Modeling the Area-Wide Integration of Male Annihilation and the Simultaneous Release of Methyl

16 Pages·2013·0.38 MB·English

ECOLOGYANDPOPULATIONBIOLOGY
Modeling the Area-Wide Integration of Male Annihilation and the
Bactrocera
Simultaneous Release of Methyl Eugenol-Exposed spp.
Sterile Males
HUGHJ.BARCLAY,1,2DONALDMCINNIS,3ANDJORGEHENDRICHS4
Ann.Entomol.Soc.Am.107(1):97Ð112(2014);DOI:http://dx.doi.org/10.1603/AN13010 D
o
ABSTRACT WepresentsimplemodelsthatdescribethefactorsinßuencingapplicationoftheSterile w
n
InsectTechnique(SIT)alone,followedbythefactorsaffectingtheintegrationoftheMaleAnnihilation lo
a
Technique(MAT)andSITsimultaneouslytoannihilatewildmalesandreplacethemwithmethyleugenol d
e
(ME)-exposedsterilemalesthatarelessresponsivetothetoxicMATbaits.TheremovalbyMAToflarge d
numbers of males to deprive females of mates is rather ineffective to suppress populations unless a fro
m
proportionoffemalesarealsoattractedtobaits.WhensimultaneouslyimplementingSITandMAT,much h
lessstringentrequirementsfortheannihilationofthewildmalepopulationapplyandthesterileßyrelease ttp
reafÞteciceanncybearreedwuhceetdhetroaopnrlyop(cid:2)o5rt%ionofotfhwatilwdhfeemnaSlIeTsiissaatptrpalcietdedaltoonMe.ATThebamitas,inwdheettehremrimnaonsttsooffthceonwtrilodl s://ac
a
malesrespondtotheMATbaitsbeforemating,andwhetheramajorityofME-exposedsterilemalesdo d
e
notrespondtothebaits.However,ahighrematingfrequencyinmatedfemales,andthereleaseofonly m
sterilemalesorbothsterilemalesandfemales,donotgreatlyaffecttheeaseofcontrol.Weconcludethat ic.o
thesimultaneousMATandSITapplicationissynergisticandthecombinationappearstobeanextremely u
p
powerfulcontrolstrategyfortephritidfruitßypeststhatareattractedtoME. .c
o
m
KEYWORDS fruitßy,Bactrocera,methyleugenol,maleannihilation,sterileinsect /ae
s
a
/a
rtic
le
/1
Theintegrationoftwoormoremethodsofinsectpest tivelyusedasa“lureandkill”technology(El-Sayedet 0
7
control as part of an integrated pest management al. 2009) to suppress (Balasubramaniam et al. 1972, /1
(IPM)approachhasremainedthedominantparadigm Qureshi et al. 1981, Vargas et al. 2010) or even to /97
ofpestcontrolforthelast50yr(FoodandAgriculture eradicate established populations of ME-responding /7
7
OrganizationoftheUnitedNations[FAO]2011),and fruitßiesofeconomicimportance(Steineretal.1965, 19
isespeciallyeffectiveandmoresustainablewhenap- Ushio et al. 1982, Koyama et al. 1984, Cunningham 4 b
pliedonanarea-widebasis(Hendrichsetal.2007). 1989,Anonymous1991,Nakamorietal.1991,Malavasi y g
However,integratedcontroltacticscanbeadditive, etal.2000). u
e
synergistic,orevensubtractiveundercertaincircum- MAT alone has often been applied successfully. st o
stances (Knipling 1979). An understanding of how However,inothercasesmostlyinvolvingwellestab- n
thesemethodsinteractisthereforecrucialtothesuc- lished populations such a stand-alone approach has 11
cessofsuchIPMprograms. failedandthefollow-upwiththeSterileInsectTech- F
e
Methyleugenol(ME)isapowerfulattractantfor nique (SIT) was needed after reduced population bru
malesofmanytropicaltephritidfruitßyspecies,es- levels were achieved following MAT application a
ry
peciallyBactroceraspp.(Drew1974,1989;Whiteand (Cunningham 1989, Steiner et al. 1970, Shiga 1989). 2
ElsonÐHarris1992;TanandNishida2012).Thisstrong This has also been necessary for Bactrocera species 02
3
attractant,mixedwithapoison,hasbeenusedsince responding to the less powerful attractant cuelure
the 1950s for purposes of monitoring and control (Matsuietal.1990,Koyamaetal.2004),orwhereless
(SteinerandLee1955,CunninghamandSuda1986) ME-sensitive strains appear to be present and wild
by attracting, trapping, and killing the males. This males have sufÞcient access to natural ME sources
MaleAnnihilationTechnique(MAT)hasbeeneffec- (Shiga1989,Anonymous1991).IntegrationwithSIT
is also an option to avoid long-term pressure on a
populationbyselectingforlessME-responsivemales
1(Retired)PaciÞcForestryCentre,CanadianForestService,506
(Cunningham1989).
W.BurnsideRoad,Victoria,BC,CanadaV8Z1M5.
