt(cid:2)Ft)(cid:8)(cid:2)sFt Ft(cid:2)1(cid:3)sVt(cid:7)Mt/(Vt(cid:2)Ft)(cid:8)(cid:2)sFt
Mt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)szMt Mt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)szMt
Bothsexesareattractedtotraps:
Femalemonogamy
Vt(cid:2)1(cid:3)aFt(cid:4)k(cid:2)syVt(cid:7)1(cid:4)zMt/yVt(cid:8)
Ft(cid:2)1(cid:3)syVt(cid:7)zMt/yVt(cid:8)(cid:2)syFt Nocontrolpossiblehere D
FeMmta(cid:2)le1p(cid:3)olyagFamt(cid:4)yk(cid:2)szMt own
VFMttt(cid:2)(cid:2)(cid:2)111(cid:3)(cid:3)(cid:3)saayVFFttt(cid:4)(cid:7)(cid:4)zMkk(cid:2)t(cid:2)/(sysyzVVMtt(cid:2)t(cid:7)1y(cid:4)Ft)zM(cid:8)(cid:2)t/(syyVFtt(cid:2)yFt)(cid:8) VFMttt(cid:2)(cid:2)(cid:2)111(cid:3)(cid:3)(cid:3)saayVFFttt(cid:4)(cid:4)(cid:7)Mkkt/(cid:2)(cid:2)(VssyztMV(cid:2)ttF(cid:7)1t)(cid:4)(cid:8)(cid:2)Mst/y(FVtt(cid:2)Ft)(cid:8) loaded
Ifonlymalesareattractedtothetraps,thenmalesareindeÞcit,limitingfemalemating.ThetermszMt/VtandzMt/(Vt(cid:2)Ft)representthe from
proportionoffemalesthatmateondaytbecauseofmalelimitation. h
ttp
OnlyMalesAreAttractedtotheBaits/Traps. Ifmalesarefewerinnumberthanvirginfemales, s
thentheconditionforanequilibriumtoexististhat ://a
1.Trappingbeforemating(maledeÞcit):Theequa- asz(cid:3)(1(cid:4)sz)(1(cid:4)sy)andtheequilibriumis: cad
tionsforthiscaseareshowninTable1.Thecondition e
forapositiveequilibriumtoexististhat:asz(cid:3)(1(cid:4)sz) V (cid:2) F[aÐ(1(cid:4)sy)]/(1(cid:4)sy),M (cid:2) aF/(1(cid:4)sz) mic
(1 (cid:4) s) and the equilibrium is at V (cid:3) F [a Ð (1 (cid:4) .o
Thisyieldsacriticalvalueforzof: u
s)]/(1(cid:4)s),M(cid:3)aF/(1(cid:4)sz),whereasFisundeter- p
mined. This equilibrium is neutrally stable because if z*(cid:3)(1(cid:4)sy)/[as(cid:2)s(1(cid:4)sy)] [4] .co
m
eachofthepopulationcomponents(V,F,andM)was The relationship between z and y at equilibrium is /a
multipliedbythesamepositivefactor,theequilibrium showninFig.1A;thereisanalmostlinearrelationship esa
wouldpersistbutthepopulationwouldnotreturntoits betweenmaleandfemaletrappingsurvivorshipsuch /a
previoussize.TheequationsforFandMcanbesolved that the two do not interact much and are almost rtic
simultaneouslyatequilibriumtoyieldacriticalvaluefor le
additive. -a
z,z*,of: Ifmalesareinexcessofvirginfemalesbecauseof bs
z*(cid:3)(1(cid:4)s)/[as(cid:2)s(1(cid:4)s)] [3] the trapping of females, all virgin females become trac
matedandtheequationsarethesameasinEquation t/1
Thisvalueofz,calledthecriticalvalue(z*),separates 1,andthebaiting/trappingofmalesdoesnotassistthe 07
successfromfailureandistheupperlimitonthepro- controlprogram. /1
portionofmalesthatdoesnothavetodieatbaits/traps 2.Trappingaftermating:Inthiscase,allthevirgins /81
everydaytocontrolthepopulation.Inthiscase,males aremated,asthetrappinghasnotyetbeeneffective /77
havetobefewerthanvirginfemalesatequilibriumfor whentheymate,sothatthecriticaltrappingrateisthe 17
thecontrolmethodtohaveanyeffect,sothereisonly sameasthatgiveninEquation2:y*(cid:3)1/s(a(cid:2)1),so 3 b
onesetofequationsrepresentingmaledeÞcit.Ifz(cid:5)z*, thatkillingmalesdoesnotassistthecontrolprogram y g
thenthepopulationwillgoextinct;ifz(cid:6)z*,thenun- and simply reduces the male population, but not ue
limitedpopulationincreaseoccurs.Ifmalesareinexcess enoughtobeofanycontrolvalue. st o
ofvirginfemales,thenallthevirginsbecomematedand Separating the Two Cases of Male Deficit and Male n
nocontrolisimposedonthepopulation. Excess. Weneedawaytodetermineifreceptivefe- 05
2.Trappingaftermating:Inthiscase,allmalesthat malesoutnumbermalesorifmalesoutnumberrecep- Ap
hinagv,eaenmdebregceaduosneathneygniuvmenbedrasyoafremaavlaeislaabnledffoermmalaets- ctiovnetfreomliasleimsptooskendo.wThhioswcasnmbaellfzoumnudstbbyesubbefsotirteutainnyg ril 20
1
emerging are the same (unless the sex ratio is not y(cid:3)1/s(a(cid:2)1)intoEquation4,byusingequilibrium 9
one-to-one), then all virgins become mated and no values.Simplifying,thevalueofzforwhich,atequi-
controlispossible.Theequationsforthiscasearethe librium,malesareequalinnumbertovirginfemales
sameasthoseinEquation1withy(cid:3)1. (thisvalueofziscalledtheseparatrix,separatingmale
excessfrommaledeÞcitthatseparatesthevalidityof
MalesandFemalesAttractedtotheBaits/Traps.
thetwosetsofequations)isz:
e
1.Trappingbeforemating:Inthiscase,thereisthe
z (cid:3)1/s(a(cid:2)2) [5]
possibility that males could be either in excess (i.e., e
males (cid:6) virgin females) or deÞcit (i.e., virgin fe- This value is independent of either V or M and
males (cid:6) males) and still control the population by assumesthatyisatthecriticalvalue,y*.Ifzislessthan
meansoftrappingoffemales.Theequationsforboth the critical value, z*, then the population will be
casesaregiveninTable1. driventoextinction.Also,ifz(cid:5)z,thenneitherfemale
e
84 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
OnlyMalesAttractedtotheBaits/Traps.
1.Trappingbeforemating:Theequationsareshown
inTable1.Thereisaneutrallystableequilibriumat
V(cid:3)aF(1(cid:4)sz)/(1(cid:4)s);M(cid:3)aF/(1(cid:4)sz),andthis
yieldsacriticalvalueforzof:
z*(cid:3)(1(cid:4)s)/as [6]
whichisonlyslightlylargerthanwithfemalemonog-
amy, as virgin females are competing with remating
females for males; thus, control is slightly easier if
femalesarepolygamous.