2Correspondingauthor,currentaddress:928OldEsquimaltRd., TheSIThasbeenwidelyandeffectivelyappliedfor
Victoria,BC,CanadaV9A4X3(e-mail:[email protected]). suppression, containment, or eradication purposes
31216ManuAlohaSt.,Kailua,HI96734. (Hendrichs et al. 2005), including against ME- and
4InsectPestControlUnit,JointFAO/IAEADivisionofNuclear
cuelure-respondingfruitßiesofeconomicimportance
TechniquesinFoodandAgriculture,InternationalAtomicEnergy
Agency,Wagramerstrasse5,P.O.Box100,A-1400Vienna,Austria. (Enkerlin2005).However,becauseofitsinverseden-
0013-8746/14/0097Ð0112$04.00/0(cid:2)2014EntomologicalSocietyofAmerica
98 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
sity dependence, SIT is generally not a stand-alone drichs2014)inwhichmaleswereassumedtobelim-
technology,requiringadditionalsuppressiontacticsto itedtotoonematingperday.Thosemodelsinturn
beabletoachieveadequatesteriletowildmaleover- were an extension of a model presented by Barclay
ßoodingratios(Knipling1979,Dycketal.2005).In- (1987) for the use of odor-baited traps to capture
tegrationofMATwithSIThassofarbeensequential, malesandfemalesofaninsectpestspecies,inwhich
rather than simultaneous, with MAT normally fol- maleshadanunlimitedcapacityformating.
lowedbySITafteraperiodofcessation(Steineretal. Barclay and Hendrichs (2014) identiÞed several
1970,Itoˆ andIwahashi1974,Shiga1989).Thereason biological factors that needed to be included in the
hasbeentoavoidtheresidualeffectofthetoxicME models. They examined the cases of 1) female mo-
orcueluresourcesthatalsoattractandeliminatemost nogamyversuspolygamy(frequentfemaleremating);
ofthereleasedsterilemales. 2)attractionofonlymalestothebaitsandtrapsversus
Nevertheless,thereisthepotentialforcombining attractionofbothsexes;3)afterbaitingortrapping,
simultaneouslytheSITwithMATapplication(Cham- males outnumber receptive females (male excess)
D
bersetal.1972,Shelly1994).Prereleaseexposureor versusreceptivefemalesoutnumbermales(maledef- o
w
directfeedingonMEcansigniÞcantlyreduce[byca. icit);4)captureoccurringbeforematingversusmat- n
two thirds in Bactrocera dorsalis (Hendel)] sterile ing occurring before capture; 5) some males mate loa
d
maleresponsetomale-annihilationbaitsortrapsinthe beforecapturetrappingandsomeaftercapture;6)the e
d
Þeld(Shelly1994).Thisopensthepossibilityofrou- abilityofmalestomatemanytimesperdayversusonly fro
tinelyincludingMEinthepostteneraldietofmaturing onceperday,and7)theexistenceofatimelagafter m
sterile males at ßy emergence and release facilities. emergence before females can mate. In most cases, h
WhilewildmaleshavetoactivelyseekoutMEsources matingbeforecapturerenderedtheMATineffective. ttp
s
in nature and would therefore continue being at- Furthermore,ifmalesoutnumberedreceptivefemales ://a
tracted and eliminated at ME baits and traps, ME- after baiting or trapping, then the model was effec- ca
d
exposed sterile males would have the advantage of tively reduced to a model of only females being e
m
havingsigniÞcantlyreducedattractiontoMEsources. trapped. Most of these same themes will be present ic
Sucha“malereplacement”approach(Robinsonand inthisarticleaswell.Weusetheterm“monogamy” .o
u
Hendrichs2005,Shellyetal.2010,McInnisetal.2011), throughouttoindicatethatthefemalesonceinsem- p
.c
simultaneouslyexchangingasigniÞcantportionofthe inatedarenotreinseminated.Thedetailsaboutsperm o
m
wildmalesinthetargetpopulationwithreleasedME-fed precedenceandsuper-inseminationareassumednot /a
sterilemales,woulddrasticallyincreasetheover-ßood- tomatterifmatingisrandom.Thefeatureofimpor- es
a
ingratioofsteriletowild(fertile)males,assumingthat tancehereispresenceorabsenceofcompetitionfor /a
wildmaleattractiontoMEbaitsandtrapsoccurslargely matingofvirginfemalesasaresultofmatedfemales rtic
beforefullsexualmaturationisreachedandmatingoc- remating. le
/1
curs(Wongetal.1989,Shellyetal.2008). TheUseofSITWithoutSimultaneousMATAppli- 0
7
PrereleaseconsumptionofME,however,provides cation.AmodelforSITalone,withnoMEexposure, /1
anothermajorbeneÞt.ME-fedsterilemalesnotonly may require six equations (Barclay and van den /97
havereducedresponsivenesstoMEsources,butalso Driessche1989):theequationsformatedfemales(F), /7
7
increased mating competitiveness compared with virginfemales(V),andmales(M)ofthewildtarget 19
ME-deprivedmales(ShellyandDewire1994,Tanand population are modiÞcations of the basic equations 4 b
Nishida1998,Shellyetal.2010,McInnisetal.2011). usedbyBarclayandHendrichs(2014),withthein- y g
ME-treatedmaleshaveenhancedaggregationbehav- clusionofthematingproportionbelow.Inaddition, u
e
ior, earlier onset of sexual attraction, and elicit in- there are three equations for sterile virgin females st o
creased female response, resulting in higher mating (W)(ifsterilefemalesarealsoreleased),sterilemales n
successwithvirginfemales(Weeetal.2007).Afur- (N), and sterilized mated females (sterile or fertile 11
ther beneÞt of ME pharmacophagy entails a role in virginfemalesmatedtosterilemalesandsterilevirgin F
e
defense, as such males are consumed less by some femalesmatedtowildmales)(G).Theequationsare bru
predators(TanandNishida1998).WhenME-fedBac- asfollows: a
ry
dtriosctuerrabepda,ptahyeayes(pDonretawneaonudslHyasneccroectke1a9r9e4c)tamlmalaetserairael Fi(cid:3)1(cid:4)sVi[Mi/Mi(cid:3)Ni](cid:3)sFi 202
3
thatdetersthepredator. G (cid:4)sV [N/M (cid:3)N](cid:3)sW (cid:3)sG
i(cid:3)1 i i i i i i
Thisarticleisasequeltoonethatinvestigatesthe
efÞcacyofMATalone(BarclayandHendrichs2014). Vi(cid:3)1(cid:4)a Fi(cid:5)k [1]
Themodelsinthisarticlehavenotbeentestedagainst W (cid:4)r
experimentaldatabecausethedatarequiredtodosofar i(cid:3)1
exceedthescopeofthisarticle.Wedo,however,apply Mi(cid:3)1(cid:4)a Fi(cid:5)k(cid:3)sMi
outage-structuredmodeltoempiricaldataonB.dorsalis.
N (cid:4)r(cid:3)(cid:2)N
i(cid:3)1 i
Here,aproportionofvirginfemalesmatewithwild
MaterialsandMethods males: M/(M (cid:3) N), and a proportion of virgin fe-
i i i
The models developed here are all extensions of males mate with sterile males: N/(M (cid:3) N). The
i i i
modelsoftheuseofMEinbaitsandtrapstocapture sterilereleaserateeverydayisrmalesandrfemales,
either males alone or both sexes (Barclay and Hen- and wild fertile insects have a daily survivorship, s,
January2014 BARCLAYETAL.:MODELOFMATANDSITFORCONTROLLINGBactrocera 99
while sterile insects survive at a rate of (cid:2)each day. FemaleMonogamyandtheReleaseofOnlySterile
Theseequationshaveoneunstablesteadystate(equi- Males. Mating Before Attraction to Baits and Traps.
librium)whenthecriticalsterilereleaserate,r*,is: Because only sterile males are being released, there
areequationsforwildfemalesfertilizedbywildmales
r*(cid:4)aF(1(cid:5)(cid:2))[as(cid:5)(1(cid:5)s)]/(1(cid:5)s)2 [2] (F),virginwildfemales(V),wildmales(M),sterile
males (N), and wild females that have mated with
Ifthepopulationfallsbelowthesteadystateandster- sterilemales(G).BecauseattractiontoMEsourcesin
ile insect releases continue, the population will be thisÞrstcasefollowsmating,thematingequationsdo
reducedtozero,whereasifthepopulationrisesabove notshowtheresultsofattractionandcaptureofthat
thesteadystateitwillexpandinunrestrictedgrowth. day, and so the Knipling-type ratio of wild males to
Theequationsarethesameforbothfemalemonog- totalmalesisgivenby[Mi/Mi(cid:3)Ni];therefore,mating
amyandpolygamy,asundertheseconditionsmales maleswillalwaysoutnumbervirginfemales,asmales
andvirginfemalesemergeinequalnumbers,sothere
arealwaysatleastasnumerousasreceptivefemales. D
In all models that follow, sterile male competitive isonlyonesetofequationstoanalyze,thatofmale ow
astbaitleitdy.isthesameasforwildmalesunlessotherwise ebxiacseesds,ianndfavalosroothfemoapleesr,aatisonmaolsstexvirrgatiniofiesmnaolremsaalrlye nload
immatureandonlyafewarereceptive.TheÞveequa- e
SimultaneousSITandMATApplicationWithAt- d
tractionofOnlyWildMalestoMEBaitsandTraps.In tionsareshowninAppendix1A.Alltheequationsin fro
Appendix1Aassumethatbothfertileandsterileadults m
themodelsthatfollow,wemustassumesomeorderof
surviveatarateseachdayfromonedaytothenext. h
events to set them up. Here, we assume that daily ThepossibilityofmaledeÞcitcanbedeterminedby ttp
eanmdefregmenacleesoofcacduurltesaacnhddraeylebaesfeoroefestiethrieleravtitrrgaicntimonaletos Þndingthepopulationsteadystateandthenequating s://a
thebaitsandtrapsormating.Tallyofadultstageswill themaleswiththevirginfemales.Theequilibriumis cad
found by assuming that there is no change in the e
therefore occur after adult emergence, then either m
mating or trapping, and then the other. Usually the variablesovertime,sothatthetimesubscriptscanbe ic
dropped.Thisgivestheequilibriumas: .o
orderwillbeunimportantsolongasitisconsistent. u
p
Thisisnotthecaseinthiscontext,however,formating V (cid:3) aF; M (cid:3) aF/(cid:6)1 (cid:4) sz(cid:7); N (cid:3) r/(cid:6)1 (cid:4) s(cid:7); .co
andattractiontobaitsandtrapseachday.Thisorder m
iscriticalforcontrolusingMEwithoutSIT(Barclay F (cid:3) sV(cid:8)M/(cid:6)M (cid:5) N(cid:7)(cid:9)/(cid:6)1 (cid:4) s(cid:7); /ae
s
andHendrichs2014),andisalsoimportantwhenusing G (cid:3) sV(cid:8)N/(cid:6)M (cid:5) N(cid:7)(cid:9)/(cid:6)1 (cid:4) s(cid:7) [3] a/a
MEWahnednSuITsedcoimnbcinoemdb,iansastieoennwbeitlhowb.aits and traps Then equating males to virgin females, we obtain: rticle
containingME,itisassumedthatbeforerelease,the M(cid:3)N(cid:4)V;thatis,aF/(1(cid:5)sz)(cid:3)r/(1(cid:5)s)(cid:4)aF.We /1
0
sstoertihliaztetdhmeaslteesrihleavmeableeesnheaxvpeossaetdistÞoeadstohuericreMofEMrEe-, cbaigngseerethbayninthspeercigtihotnsitdhea,tatsheaFl/e(ft1h(cid:5)ansdz)si(cid:10)deaiFs,a(lswinacyes 7/1/9
quirementsasacriticalprecursortopheromonepro- 1(cid:5)sz(cid:11)1.0),andalsor/(1(cid:5)s)(cid:10)0.Thus,theequality 7/7
duction (Tan and Nishida 2012) and are no longer neverholds,andmalesarealwaysinexcessatsteady 71
9
attractedtoMEsources(Shelly1994),whilethewild state. 4
malesareattracted. Fromequations3,wecansubstitutetheexpressions by
Weexamineallcombinationsofthecasesoffemale forV,M,andNintotheequationforFandsolvefor gu
e
monogamy versus female polygamy, the release of r.Doingthis,weobtainthecriticalvalueofr(sterile s
onlysterilemalesversusthereleaseofbothsexesin male release rate), r*, that will separate population t o
n
equalnumbers,matingbeforeattractionandcapture growthfrompopulationcollapse.Thisgives 1