2.Trappingaftermating:Contrarytothesituation
D
withfemalemonogamy,thereappearstobeapossi- o
w
bilitythatmalesmaybefewerthanvirginandmated n
femalescombined,eveniftrappingoccursaftermat- lo
a
ing.ThisleadstotheequationsshowninTable1,and de
sthz)e/re(1is(cid:4)asn)e(u1tr(cid:2)allys(cid:4)stasbzl)eaenqduiMlib(cid:3)riuamF/a(t1V(cid:4)(cid:3)sz)a.FT(h1es(cid:4)e d from
equationsyieldV(cid:2)F(cid:3)aF/(1(cid:4)s).Ifmalesaretobe h
lessthanfemales(allfemales,astheyarepolygamous), ttp
s
tahFe/n(1M(cid:4)(cid:5)s)V,(cid:2)whFi,cwhhreicdhuccaenstboeswzr(cid:5)ittes,naansdaFth/i(s1is(cid:4)alswz)ay(cid:5)s ://ac
a
trueforpositiveM,V,F,s,andz((cid:5)1).Thus,inthecase de
offemalepolygamy,capturingonlymalesmightyield m
ic
controlwhentrappingoccursbeforemating.However, .o
u
thevalueofthecriticalrateforzis: p
.c
z*(cid:3)[(1(cid:2)s)(1(cid:4)s)(cid:4)as]/s(1(cid:4)s) [7] om
Fig.1. Isoclinesofcapturelevelsofmalesandfemales
at methyl eugenol (ME) baits/traps that will result in andthisisonlypositiveforas(cid:5)(1(cid:4)s)(1(cid:2)s).From /ae
s
pffbeeoemmtpwuaalleleeaesstnioaorutnehtneeaultitsmmruarbicnvetiaervtdomiortansolh,esiasups.cssThuohmbfeaimcnitugasrl/vetthesraas(tpszhsa)oapnawrndotdphthoefaerittmnirotaeenlcreeaoscpft(ttiiyohv)nee, tEa(o1sq(cid:6)bu(cid:4)ea1tsei)(cid:4)onn(cid:5)cso,1asu,osnw(cid:5)ttheear1neto(cid:4)dthteiisns2t,chnwraaittthiuticchraehel,cisasoysnaswttreraiomtnhlgriteashtonoenosrelteyqlpivukaieiraralbeymlseeetvhftoeearrrt a/article-a
thatis,proportionsoftheirpopulationsthatdonothave b
to visit and die at the baits/traps each day to achieve valuesthepopulationwouldbeonthebrinkofcol- stra
populationelimination.Withsomefemaleattractiontothe lapseevenwithoutcontrol. c
baits/traps,therequiredcapturerateforeliminationofthe t/1
population is uniformly less than with male attraction MalesandFemalesAttractedtotheBaits/Traps. 07
/1
alone.Parametersaandsaremeandailyfertility,prorated 1.Trappingbeforemating:FormaledeÞcit,wehave /8
by the preadult survivorship, and daily survivorship of 1
the equations shown in Table 1, and yield a critical /7
adults,respectively.(A):Femalesaremonogamous;(B): 7
valueforzof: 1
Femalesarepolygamous. 7
3
z*(cid:3)(1(cid:4)sy)/as [8] b
y
g
Therelationshipbetweenz*andyisshowninFig.1B. u
normalecaptureneedstobeasefÞcientaseitherone e
Formaleexcess,wehavethesameequationsasinEqua- s
a1c(cid:4)tinsg.Hinoiwsoelavteiro,ni.tIctacnanalbsoesbheoswhnowthnatthzea,t(cid:6)aszm*iufsatsb(cid:6)e tio2n.1T,raanpdptihnegcaafpteturrmeaotfinmga:leHsehraesangoaeinff,emctaolensccoanntrboel. t on 0
(cid:6)1(cid:4)sforthepopulationtosurvivewithoutcontrol, eithergreaterorlessinnumberthanfemales,because 5 A
bsoiotlhoagtictahlerienqeuqiureamliteynztse.(cid:6)z*issatisÞedasaresultof bceopthtivseexteosmaraetinbge.ingtrappedandallfemalesarere- pril 2
0
Equation5canalsobederivedbylettingyV(cid:3)zM 1
9
andlettingy(cid:3)1/s(a(cid:2)1)andthensolvingforz.As ThecaseofmaledeÞcitisshowninTable1and
yieldsaneutrallystableequilibriumat:
both males and females visit the baits/traps before
mating,thenumbersatmatingarereducedfromVto V(cid:3)aF(1(cid:4)sz)/(1(cid:4)sy)(1(cid:2)sy(cid:4)sz)
yVandMtozMforfemalesandmales,respectively.If
and
z(cid:6)1/s(a(cid:2)2),thenmalesplaynopartincontrolling
thepopulation.Ifz(cid:5)1/s(a(cid:2)2),thenthetrappingof M(cid:3)aF/(1(cid:4)sz)
femalesdoesnotneedtobeasintenseasifmaleswere Thecriticalvalueofzisat:
notbeingtrapped(Fig.1A).
z*(cid:3)[(1(cid:2)sy)(1(cid:4)sy)(cid:4)asy]/s(1(cid:4)sy) [9]
FemalePolygamy.Hereweassumethatbothvirgin
andmatedfemaleswillmatedailywhenevertheop- whichispositiveforasy(cid:5)(1(cid:2)sy)(1(cid:4)sy),which
portunityexists. againissatisÞedforonlyarelativelynarrowinterval
January2014 BARCLAYANDHENDRICHS:MODELOFMATFORCONTROLLINGBactrocera 85
Table2. Equationsforthemodelsinvolvinglimitedmalematingfrequency
Somematingbeforetrapping Multiplemalematingperday
Vi(cid:2)1(cid:3)aFi(cid:4)k(cid:2)sVi(cid:7)1(cid:4)(q(1(cid:4)z)Mi(cid:2)zMi)/Vi(cid:8) Vi(cid:2)1(cid:3)aFi(cid:4)k(cid:2)sVi(cid:7)1(cid:4)nzMi/Vi(cid:8)
Fi(cid:2)1(cid:3)sVi(cid:7)(q(1(cid:4)z)Mi(cid:2)zMi)/Vi(cid:8)(cid:2)sFi Fi(cid:2)1(cid:3)sVi(cid:7)nzMi/Vi(cid:8)(cid:2)sFi(cid:3)snzMi(cid:2)sFi
Mi(cid:2)1(cid:3)aFi(cid:4)k(cid:2)szMi Mi(cid:2)1(cid:3)aFi(cid:4)k(cid:2)szMi
VirginfemalescannotmateimmediatelyonadultemergenceÑFemalemonogamy
Attractionofonlymales;maledeÞcit
V(cid:3)aF(cid:2)asF(cid:2)as2F(cid:2)...(cid:2)as(cid:4)(cid:4)1F(cid:2)as(cid:4)F(cid:7)1.0(cid:4)zM/as(cid:4)F(cid:8)(cid:2)as(cid:4)F(cid:7)1.0(cid:4)zM/as(cid:4)F(cid:8)/(1(cid:4)s)
F(cid:3)as(cid:4)F(cid:7)zM/as(cid:4)F(cid:8)(cid:2)sF
M(cid:3)aF(cid:2)szM
Attractionofbothmalesandfemales
MaledeÞcit
V(cid:3)aF(cid:2)aF(sy)(cid:2)aF(sy)2(cid:2)...(cid:2)asF(sy)(cid:4)(cid:7)1.0(cid:4)zM/aF(sy)(cid:4)(cid:8)(cid:2)as(cid:4)F(cid:7)1.0(cid:4)zM/as(cid:4)F(cid:8)/(1(cid:4)s)