1
atbaitsandtrapsversusthereafter,andthecasesof F
r*(cid:4)aF[as(cid:5)(1(cid:5)s)]/(1(cid:5)sz) [4] e
maleexcessversusmaledeÞcit.Therearethus16cases b
toinvestigate,althoughinsomecasesmalesareeither Equation 4 deÞnes a curve along which are pairs of rua
alwaysinexcessoffemalesoralwaysindeÞcitwhen valuesofrandzthatwillallbecriticalpairs;thiscurve, ry 2
atequilibrium,andthusonesetofequationsdoesnot called an isocline, separates success from failure. At 02
apply.Asthereismuchduplicationandoverlapamong anypointalongtheisocline,theequilibriumismain- 3
the16cases,theÞrstfourcaseswillbedevelopedin tained;ifrisincreasedorifzisdecreased,thenthe
somedetailandthentheequationsandresultsforall population will collapse; if r is decreased or if z is
16casesarepresentedintablesandappendices.The increased, then the population will increase un-
attractionandcaptureofbothwildandsterilemales checked.Figure1ashowsther–zisoclinesfromequa-
andtheattractionandcaptureofwildmalesaswellas tion4forfoursetsofparametervalues:a(cid:4)2,s(cid:4)0.7;
virgin and remating females are then examined as a(cid:4)2,s(cid:4)0.9;a(cid:4)10,s(cid:4)0.7;anda(cid:4)10,s(cid:4)0.9.
special cases for comparison with the attraction of Mating Occurs After Attraction to Baits and Traps.
onlywildmales.Finally,weexaminethedelayedmat- Forthecaseofmatingfollowingattractiontobaitsand
ingoffemalesresultingfromtherequirementofsexual traps,thefertilityratiois[zM/zM (cid:3)N],wherezis
i i i
maturationforthecasesoffemalemonogamytogether the survivorship of males following visits to ME
with1)theattractionandkillingofonlymales,or2) sources.Thus,thereareonlyzMwildmalesremaining
theattractionandkillingofbothsexes. formatingafterattractionandcapture.Ifmalesarein
100 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
1000
a
MATING BEFORE TRAPPING
800
S (r)SES 600 aa == 22,, ss == 00..79
A
E a = 10, s = 0.7
EL 400 a = 10. s = 0.9
R
E E
L
RI 200
E
T
S
0
D
o
w
0.0 0.2 0.4 0.6 0.8 1.0 1.2 n
lo
1000 ad
b e
800 MMAATTIINNGG AAFFTTEERRTTRRAAPPPPIINNGG d fro
S (r) m h
EASELE RELE 460000 aaaaa ==== 1122200,,,,, sssss ==== 00000.....79997 ttps://academ
RI 200 ic
E .o
TT u
S p
0 .c
o
m
/a
e
0.0 0.2 0.4 0.6 0.8 1.0 1.2 s
a
TRAPPING SURVIVORSHIP (Z) /a
rtic
Fig.1. TheinteractionofMEbaitingandcapturesurvivorship,(z),andME-exposedsterilemalereleases,(r),required le
toeliminatethepestpopulationforthecaseinwhichfemalesaremonogamousandonlymalesarereleased.(a)matingoccurs /1
0
beforeattractionandcapture;(b)attractionandcaptureoccursbeforemating.Thefoursetsofparametervaluesforfecundity 7
(a)andsurvivorship(s)are:a(cid:4)2,s(cid:4)0.7;a(cid:4)2,s(cid:4)0.9;a(cid:4)10,s(cid:4)0.7;anda(cid:4)10,s(cid:4)0.9. /1/9
7
/7
excessofreceptivefemales,thentheequationsareas number of fertile matings between wild males and 71
9
showninAppendix1A.Thecriticalvalueofris: females is now determined not by the ratio of wild 4
b
r*(cid:4)zaF[as(cid:5)(1(cid:5)s)]/(1(cid:5)sz) [5] mwialldemstaoleasl.lTmhaules,s,wbhuetnrawthiledrmsiamlepslyarbeyinthdeenÞucimtibnetrhoisf y gu
e
which is just z times the corresponding value from specialcase,sterilemalereleaseshavenoeffectonthe s
equation4,wherematingprecedestrapping. growthpotentialofthepopulation;alltheydoisgen- t on
ToassessthepossibilityofmaledeÞcit,weequate erateasubpopulationofsterile-matedfemalesthatdo 1
1
males(zM(cid:3)N)andvirginfemales(V),andobtainthe nothaveanyeffectonpopulationgrowth. F
e
separatrix,r: Abovethevalueofrthatresultsinequalityofmales b
e andvirginfemales(i.e.,theseparatrix),theremovalof rua
re(cid:4)aF(1(cid:5)s)(1(cid:5)z(1(cid:3)s))/(1(cid:5)sz) [6] wildmalesbymaleannihilation(MAT)assistssterile ry 2
twohtaelrmeraeleisstahnedvareluceepotfisvteerfielemmalaelsetroelbeeaseeqsutahlaattaellqouwis- mefafelectrsebleeacsoemsienssturopnpgreesrsainsgththeenpuompbuelartoiofnst,earnildetmheaslee 023
librium.Thesevaluesofrandzassumethatthesystem releasesbecomesgreater.Wecanalsosubstitutez(cid:4)
isatequilibrium,sowecanequatethevaluesofrin (1(cid:5)s)/[as(cid:3)s(1(cid:5)s)]backintoequation6toobtain
equations5and6,asthisvalueisnotonlyacritical thevalueofr intermsofonlythebiologicalparam-
e
valuebutalsotheparticularcriticalvaluethatmakes eters,andwegetr (cid:4)aF(1(cid:5)s)[as(cid:5)(1(cid:5)s)]/as.It
e
totalmalesequalreceptivefemales.Doingthisgives isfortunatethatthelevelsofsterilemalereleasesthat
z(cid:4)(1(cid:5)s)/[as(cid:3)s(1(cid:5)s)].However,thisisthesame havenoeffectareseldomrealized,astheyrepresent
as that for z* with no sterile releases (Barclay and averysmalloverßoodingratio.
Hendrichs2014). IntheequationsforthemaledeÞcitcase,thefactor
Itappearsthatbelowthepointofequalityoftotal [(M (cid:3)N)/V]givestheproportionofvirginfemales
i i i
malesandreceptivevirginfemales,theeffectsofthe that manage to mate (as mating is limited by the
matingratio(M/M(cid:3)N)arelostinthismodel.Here, numberofmales)andthenmultiplyingbysV gives
i
femalesoutnumbertotalmalesatequilibriumandthe the number that mate, which is just the number of
January2014 BARCLAYETAL.:MODELOFMATANDSITFORCONTROLLINGBactrocera 101
Table1. Valuesofsurvivorship(z)afterattractingandkillingwildmaleswithMEtogetherwithsimultaneoussterilereleases,(r*),
tocausecollapseofthepestpopulation
Sterilereleases
Excess DeÞcit
SITonly r*(cid:4)aF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)s)
A.MEsourcesattractonlywildmales
Mo M Mate r*(cid:4)aF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) None
Mo M Attr. r*(cid:4)zaF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) z(cid:4)(1(cid:5)s)/(cid:8)as(cid:3)s(1(cid:5)s)(cid:9)
Mo M&F Mate r*(cid:4)aF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) None
Mo M&F Attr. r*(cid:4)zaF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) r*(cid:4)aF(cid:8)sz(a(cid:3)1(cid:5)s)(cid:5)(1(cid:5)s)(cid:9)
(1(cid:5)s)(1(cid:5)sz)
Po M Mate r*(cid:4)aF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) None
Po M Attr. r*(cid:4)zaF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)sz) r*(cid:4)aF(cid:8)asz(cid:5)(1(cid:5)s)(cid:9)/s(1(cid:5)sz)
Po M&F Mate None r*(cid:4)aF(cid:8)as(cid:5)(1(cid:5)s)(1(cid:3)s(cid:5) D
sz)(cid:9)/(1(cid:5)sz) o
Po M&F Attr. None r*(cid:4)aF(zas(cid:5)(1(cid:3)s))/(1(cid:5)sz) wn
B.MEsourcesattractbothwildandsterilemales lo
Mo M Mate r*(cid:4)aF(1(cid:5)su)(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/(1(cid:5)s)(1(cid:5)sz) None ad
Mo M Attr. r*(cid:4)zaF(1(cid:5)su)(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/u(1(cid:5)s) z(cid:4)(1(cid:5)s)/(cid:8)as(cid:3)s(1(cid:5)s)(cid:9) ed
C.MEsourcesattractwildmalesandreceptivefe(m1a(cid:5)lessz) from
MMoo MM MAtattre. rr**((cid:4)(cid:4)1(cid:5)azaFF(sz(1)1(cid:5)(cid:5)s)s)(cid:8)a(cid:8)ass(cid:5)y(cid:5)(11(cid:5)(cid:3)sysy)(cid:9)(cid:9)//((11(cid:5)(cid:5)sszy)) zNo(cid:4)ne(1(cid:5)sy)/(cid:8)as(cid:3)s(1(cid:5)sy)(cid:9) https
(1(cid:5)sy) ://a
D.IncreasedsterilemalematingsuccessafterMEfeeding c
Mo M Attr r*(cid:4)azF(cid:8)as(cid:5)(1(cid:5)s)(cid:9)/c(1(cid:5)sz) z*(cid:4)(1(cid:5)s)/(cid:8)as(cid:3)s(1(cid:5)s)(cid:9) ad
e
E.Femalematingisdelayed m
Bothmalesandfemalesattracted ic
Mo M Attr (1(cid:5)sy)(cid:8)zaF(1(cid:5)s)(cid:3)r(1(cid:5)sz)(cid:9)(cid:4) z*(cid:4)(1(cid:5)sy)/(cid:8)a(cid:3)s(1(cid:5)sy)(cid:9) .o
a(sy)(cid:6)zaF(1(cid:5)s) up
Onlymalesattracted .c
Mo M Attr z*(cid:4)r/{aF(cid:8)as(cid:6)(cid:5)(1(cid:5)s)(cid:9)(cid:5)sr} z*(cid:4)(1(cid:5)s)/(cid:8)as(cid:3)s(1(cid:5)s)(cid:9) om
/a
e
Herereproductionisdaily.Symbolsareasfollows:Mo,femalesmonogamous;Po,femalespolygamous;Mate,matingisbeforeattractionto s
a
traps;Attr.,attractiontoMEbaitsandtrapsisbeforemating;M,onlysterilemalesarereleased;M&F,sterilemalesandfemalesarereleased; /a
Excess,malesareinexcessofreceptivefemales;DeÞcit,malesarelessthanreceptivefemales.“None”referstothenonexistenceofthiscondition. rtic
le
/1