F(cid:3)aF(sy)(cid:4)(cid:7)zM/aF(sy)(cid:4)(cid:8)(cid:2)syF D
M(cid:3)aF(cid:2)szM o
w
Maleexcess n
V(cid:3)aF(cid:2)aF(sy)(cid:2)aF(sy)2(cid:2)aF(sy)3(cid:2)...(cid:2)aF(sy)(cid:4)(cid:4)1(cid:3)aF(cid:7)1(cid:4)(sy)(cid:4)(cid:8)/(1(cid:4)sy) lo
F(cid:3)aF(sy)(cid:4)(cid:2)syF ad
M(cid:3)aF(cid:2)szM ed
fro
Threespecialcasesaremodeled:1)somemalesmatebeforetrappingandsomeaftertrapping,2)malescanmatemultipletimesperday, m
3)virginfemalesrequireamaturationtimebeforemating. h
ttp
tarbaocvtiengy*an(cid:3)d1k/isl(liang(cid:2)o1n)ly, tfheemcarlietsic.aTlhvuaslu,ecownhtreonl abty- kgiivlliendgqea(cid:3)ch[(d1a(cid:4)y)s.)(E1q(cid:4)uastzio)n(cid:4)1a1szc]a/nas(b1e(cid:4)sozl)v,eadndfothreqn, s://a
c
usingMEbaits/trapsiscompatiblewithtrappingafter themaximumvalueofqisgivenby: ad
e
mating for a restricted range of control parameters, m
whichrequiresthattheattractionoffemalestotheME qm(cid:3)(1(cid:4)s)/as [12] ic.o
sourcesisfairlyeffective.Thecaseofmaleexcessdoes u
p
notyieldcontrol. Inanumericalexampleusinga(cid:3)10ands(cid:3)0.9,we .co
SeparatingtheTwoCasesofMaleDeficitandMale Þndthatq*(cid:3)0.1/9.0(cid:9)0.011.Thisisidenticaltothree m
Excess.TheequationsformaledeÞcitaboveimplythat decimals with the value of z* (when q (cid:3) 0) given /ae
stMhme(cid:3)asleleatsFwt/vo(a1lg-uisveze)osfaztnrd(cid:3)apFpyi(cid:2)n(cid:3)gVf1o/(cid:3)rs(waahF(cid:2)i/c(h11t)r(cid:4),aapsnpydi)n.tghEoiqsfumisatatilhneegs pqmrae(lvpeisroouapstolybr.taiFoitinsg/utmrraeapt2siAn)gsahnbodewfozsrt(ehsueartivtnritavecartlaiocontfi/omcnaabpleetustwraeefteeonrf sa/article
contributestopopulationcontrol.Thus,theseparatrix baiting/trapping) for four values of the trapping of -ab
isgivenbyze: femaleswhenfemalesaremonogamous;Fig.2Bshows stra
ze(cid:3)y*(cid:3)1/s(a(cid:2)1) [10] tThheesfaomuervinatlueerascotfiotnrafpoprinthgeocfafseemoaflefesmarael:e1p)onlyognaem,2y). ct/10
asinEquation2.Thus,thetrappingoffemalesmustbe one-thirdofthedifferencebetween1.0andthecrit- 7/1
sufÞcient to eliminate the population alone in the ical value (y*), 3) two-thirds of the difference be- /8
equilibriumsituation.Inthetransientsituation,both tween1.0andthecriticalvalue(y*),andthecritical 1/7
malesandfemalesmaycontributetopopulationcon- 7
value,y*.Itisclearthatifbothfactorsareoperating, 1
troloreliminationbyimposingexcessmortalityover theconstraintsonthecontrolsystemaremorestrin- 73
thatrequiredforanequilibriumtobemaintained. b
gentthanifallbaiting/trappingoccursbeforemating. y
SomeMatingOccursBeforeCaptureofMalesand g
ThisshowsthatforMEbaits/trapstobeeffectivein u
SitomseeemAsftelirk.eTlyhisthsiattuasotimonemmaaytianpgpraonxdimtarateppreinaglitmy,aays canondtkroillllimngusatnbiensceocntspidoeprualbalteio,nan,tdheifrmatoereofthatatnraactfieown est on
occur together, as mating normally occurs at dusk, malesmatebeforevisitingtheMEsources,themethod 05
althoughmostmaletrappingoccursthroughouttheday appears likely to fail. However, if females are also A
beforemating.Inthiscase,therewillbeamixtureofthe p
equationsshowninTable1andEquation1.Assumethat attractedandkilled,thesituationismoreoptimistic.In ril 2
a proportion q of the (1 (cid:4) z)M males that visit the fact,thisappearstooccurandcanbesigniÞcantÑitis 01
theso-called“pseudomale”responsebyfemaleswhen 9
baits/trapswillmatebeforebeingattractedtotheseME
males are rare and females are old and virgin (see
sources,whereasaproportion(1(cid:4)q)visitthebaits/
Nakagawaetal.(1970)andMcInnisetal.(1994)for
trapswithoutmating.Thenamixtureofthetwomodels
CeratitiscapitataWied.withthemaleluretrimedlure;
willapply.Theequationsformalemonogamyandat-
unpublished Þeld observations (Cunningham,
tractionofonlymalestobaits/trapsareshowninTable
McInnis, and others) for B. dorsalis with ME; and
2.Thecriticalvalueofzatequilibrium,z*,isgivenby:
Þeldcagestudies(Bull2010)forBactroceratryoni
z*(cid:3)[(1(cid:4)s)(cid:4)asq]/[as(1(cid:4)q)(cid:2)s(1(cid:4)s)] Froggatt). Thus, perhaps, if ME could be used in
conjunctionwithafemalelure,attractioncouldbe
[11]
enhanced and MAT control may be more feasible.
Then the largest value of q (q ) compatible with Theotherthreeaforementionedmodelsyieldsim-
m
controliswhenz(cid:3)0(i.e.,allmalesareattractedand ilarresultsformixedorderoftrappingandmating,
86 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
attracted but somehow were not captured at ME
sources,thusescapingbeingkilled.Thisissimilarto
thesituationinwhichsomemalesmatebeforebeing
attracted to the traps inasmuch as they get to mate
before dying. In terms of the model, the two are
identical, except for the fact that males that escape
beingkilledsurviveuntiltheydienaturally,buthaving
mated,thisdoesnotaffectthefemalefecunditymuch
differentlyfrommalematingbeforevisitingMEbaits/
traps.