survivingmales(Appendix1A).Solvingforthecritical populationequilibrium.AsufÞcientlylargevalueofr 0
valuegivesz*(cid:4)(1(cid:5)s)/[as(cid:3)s(1(cid:5)s)],whichisthe willyieldmaleexcess,unlesssterilefemalesarealso 7/1
same as with no sterile releases, and we see that at beingreleased. /97
equilibriumsterilemalereleasesplaynopartinthis Remaining Combinations of Female Mating Pat- /7
7
condition. terns,SterileSexesReleased,andOrderofMatingand 19
For the case of attraction to ME sources before Capture.Inadditiontothecasesdescribedabove,the 4 b
mating,therelationshipbetweenr*,z,andtheother remainingcaseshavebeeninvestigated,butonlythe y g
parameters is graphed in Fig. 1b for four sets of pa- resultsareshown.Wehaveexaminedthecasesforall u
e
rametervalues.AlthoughFig.1aandbresembleeach combinationsoffemalemonogamyorpolygamy,the st o
other,notethatthecurvesinFig.1bstartnearzeroon release of only sterile males or the release of both n
theleft,whilethoseinFig.1astartabovezero.Thus, sterile males and females in equal numbers, mating 11
ifmatingoccursbeforecapture,thenSITwillneedto beforevisitingthebaitsandtrapsorattractiontoME F
e
be used as well to assist MAT application because sourcesbeforemating,andmaleexcessormaledeÞcit. bru
baitingandtrappingcannotcontrolthepopulationby Theequationsareshown(wheretheyapply)inAp- a
ry
itself,whereasifattractionoccursbeforemating,then pendix1A;incaseswhereoneofmaleexcessormale 2
thereexistarangeofcapturemortalitiesatwhichSIT deÞcit never occurs, the equations are omitted and 02
3
isnotrequired,butisstilluseful.Theassistancethat “Noequations”isprintedintheirplace.Thesepara-
theSITprovidestotheapplicationofMATdepends trices and conditions for male excess or deÞcit are
ontheparametersvalues. showninAppendix2,andthecriticalvaluesofsterile
Thedistinctionbetweenmaleexcessandmaledef- releaserates,r,andMEcapturesurvivorships,zandy,
icitatequilibriumwilldeterminewhichsetofequa- areshowninTable1.
tions to use. The determination of male excess or
deÞcitwilldependonthepairofz–rcriticalvaluesthat 1. OrderofmatingandMEattractionandcapture:As
aretobeused.AmaledeÞcitcasecanbeturnedinto mentioned above, the difference between the
amaleexcesscase(wherebothexist)byswitchingto equations for mating before baiting and trapping
a different pair of r–z values with sufÞciently high versusattractionandcapturebeforemating,isthat
valueofsterilereleases,r.Inpractice,forpestcontrol formatingbeforeMEcapture,themales(M)have
purposes, sterile male release rates will usually be not been reduced by the factor z, whereas after
considerablyabovethosesimplyrequiredtoestablish baitingandtrapping,themalesonlysurviveatarate
102 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
1000 containedintheequationforsterilefemales(G).
Mo-M-B ThecomparisoninthemaledeÞcitequationsthen
Mo-M-A includes W in the denominator, so that M/V be-
S (r) 800 MMoo--MM&&FF--BA comesM/(V(cid:3)W)forthemonogamouscases.The
ASE 660000 PPoo--MM--BA majordifferencebetweenthereleaseofonlyster-
LE Po-M&F-B ilemalesandthereleaseofbothsterilesexesisin
E Po-M&F-A thedeterminationofmaleexcessordeÞcit,and
R
E 400 the distinction had little affect on the isoclines
L
RI or the critical values of r and z. There is little
E
ST 200 difference in the required capture efÞciency or
sterile release rates (apart from the unnecessary
releaseoffemales).
0
0.0 0.2 0.4 0.6 0.8 1.0 4. Femalematingpatterns:Theequationsforfemale D
polygamyhaveallthefemales(V,W,F,andG)in o
TRAPPING SURVIVORSHIP (z) w
thetermsinthedenominatorsoftheequationsfor n
Fig. 2. Isoclines for the eight models consisting of all F,G,V,andW(ifpresent)ratherthanjustVand loa
combinationsoffemalemonogamy(Mo)orpolygamy(Po), d
W(ifpresent).Itisfurtherassumedthattheonly e
thereleaseofonlyME-exposedsterilemales(M),orboth d
ME-exposed sterile males and females in equal numbers effect of polygamy is to deny virgin females the fro
(M&F),andmatingbeforeMEattraction(B)orMEattrac- certainty of mating when they are receptive; re- m
tailolnthaendpaciarsptoufrveablueefosroefmcaaptitnugre(Asu)r.vEiavcohrsihsoipcl(inz)earenpdrestseerniltes mstaattuins.gHisearessuamlsoe,dtnhoetmtoajcohradnigfefertehnecirefebceutwndeietny https
releases (r) that will hold the population at equilibrium, female monogamy and polygamy is in the deter- ://a
assumingthewildfemalesubpopulationis10individuals;an minationofmaleexcessordeÞcit,andthedistinc- ca
increaseinroradecreaseinzwillcauseeliminationofthe tionhadlittleaffectontheisoclinesorthecritical de
population.Parametervaluesusedintheeightmodelsarea(cid:4) m
5.0forfecundityands(cid:4)0.8forsurvivorship.Theisoclines vrealleuaesseooffrboanthdsze.xeBsoitnhcfreemasaelethpeoclyhgaanmceysaonfdmtahlee ic.o
formtwogroups:theupperrepresentsmatingbeforeME u
deÞcitatequilibrium.OnthelowerleftinFig.2, p
attractionandcapture(B)andthelowerMEattractionand .c
capture(A)beforemating. there are slight differences in z* because of the om
differencebetweenfemalemonogamyandpolyg- /a
amy, but these differences are only slight, being es
a
zMformating.Isoclinesofcriticalvaluesofrandz 0.0481and0.050. /a
are shown in Fig. 2 for the eight cases of female rtic
matingpatterns(monogamyvs.remating),sterile SITApplicationinCombinationWithAttractionof le
/1
malesorbothsexesreleased,andtheorderofmat- BothFertileandSterileMalestoBaitsandTraps.The 0
ingandcapturefortheparametervaluesused(a(cid:4) equationsforthiscaseareshowninAppendix1Band 7/1
5.0ands(cid:4)0.8).Theseeightisoclinesfallneatlyinto involvesurvivorshipsfrombaitingandtrappingofzfor /97
twogroups:fourformatingoccurringbeforeME wild males and u for sterile males. When the SIT is /7
7
attraction(uppercurves)andfourforMEattrac- used simultaneously with baits and traps containing 19
tionoccurringbeforemating(lowercurves).There ME,itisassumedinthismodelthatthesterilizedmales 4 b
issomuchoverlapinthecurvesthatthosewithin havebeenexposedtoME,butthatsomesterilemales y g
eitherofthesegroupsarealmostindistinguishable. are still attracted to the baits. We develop only the u
e
All curves meet at the upper right, where male scenariooffemalemonogamyandthereleaseofonly st o
annihilation does not occur (SIT is the sole sup- sterile males. There are two cases, depending on n
pression method) and where the distinction be- whether the males are attracted before mating, or 11
tweentheorderofmatingandMEattractionvan- subsequenttomating. F
e
ishes. Thesurvivorshipszanduwouldbeexpectedtobe bru
2. MaleexcessandmaledeÞcit:Someofthesecurves relatedviathebaitingeffort,asanincreaseinamount a
ry
weregeneratedusingequationsforbothmaleex- ofbaitwouldgenerallyincreasemortalityofboththe 2
cess and male deÞcit, whereas some were gener- wildandsterilemales.Thus,wehavemadethemor- 02
3
atedusingonlyoneofthese,astheotherwasnot tality of sterile males a constant multiple of that for
applicable.Thesmoothnatureofallthecurves(not wildmales.Themortalitiesare(1(cid:5)z)and(1(cid:5)u),
shown)indicatesthattheisoclinesformaleexcess sowelet(1(cid:5)u)(cid:4)c(1(cid:5)z)andthenmanipulateu
and for male deÞcit were similar, as there are no bymeansoftheequation:u(cid:4)1(cid:5)c(1(cid:5)z),sothat
sharpchangesofdirection,andalsotheymeeteach whenzisvariedtoassessitseffects,uisalsovaried
otheratthepositionsofseparation(theseparatri- accordingly.
ces,r).Inaddition,alloftheseparatricesthatwere TheisoclinesforthismodelareshowninFig.3for
e
calculated were close to the bottom left of the eachorderofmatingandMEattractionandforvalues
graphs, near the critical z values with no sterile ofF(cid:4)10,a(cid:4)5.0,ands(cid:4)0.8andofwildmalebaiting
releases. survivorship,z,from1.0tothecriticalvalue,z*,andfor
3. The release of both sexes: The equations for the threevaluesofcdeterminingtrappingsurvivorshipof
release of both sexes have an extra equation for sterilemales,u,suchthatthesterilecapturemortality
sterile virgin females (W) and a term for W is isaproportion(0.0,0.2,or0.4)ofthecapturemortality
January2014 BARCLAYETAL.:MODELOFMATANDSITFORCONTROLLINGBactrocera 103
1000 1000
M; C = 0.0
MM; CC = 00.22
M; C = 0.0
ASES (r)A 686000000 AMAA;;;; CCCC ==== 0000....2044 ASES (r)A 686000000 MAMAA;;;;; CCCCC ==== 00000.....02224
LE LE A; C = 0.4
E E
R R
E 400 E 400
RIL RIL
STE 200 STE 200