MalesCapableofMultipleMatingsPerDay
D
Aspreviouslyshown,ifmaleshaveunlimitedmale ow
mating capability and can always inseminate all the nlo
receptive females, the population cannot be con- a
d
trolled by deployment of ME baits/traps unless fe- ed
malesarealsocapturedandkilled.Nowassumethat fro
eachmaleiscapableofmatingntimesperday.Inthat m
h
case,theÞrstsetofequationsinTable1canbemod- ttp
iÞedtoincludethefactthateachmalecanmatewith s
nfemales,sothatthefractionthatmalesrepresentof ://a
c
the total virgin female population becomes nzM/V a
i i d
becauseeachmalehereisworthnmalesthatcanonly em
mateonceperday.Theconditionforanequilibrium ic
toexististhatasnz(cid:3)(1(cid:4)sz)(1(cid:4)s)anditgivenby .ou
V* (cid:3) F* [a (cid:4) (1 (cid:4) s)], M* (cid:3) F* (1 (cid:4) s)/snz. The p.c
o
criticalvalueofz,z*,isgivenby: m
/a
z*(cid:3)(1(cid:4)s)/s[an(cid:2)1(cid:4)s] [13] e
Fig.2. IsoclinesofcapturelevelsofmalesatMEbaits/traps sa
thatwillresultinpopulationeliminationwhenaproportion,q, Comparingthiswiththecasewheren(cid:3)1,wenote /a
ofinsectsthatvisitthetrapsmatebeforevisitingthetrapsand thatthecriticalvalueforzisamuchlowervaluehere, rtic
1(cid:4)qthatvisitthetrapswithoutmating.(A):Femalesare asnisinthedenominator(inmostcases,(cid:9)1/ntimes le-a
ametatocrnhaoctpgiaaomnneoltuorse;tp(hrBee)s:ebFnateitms1/)atrleyasp(cid:3)sar(e1f.up0lo,llsyfogeamtmhaaolteutssh.ueTrrhveieviasfol)nu;or2lf)inemeysa(cid:3)ilne mthaelevaclaupetuorfez*suwrvitivhonrs(cid:3)hip1)m.Tushtubs,etmouacchhielevsescifomntarloels, bstrac
0.67(1.0(cid:4)y*)(cid:2)y*;3)y(cid:3)0.33(1.0(cid:4)y*)(cid:2)y*;and4)y(cid:3)y*, arecapableofmultiplematingseachday,compared t/1
thecriticalvalueoffemalesurvivorshiptoachieveeradication. withmalesmatingonlyonceeachday.Inthiscase,the 07
captureoffemalesismadeevenmoredesirable. /1
/8
andtheequationsforthecriticalmaletrappingrates 1
/7
areshowninTable3. 7
VirginFemalesCannotMateImmediatelyonAdult 1
7
Emergence 3
b
SomeMalesEscapeAttraction y
In some species, a period of maturation including g
u
AproportionofmalesmaynotbeattractedtoME, nutritionalfeedingmustoccurbeforevirginfemales e
s
have fed already on a natural ME source, or were canmateandlayeggsorbeforemalescanmate.This t o
n
0
Table3. Modelsshowingthefourcombinationsof1)femalemonogamyversuspolygamy,and2)thetrappingofonlymales(“Trap 5
Males”)versusthetrappingofbothmalesatarate(1(cid:2)z)andfemalesatarate(1(cid:2)y)(“TrapM&F”) A
p
Model z* z (q ) ril 2
e m 0
Monogamy (1(cid:4)s)/(cid:7)as(cid:2)s(1(cid:4)s)(cid:8) Ð (1(cid:4)s)/as 19
Trapmales
Monogamy (1(cid:4)sy)/(cid:7)as(cid:2)s(1(cid:4)sy)(cid:8) 1/s(a(cid:2)2) (1(cid:4)sy)/as
TrapM&F
Polygamy (1(cid:4)s)/as Ð (1(cid:4)s)/(cid:7)as(cid:4)s(1(cid:4)s)(cid:8)
Trapmales
Polygamy (1(cid:4)sy)/as 1/s(a(cid:2)1) (1(cid:4)sy)/(cid:7)as(cid:4)s(1(cid:4)sy)(cid:8)
TrapM&F
Thecriticalsurvivorship(z*)isshownaswellasthesurvivorship(z)thatcausesmalestobeequaltoavailablefemalesatequilibriumand
e
themaximumproportionofcases(q )inwhichmatingoccursbeforecaptureforcontroltobepossible.Thetwocasesoftrappingonlymales
m
donotyieldanyvaluesforwhichmalesequalavailablefemalesatequilibrium,asthemalesmustbelessthantheavailablefemalesatequilibrium.
January2014 BARCLAYANDHENDRICHS:MODELOFMATFORCONTROLLINGBactrocera 87
sof0.7and0.9,aswellasvaluesof(cid:4)rangingfrom1to
10.Figure3showsthatcriticalsurvivorshipfromtrap-
pingincreaseswiththedelayinmating,indicatingthat
controlbecomeseasierasthedelayinmatingbecomes
longer.Also,wecansolveforthevalueofzatwhich
malesequalsreceptivefemales.Itisfoundbyequating
zMandy(sy)(cid:4)(cid:4)1aF,sozaF/(1(cid:4)sz)(cid:3)(sy)(cid:4)aF/s,
giving:
z (cid:3)(sy)(cid:4)/s[1(cid:2)(sy)(cid:4)] [16]
e
We can see that when (cid:4)(cid:3) 1 and y (cid:3) y*, we get
z (cid:3)1/s(a(cid:2)2),asaforementioned.FormaledeÞcit,
e
wegettheequationsshowninTable2,andfromthe D
o
secondtwoequationsweobtainthecriticalvalueofz, w
n
z*,of: lo
a
Fig.3. Effectsofamatinglaginfemales(horizontalaxis) z*(cid:3)(1(cid:4)sy)/[as(cid:2)s(1(cid:4)sy)] [17] de
d
osunrvtihveorvshaliupe(voefryti*c,althaexisc)r.itTichaelfvoaulruecufrovresfesmhoawlevtaralupepsinogf whichisthesameaswithonlyatimelagof1day.This fro
m
y*forfourcombinationsoftheparametersaands,being5.0 illustratesthefactthataÞxedamountofmalemor- h
and10.0foraand0.7and0.9fors. talityisrequiredtobringthepopulationtoitsunstable ttp
canbemodeledbyintroducingatimelagintoEqua- eTqhueilmibirniiummu,mwhvealtuheeroffesmuravleivmorasthinipgriseqdueilraeydedtooarlnloowt. s://ac
tion1,withy(cid:3)z(cid:3)1.0,suchthatthevirginfemales thepopulationtosurviveisgivenby(s(cid:4)(cid:4)1(cid:3)(1(cid:4) ad
e
ofage(cid:4)willmateifmalesareavailable.Theparent s)/as),whereasEquation17givestheamountofsur- m
modelwithnotrappingthenbecomes: vivorshipaftertrappingtoachievetheequilibrium.As ic.o
(cid:4)increases,eventuallyavalueisreachedatwhichthe u
p
V(cid:3)aF(cid:2)asF(cid:2)s2aF(cid:2)s3aF(cid:2)...(cid:2)s(cid:4)(cid:4)1aF respondingfemalepopulationisreducedtothesame .c
o
sizeasthemalepopulationrequiredforcontrol. m
(cid:2) aF[1(cid:4)s(cid:4)]/(1(cid:4)s) Thetwocasesforfemalepolygamyaresomewhat /ae
s
F (cid:2) s(cid:4)aF(cid:2)sy F morecomplicated,butthecriticalvaluesofzare: a/a
M (cid:2) aF(cid:2)s M [14] z*(cid:3)(1(cid:4)sy)[y(cid:2)a(sy)(cid:4)(cid:4)1] rtic
le
Forthismodel,populationpersistencerequiresthat /{s(1(cid:4)sy)[y(cid:2)a(sy)(cid:4)(cid:4)1](cid:2)a2(sy)(cid:4)} [18] -ab
as(cid:4)(cid:6)1(cid:4)s. fortheattractionofbothsexes,andthesameequation stra
attWracetÞiornstocfoonnsildyemrtahleesctaosethoeffbeamitsa/letrmaposnaongdamexyteanndd wiItnhysu(cid:3)mm1aforyr,thaetaimttreaclatigoninoffeomnlaylemmalaetsi.ng assists ct/10
7
themtoincludedelayedmating.ThisisamaledeÞcit controlbytrappingfemalesbuthaslittleornoeffect /1
casebecausemaleexcesscannotcontrolthepopula- /8
oncontrolbytrappingofmales.Thispresumablyre- 1
tionwhenfemalesarenotattractedtothetraps.The ßectstheneedtoreducethemalestoalevelthatwill /77
equationsareshowninTable2,andtheequilibrium 1
reducereproducingfemalestoadegreethatwillnot 7
equationsforFandMtogetheryieldthecriticalvalue 3
allow one-for-one replacement of individuals in the b
ofz,z*,forcontrol: y
population,andthislevel(ofmales)willnotdepend g
z*(cid:3)(1(cid:4)s)/[as(cid:2)s(1(cid:4)s)], on how many females die from other causes before ue
s
mating;thecriticallevelisdeterminedbytheavailable t o
thesameaswithnodelay(Equation3)inmating. males,notbythevariouscausesofdeathofthefemales n
Forthecaseofattractionofbothmalesandfemales thatdonotgettoreproducesuccessfully.Thismodel 05
tobaits/traps,wehavethetwopossibilities:maleex- isfurtherdevelopedintheage-structuredmodellater Ap
cessormaledeÞcit.Theequationsformaleexcessat inthetext. ril 2
equilibrium are also shown in Table 2 and give the 0
equilibriumconditions:V(cid:2)F(cid:3)aF/(1(cid:4)sy). 19
FromtheequationforFweget(1(cid:4)sy)(cid:3)a(sy)(cid:4) ConclusionsFromtheSimpleModels
giving: ReferringtoTable3andFig.1,itisseenthatthe
a(sy)(cid:4)(cid:2)sy(cid:4)1(cid:3)0 [15] trapping of both sexes is more efÞcient than trap-
pingonlymales.Thisisbecausethecriticalvalueof
Thus,maletrappingplaysnopartincontrolofthe male survivorship for control (z*) can be greater
populationifmalesareinexcessofreceptivefemales. when also some females are captured than when
When(cid:4)(cid:3)1,wehavetheEquation1,andEquation15 only males are trapped. Also, the technique works
reducestoy*(cid:3)1/s(a(cid:2)1),aspreviously.Ingeneral, somewhatmoreeffectivelyonspeciesinwhichfe-
thismodelisnotanalyticallysolvableforyfortimelags malesarepolygamousthanonmonogamousspecies,
(cid:6)2 d. Figure 3 shows numerical solutions for y* in asvirginandrematingfemaleshavetocompetefor
Equation15forvaluesofaof5and10andvaluesof the declining number of males. In addition, the
88 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
Table4. Age-structuredpopequationsshowingeggs(E),larvae(L),pupae(P),virginfemales(V),matedfemales(F),andmales(M)
Preadultstages
Eggstages Larvalstages Pupalstages
E1,t(cid:2)1(cid:3)(cid:10)kfFi,tmxhx L1,t(cid:2)1(cid:3)Eke,t P1,t(cid:2)1(cid:3)qklLkl,t
Ei(cid:2)1,t(cid:2)1(cid:3)Ei,t Li(cid:2)1,t(cid:2)1(cid:3)qiL Pi(cid:2)1,t(cid:2)1(cid:3)wPi,t
Eke,t(cid:2)1(cid:3)Eke(cid:4)1,t Lkl,t(cid:2)1(cid:3)qLkl(cid:4)1,t Pkp,t(cid:2)1(cid:3)wPkp(cid:4)1,t
Adultstages
Virginfemales Matedfemales Males
V1,t(cid:2)1(cid:3)wkpPkp,t F1,t(cid:2)1(cid:3)(cid:11)kv(cid:4)1siykv(cid:4)1V1,t M1,t(cid:2)1(cid:3)wkpPkp,t
Vi(cid:2)1,t(cid:2)1(cid:3)siyVi,t Fi(cid:2)1,t(cid:2)1(cid:3)siyFi,t Mi(cid:2)1,t(cid:2)1(cid:3)sizMi,t
Vkv,t(cid:2)1(cid:3)Skv(cid:4)1yVkv(cid:4)1,t Fkf,t(cid:2)1(cid:3)Skf(cid:4)1yFkf(cid:4)1,t Mkm,t(cid:2)1(cid:3)Skm(cid:4)1zMkm(cid:4)1,t
D
Thevariableswithsubscriptsioccurfrom1totheirrespectiveupperlimitsminusone. o
w
x,aEni,dtihstihsethneuhmabtcehraobfileitgygosfinegaggseocflaagsesiclaatsstixm.Aeltsoan,Ldt,hPes,uVm,foFre,agngdagMecalarsest1heisatdaukletnnufrmobmer1stionkafg;emcxlaissstihaettfiemcuentd(iitnydoafyasd).uPltaraagmeectlearsss nlo
x i,t i,t i,t i,t i,t a
ke,kl,kp,kv,kf,andkmarethemaximumnumbersofdaysoccupiedbyeggs,larvae,pupae,virginfemales,matedfemales,andmales,respectively. d
Parametersqandwarelarvalandpupaldailysurvivorship;sisthenaturaldailysurvivorshipofmalesandfemales;andyandzarethe ed
survivorshipsoffemalesandmales,respectively,afterdailyvisitstotheMEbaits/traps. fro
m
h
constraintsonwhethermaletrappingoccursbefore dynamic form, they are cumbersome and have too ttp
oraftermatingarelessstringentwhenfemalesare many parameter values to easily deal with. We will s
alsotrappedandarepolygamous.Thus,speciesthat develop the equations in equilibrium form as a pre- ://a
c
arepolygamousandinwhichfemalesrespondtoME cursortothelife-tabletreatmentofB.dorsalisdonein a
d
aremostamenabletobeingcontrolledbyMEtrap- alatersection.Welaterusetheequilibriumtoderive em
ping. If female mating is delayed as a result of re- criticalvaluesofz,z*,andofy,y*,thesurvivorshipsof ic
quiredmaturation,thenreductionofthemalepop- malesandfemales,respectively,afterdailyattractionto .ou
p
ulationtothesmallerreceptivefemalecohortwill sourcesbaitedwithME.Muchofthemessinessofthe .c
o
beevenmoredifÞcultthanifmatingisimmediate equationsdisappearsintheequilibriumformulation;in- m
onadultemergence;however,controlbyattraction deed,thewholeeffectofthepreadultcomponentscan /a
e
offemalesbecomeseasierasaresultofthedelayin bereducedtoonenumber:preadultsurvivorship(here s
a
mating, because natural mortality during the mat- labeled(cid:3)),whichmeasurestheproportionofeggsthat /a
urationperiodreducesthesizeofthefemalepop- surviveuntiltheyemergeasadults.Wealsoneedmean rtic
ulation, and thus effectively also the total fertility dailyfertility((cid:5)),called“meanfertileeggsperßy-day” le-a
rate. Nevertheless, the success of this control byCarey(1989),and(cid:5)is: bs
methoddependsontherebeinglittlematingbefore tra
trappingeachday.Mostofthemodelsindicatethat (cid:4)(cid:3)(cid:10)lxmxhx/(cid:10)lx [19] ct/1
mating before trapping will probably render this 0
7
controlmethodineffective. wherel isthesurvivorshipfromagextoagex(cid:2)1, /1
x /8
m is the fecundity of age class x, and h is the 1
AnAge-StructuredVersionoftheModel haxtchability of eggs laid by age class x aduxlts. The /771
order of events implicit in the equations for this 7
3
VirginBactrocerafemalesareanautogenous,requir- modelis:overnightmortality,morningoviposition, b
y
ingproteinbeforetheybecomereceptivetomating, andeveningmating.Inaddition,theorderoftrap- g
andthisrequiressometimetoobtain(DrewandYuval ping and mating is important here, as has been ue
s
2000).Toaccommodatethisfeatureeasily,weneedan demonstratedinthesimplermodelspreviously,so t o
age-structuredmodelinwhichatleastthevirginfe- theorderadoptedfortheage-classmodelswillbe n
0
malesneedtobetalliedbyageclassuntilanappro- trappingÞrst,andthenmating. 5
A
priatetimehaselapsedforthemtoacquireproteinand Theequilibriumvaluesofeggsarefoundbydrop- p
becomereceptive.B.dorsalisfemalesfromstrainsthat ping the time subscript, t; the remaining subscripts ril 2
havebeen,forgenerations,undermass-rearingcon- denote age. There is no mortality shown in the egg 0
1
ditionsinthelaboratoryrequireabout6dafteremer- stage,astheonlyeggsweconsiderarethosethatwill 9
gencebeforetheyarereceptivetomating,andthisis hatch.ThesymbolsinTable5areasfollows:keisthe
under ideal nutritional and temperature conditions numberofdaysrequiredforegghatch;klandkpare
(Vargasetal.1984);inthewild,virginfemalesmay thenumberofdaysforlarvalandpupaldevelopment,
sometimesrequireupto29dforsexualmaturation. respectively;kv,kf,andkmarethelifespansforvirgin
Thisallowsmuchmoremortalitytooccurthanifthey females,matedfemales,andmales,respectively.The
wereimmediatelyreceptiveonemergence.Theage- subscriptTdenotesthetotalforthelifestageand(cid:10)(kf)
structuredequationsforthepreadultandadultstages denotesthesumfromi(cid:3)1tokfoftheexpressionto
are shown in Table 4. These equations are entirely therightofthesummationsign,i.e.,F m h,and(cid:5)is
i i i i
density-independent. They allow for age-dependent theproductm h (thefecundityandegghatchability
i i
survivorshipoflarvaeandpupaeandage-dependent ofadultsofageclassi).Thesumistakenoverthekf
fecundity and survivorship of adults. In their fully matedfemaleageclasses.Theequilibriumforlarvae
January2014 BARCLAYANDHENDRICHS:MODELOFMATFORCONTROLLINGBactrocera 89
Table5. Equilibriumvaluesofthevariablesinthefullage-structuredmodel,showingeggs(E),larvae(L),pupae(P),virginfemales
(V),matedfemales(F),andmales(M)
Eggs:E (cid:3)E (cid:2)E (cid:2)E (cid:2)...(cid:2)E (cid:3)keE,(cid:3)ke(cid:10)(kf)F m h (cid:3)ke(cid:10)(kf)(cid:5)F
Larvae:TL (cid:3)1L (cid:2)2q L3(cid:2)q q L k(cid:2)e ...(cid:2)(1(cid:11)(kl(cid:4)1)q)Li (cid:3)i(1i (cid:2)q (cid:2)q qi (cid:2)i ...(cid:2)((cid:11)(kl(cid:4)1)q)L (cid:3)(1(cid:2)q (cid:2)q q (cid:2)...(cid:2)
(cid:11)(kl(cid:4)1T)q)E1 1 1 1 2 1 i 1 1 1 2 i 1 1 1 2
Pupae:P (cid:3)iP (cid:2)kew P (cid:2)w w P (cid:2)...(cid:2)((cid:11)(kp(cid:4)1)w)P (cid:3)(1(cid:2)w (cid:2)w w (cid:2)...(cid:2)(cid:11)(kp(cid:4)1)w)P (cid:3)(1(cid:2)w (cid:2)w w (cid:2)...(cid:2)
(cid:11)(kp(cid:4)T1)w1)q L1 1(cid:3)(1(cid:2)1 w2 (cid:2)1 w w (cid:2)...(cid:2)(cid:11)(kp(cid:4)i 1)1w)((cid:11)(kf)q1)E1. 2 i 1 1 1 2
Virginfemalesi:Vkl(cid:3)kVl (cid:2)V (cid:2)1V (cid:2)1...2(cid:2)V (cid:3)(1(cid:2)sy(cid:2)si sy2(cid:2)siss kye3(cid:2)...(cid:2)ykv(cid:4)1(cid:11)(kv(cid:4)1)s)V (cid:3)(1(cid:2)sy(cid:2)ssy2(cid:2)
sss y3(cid:2)...(cid:2)T ykv1(cid:4)1(cid:11)2(kv(cid:4)31)s)((cid:11)(kp)kvw)P 1 12 123 i 1 1 12
(cid:3)(112(cid:2)3sy(cid:2)ssy2(cid:2)sss y3(cid:2)...(cid:2)i ykv(cid:4)1(cid:11)(ikv(cid:4)11)s)(cid:6)E
Matedfem1ales:1F2(cid:3)F 1(cid:2)2F3 (cid:2)F (cid:2)...(cid:2)F (cid:3)(1(cid:2)syi(cid:2)sskey2(cid:2)ss s y3(cid:2)...(cid:2)ykf(cid:4)1(cid:11)(kf(cid:4)1)s)F
(cid:3)(1(cid:2)sy(cid:2)ssTy2(cid:2)1...(cid:2)2ykf(cid:4)31(cid:11)(kf(cid:4)1)kfs)(ykv(cid:11)(1kv)s)(cid:6)12E 12 3 i 1
Males:M1(cid:3)M1(cid:2)2 M (cid:2)M (cid:2)...(cid:2)M (cid:3)(1i(cid:2)sz(cid:2)ssiz2(cid:2)ksess z3(cid:2)...(cid:2)zkm(cid:4)1(cid:11)(km(cid:4)1)s)M
(cid:3)(1(cid:2)sTy(cid:2)s1sy2(cid:2)2sss3z3(cid:2)...(cid:2)kzm(km(cid:4)1)(cid:11)1(km(cid:4)11)2s)(cid:6)E123 i 1
1 12 123 i ke
D
kvT;ih.ee.,n(cid:10)o(tkavt)io(cid:11)n(hkve)resii(cid:3)sso(1m(cid:2)ewsh1yat(cid:2)nosn1ss2tya2nd(cid:2)arsd1,s2asn3dy3(cid:10)(cid:2)(kv.).(.(cid:11)(cid:2)(kyvk)vsi(cid:11))(m[seuapin]skvt)hsei)s,uamndofsitmhielaprlryodfourcttshoefoatdhuelrtpsouprvcivoomrsphoinpesnfrtos.mi(cid:3)1toi(cid:3) ownlo
(cid:6)iEsi(cid:6)is(cid:3)th(e(cid:11)n(ukmp)bweir)o(f(cid:11)eg(kgls)iqni)a.geclassi;mxisthefecundityofadultageclassx,andhxisthehatchabilityofeggsofageclassx.Theparameter aded
fro
isshowninTable5andinvolvesthelarvalsurvivorship showrecruitsandthenextthreeshowtotalsatequi- m
q,beingthesurvivorshipfromstageitostagei(cid:2)1. librium.Also,femalesdonotmateuntildaykvoftheir h
i LT (cid:2) (1(cid:2)q1(cid:2)q1q2(cid:2)...(cid:2)(cid:11)(kl(cid:4)1)qi)Eke. lifeMaasleasdiunltEs.xcessofReceptiveFemales. ttps://a
c
paWrameeatsesrusmdeoanoctovnasrtya.nItnteadmdpiteiroantu,(cid:11)re(kslo(cid:4)t1h)aqt itshtehsee V1(cid:3)(cid:6)(cid:5)FT ade
i m
rspiirmgohidtlauorcfltythfreomproid(cid:3)uc1ttsoigknl.(cid:4)Th1eoefquthileiberxiupmresfosiropnutpoatehies F1(cid:3)skvyMV1(cid:3)kv(cid:3)(cid:6)y(cid:5)kvFVT1(cid:11)(kv)si [21] ic.oup.c
PT (cid:2) (1(cid:2)w1(cid:2)w1w2(cid:2)...(cid:2)(cid:11)(kp(cid:4)1)wi)((cid:11)(kl)qi)Eke VT(cid:3)V1(cid:10)(kv(cid:4)1)yj(cid:11)(j)si om/a
as P1 (cid:3) qkl Lkl and also Lkl (cid:3) L1 (cid:11)(kl (cid:4) 1) qi. The FT (cid:2) F1(cid:10)(kf(cid:4)1)yj(cid:11)(j)si esa
1prpoudpuacltsa,g(cid:11)e(cklpa(cid:4)ss1e)swani,dantdhe(cid:11)k(lkll)aqrviaarleagtaekcelnasosvees.rAkpls(cid:4)o, MT (cid:2) M1(cid:10)(km(cid:4)1)zj(cid:11)(j)si [22] /artic
thetotalpreadultsurvivorship,(cid:3),is: 1,kInf(cid:4)thi1s,coarskem,th(cid:4)e1suimnsEaqrueattiaoknen22f,roanmdjth(cid:3)e0ptroodkuvc(cid:4)ts le-ab
(cid:3)(cid:3)((cid:11)(kp)wi)((cid:11)(kl)qi) [20] aretakenfromi(cid:3)0toj,withs0(cid:3)1.0.Theseequations stra
femThaleese(qFu)il,ibanridamfoarletost(aMl v)iragriensfimemilaalrelyst(hVe)s,ummasteodf fcoalnlobwes:solved numerically for y* by substitution as ct/10
7
theirindividualageclassesuptotheirmaximumval- FT(cid:3)(cid:10)(kf(cid:4)1)yj(cid:11)(j)siF1(cid:3)((cid:10)(kf(cid:4)1)yj(cid:11)(j)si)((cid:11)(vk)yjsi)V1 /1/8
suteasg.eTbheefsoereeqhuailticbhriinaga,reexaclleipnttfeorrmtshoefeEgkges,,thwehliacshteagreg (cid:2) (cid:12)(cid:10)(kf(cid:4)1)yj(cid:11)(j)si)((cid:11)(kv)yjsi)(cid:6)(cid:5)FT. 1/77
TinhetesermesquoaftiFoTn,stahsesutmotealthnautmalbleadruolftamgeatseadrefeemquaalellsy. Cexapnrceeslsliionngethqeuaftaecdtotros1F.T0othnaetitchaenrbseidseorlveesudltnsuimnearn- 173 b
attractedtotraps/baits;ifthisisnotthecase,thenthe icallyfory.Thisexpressionis: y g
pfarcotdoursctysioanfdthzeicianndibvieduchalanaggeed-dteop(cid:11)enydie,notrs(cid:11)urzvii,vtohre- (cid:6)(cid:5)((cid:10)(kf(cid:4)1)yj(cid:11)(j)si)((cid:11)(kv)yjsi)(cid:3)1.0 [23] uest o
shipsinthesameformulationsastheadultsurvivor- Life-TableEquivalence.Wecanrelatethistostan- n
0
ships, s. The evaluation of these equilibria requires dard life-table symbology by noting that if l is the 5
considerableknowledgeoftheeffectsofageonthe survivorship (i.e., proportion still alive) fromxovipo- Ap
varioussurvivorships.Life-tableanalysiswillbeuseful sitiontototalagex,andifthepreadultstagestotale ril 2
inprovidingsomeofthisinformation,butthepreadult days, then the day of emergence of adults is e, and 0
1
survivorshipscanbereplacedbyanoverallmeasure- l (cid:3)(cid:6);also,thesurvivorshipsoftheadultstagesare 9
e
mentofpreadultsurvivorship((cid:6)),ifthatisavailable. le(cid:2)1,le(cid:2)2,le(cid:2)3,...etc.Inthesymbologyusedearlier,
Inthatcase,theequilibriaaremodiÞed,as(cid:6)(cid:3)((cid:11)(kp) survivorshipoftheadultstagesares forthesurvivor-
i
r)((cid:11)(kl)q),andsoV (cid:3)M (cid:3)(cid:6)(cid:5)(cid:10)(kf)F,andF (cid:3) shipfromtheithadultagetothei(cid:2)Þrstadultage.
Vsui1myskva(cid:11)n(dkvp)isrio.dInucatdsdcitaino1nb,ethee1vaaflouraetmedenwtihoenniedthvearcio1ouns- lTeh(cid:2)us3,(cid:3)le(cid:6)(cid:2)s11(cid:3)s2(cid:6)s3s(cid:3)1(cid:3)s3sl1el(cid:2)e,2le..(cid:2).e2t(cid:3)c.,(cid:6)sos1ths2a,t(cid:3)(cid:6)(cid:10)s2(klfe(cid:4)(cid:2)11),
stants are known. The difÞculty comes in trying to (cid:11)(j) s (cid:3) (cid:6)[s (cid:2) s s (cid:2) s s s (cid:2) ... (cid:2) s s ...
i 1 1 2 1 2 3 1 2
obtainaclosed-formexpressionfory*andz*. skf(cid:4)1](cid:3)le(cid:2)1(cid:2)le(cid:2)2(cid:2)le(cid:2)3(cid:2)...(cid:2)le(cid:2)1(cid:2)kf(cid:4)1
PopulationEquations.Wecanformulatetheequi- sothatEquation23canbewrittenas:
libriumpopulationequationsaswasdoneinEquation (cid:5)((cid:10)(kf)yxl )ykvl (cid:2) (cid:6),where1 (cid:7) x (cid:7) kf
1, with the equations not shown conforming to the e(cid:2)x e(cid:2)kv
equationsshowninTable4.TheÞrstthreeequations [24]
90 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
Table6. Age-structuredpopequationsshowingearliestageclassandtotalvirginfemales(V andV ),matedfemales(F andF ),
1 T 1 T
andmales(M andM )
1 T
Maleexcess MaledeÞcit
Recruits
V (cid:3)(cid:6)(cid:5)F V (cid:3)(cid:6)(cid:5)F
1 T 1 T
F (cid:3)syV (cid:3)skvykvV F (cid:3)szM
1 kv 1 1 T
M (cid:3)(cid:6)(cid:5)F M (cid:3)(cid:6)(cid:5)F
1 T 1 T
Totals
V (cid:3)V (1(cid:4)skvykv)/(1(cid:4)sy) V (cid:3)(1(cid:4)skvykv)/(1(cid:4)sy)V s (cid:2)s (V (cid:4)M)(cid:10)(cid:13)siV
T 1 T 1 i kv kv T 1
F (cid:3)F/(1(cid:4)sy) F (cid:3)1/(1(cid:4)sy)F
T 1 T 1
M (cid:3)M/(1(cid:4)sz) M (cid:3)1/(1(cid:4)sz)M
T 1 T 1
Parameters:sisthenaturaldailysurvivorshipofmalesandfemales,kvisthenumberofdaysforvirginfemalestobereadytomateafter
emergence,andyandzarethesurvivorshipsoffemalesandmales,respectively,afterdailyvisitstotheMEbaits/traps.Maleexcess:males D
aremorenumerousthanreceptivefemales;maledeÞcit:malesarelessnumerousthanreceptivefemales. o
w
n
lo
a
andthiscanbesolvednumericallyforyoncethedaily ricifonetalliessurvivorsbydays.Inaddition,ifthere de
d
surMviavloesrsFheiwpseranTdhaonthReerceppatriavmeeFteemrsalaerseknown. ifsemanaluesp,ptehrenlimthiettsoumthfeorloVnTge(vTiatybloef1a)dbueltcommaeless(a1n(cid:4)d from
V1(cid:3)(cid:6)(cid:5)FT sskkvmyzkkvm))//((11(cid:4)(cid:4)ssyz)).anIfdththeeresiusmnofourpMpeTrbliemciot,mtehsen(1th(cid:4)e http
s
F1(cid:3)skvz MT tshuimsscacsoer,rwesepognetdeinxgpraersesi1o/n(s1f(cid:4)orsyy)ananddz1b/y(1co(cid:4)nsstzr)u.cItn- ://ac
a
M (cid:3)(cid:6)(cid:5)F [25] ingtheappropriateequationsWiththesesimpliÞca- d
1 T e
tions,weobtaintheequilibriumequationsshownin m
V (cid:2) V (cid:10)(kv(cid:4)1)yj(cid:11)(j)s (cid:2)s (V (cid:4)M )(cid:10)(cid:13)(cid:11)(j)s V ic
T 1 i kv kv T i 1 Table6formalesbeinginexcessofreceptivefemales. .o
F (cid:3)F (cid:10)(kf(cid:4)1)yj(cid:11)(j)s HeretheÞrstthreeequations(Table6;MaleExcess) up
M T(cid:2) M1 (cid:10)(km(cid:4)1)zj(cid:11)(j)is [26] represent equilibrium recruitment and the last three .com
T 1 i representequilibriumtotalsofthethreecomponents:V, /a
Thesolutionoftheseequationscanbefoundnu- F,andM.Thisyieldsanequationforthecriticalvalueof e
s
mericallywhentherelevantcoefÞcientsareknown. ythatwouldholdthepopulationincheck;itappears a/a
FromEquations25and26weequate: impossible to solve explicitly, but it can be shown by rtic
takingderivativesthatthevalueofy*isapositivefunc- le
FT (cid:2) F1((cid:10)(kf(cid:4)1)yj(cid:11)(j)si) tionoftheageofmatingofvirginfemales,kv,andsample -ab
(cid:3)(skvzMT)((cid:10)(kf(cid:4)1)yj(cid:11)(j)si) numericalsolutionsareshowninFig.4.Here, stra
c
(cid:3)(skvz)M1((cid:10)(km(cid:4)1)zj(cid:11)(j)si)((cid:10)(kf(cid:4)1)yj(cid:11)(j)si) (cid:12)skvykv)(cid:6)(cid:5)/(1(cid:4)sy)(cid:3)1 [29] t/1
0
(cid:3)(skvz)((cid:6)(cid:5)FT)((cid:10)(km(cid:4)1)zj(cid:11)(j)si)((cid:10)(kf(cid:4)1)yj(cid:11)(j)si). 7/1
Aobgtaaiinn:wecancancelthefactorsFTonbothsidesto /81/7
7
1
(cid:12)s z)((cid:6)(cid:5))((cid:10)(km(cid:4)1)zj(cid:11)(j)s)((cid:10)(kf(cid:4)1)yj(cid:11)(j)s)(cid:3)1.0 73
kv i i b
[27] y g
u
andsolveforyandznumerically. es
By using the equivalence of life-table symbology t o
n
shownpreviously,Equation27canbewrittenas: 0
5
(cid:5)skvz((cid:10)(km)zxle(cid:2)x)((cid:10)(kf)yxle(cid:2)x)(cid:3)(cid:6) [28] April 2
0
1
where x goes from 1 to km and kf in the two sums. 9
Equation28willbeusedinthecomputationofyand
zforB.dorsalislater.
Fig.4. IsoclinesoftrappingsurvivorshipoffemalesatME
baits/traps using the simpliÞed age-structured model that
ReductiontoaSimplerModel will result in control of the population, assuming that a
proportionofthefemalesareattractedtosuchbaits/traps
TheEquations23and24,aswellas27and28,cannot
andthatmalesoutnumberreceptivefemales.Inthissitua-
besolvedanalyticallyforyandz,sotogetanapprox-
tion, trapping of males does not contribute to population
imationtothecriticalvaluesofyandz,wecansimplify
control.Valuesofy*areshownfornumbersofdaysdelay
theequationsIfadultshaveaconstantprobabilityof (1Ð20)inmatingofvirginfemalesforfourcombinationsof
dyingovertime(sothats1(cid:3)s2(cid:3)...(cid:3)si(cid:3)s),then theparametersaands.Adelayinmatingsubstantiallyre-
thesurvivorshipcurvewillbeexponential,orgeomet- ducesthecriticalvaluey*requiredforeradication.