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
D
MALE TRAPPING SURVIVORSHIP (z) MALE TRAPPING SURVIVORSHIP (z) ow
Fig. 3. A proportion, (u), of the ME-exposed sterile Fig.4. Aproportion,(y),ofthefemalesareattractedto nlo
malesareattractedtotheMEsourcesandkilled.ThisÞg- MEsourcesandkilled.ThisÞgureshowsforfemalemonog- ad
ure shows for female monogamy and the release of only amyandthereleaseofonlyMEexposedsterilemales,the ed
sterile males, the interaction among the critical values of interactionamongthecriticalvaluesofsterilereleases,(r*), fro
sterile releases, r*, and wild male survivorship from ME andwildmalesurvivorshipfromMEbaiting,(z),forthree m
baiting,(z),forthreevaluesofu,representing0.0,0.2,and valuesofy,representing0.0,0.2,and0.4timesthecapture h
0.4timesthecapturemortalityofwildmales(representedby mortalityofwildmales(representedbytheconstantc).M, ttp
s
tahtteraccotniosntanantdc)c.aMpt,umreatbinegfobreefmoraetiantgtr.aHcteiroenFto(cid:4)tr1a0p,sa;A(cid:4),M5.0E, mfoaretinmgabtienfgo.rHeeartteraFct(cid:4)ion10;,Aa,(cid:4)ME5.0a,ttarnacdtiso(cid:4)na0n.8d.capturebe- ://ac
ands(cid:4)0.8. ad
e
m
be substituted for z in the equations for any of the ic
ofthewildmales.Itisevidentthattheattractionofa models.Consideringthemodelforfemalemonogamy .ou
proportionofsterilemalestoMEsourcesnecessitates withthereleaseofonlymales,wecansubstitutebfor p.c
aconsiderableincreaseinthesterilereleaserate,es- zinequationsforfemalemonogamyandthereleaseof om
peciallyathighercapturemortalities,andthisincrease sterile only males (Appendix 1A) for mating before /a
isgreaterformatingprecedingMEattractionthanfor attractiontobaitsandthoseforattractionbeforemat- es
a
MEattractionprecedingmating. ing.Usingequations4and5forthecriticalvalues,r*, /a
SITApplicationinCombinationWithAttractionof weget: rtic
BothWildMalesandReceptiveFemalestoBaitsand le
Traps.TheequationsforthiscaseareshowninAp- r* (cid:3) aF (cid:8)as (cid:4) (cid:6)1 (cid:4) s(cid:7)(cid:9)/(cid:6)1 (cid:4) s b(cid:7) /10
7
ptreanpdpiixng1Cofanzdfoinrvwolivldesmuravleivsoarnshdipysfforormrebcaeiptitnivgeafned- (cid:3) aF (cid:8)as (cid:4) (cid:6)1 (cid:4) s(cid:7)(cid:9)/(cid:8)1 (cid:4) s (cid:5) sp(cid:6)1 (cid:4) z(cid:7)(cid:9) /1/97
males.Thiscaseisalsoanalyzedonlyforthecaseof [7] /7
7
fmeamleasle. Tmhoenfoegmamaley saunrdvivtohreshriepl,eyas,eisotfreoantelyd isnterthilee formatingbeforeattraction,and 194 b
samewayasuabove,sothatx(cid:4)1(cid:5)c(1(cid:5)z)forthree r* (cid:3) b aF (cid:8)as (cid:4) (cid:6)1 (cid:4) s(cid:7)(cid:9)/(cid:6)1 (cid:4) s b(cid:7) y g
valuesofcinthemodelmanipulations.Theisoclines u
fm5o.0ar,ttianhngidsamsn(cid:4)dodM0e.El8aaartnetdrasohcftoiwwoninldainnmdFaflioegr.svu4arlfvuoievrsoeorasfchFhip(cid:4)o,rz1d,0ef,rraoo(cid:4)mf (cid:3) aF (cid:8)1 (cid:4) p(cid:6)1 (cid:8)(cid:4)1 (cid:4)z(cid:7)(cid:9)s(cid:8)a(cid:5)s (cid:4)sp(cid:6)(cid:6)11 (cid:4)(cid:4) zs(cid:7)(cid:7)(cid:9)(cid:9)/ [8] est on 11
1.0 to the critical value, z*, and for three values of forattractionbeforemating.Theeffectsofthispro- F
e
baitingsurvivorshipoffemales,y,suchthatthefemale portionofwildmalesnotbeingattractedtotheME bru
capturemortalityisaproportion(0.0,0.2,or0.4)ofthe baitsandtrapsisshowninFig.5forproportionsthat a
ry
capturemortalityofthewildmales.Itisevidentthat escapebeingkilledattrapsof0.0,0.2,and0.4ofthe 2
evensomeattractionandcaptureoffemalesconsid- wildmalesandforvaluesofF(cid:4)10,a(cid:4)5.0,ands(cid:4) 02
3
erablyenhancesthecontrolprogramandreducesthe 0.8.ItcanbeseenfromFig.5thattheavoidanceof
number of sterile males that need to be released, being captured at ME sources by some wild males
especiallyathigherMEcapturerates. increasestherequiredsterilereleaserate,andthisis
AProportionofWildMalesHasAccesstoaNatural greaterasbaitingintensityisgreater,neartheleftsides
MESourceandDoesNotRespondtoMEBaitsand ofthecurves,buttheeffectsarenotdrastic.Forthese
Traps.Ifacertainproportionofwildmalesdoesno parametervalues,sterilemalereleaseratescanstillbe
longerrespondtoMEbaitsandtrapsbecauseofprior lowerevenwith40%ofthewildmalesescapingthe
exposuretonaturalMEsources,thenthatproportion baiting if attraction to ME sources is before mating
will not be killed and will therefore be available to thananyofthevaluesifmatingisÞrst.
mate.LettheproportionthatdoesnotrespondtoME Increased Sterile Male Mating Success After ME
be1(cid:5)p,thentheproportionkilledattrapswillbe Exposure or Feeding. As a result of certain mass-
p(1(cid:5)z)andtheproportionthatsurvivethebaiting rearing,handling,sterilization,andreleaseprocesses,
andtrappingeachdayisb(cid:4)1(cid:5)p(1(cid:5)z).Thiscan sterilemalesmaynotbeassuccessfulinmatingwith
104 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
1000 20000
M; C = 1.0 18000
ASES (R)LEA 668000000 MAMAA;;;;;; CCCCC ===== 10000.....08686 EASES (r)ELE 11110246000000000000 CCC === 100...057005
E R 8000
E R 400 LE 6000
RILSTER 200 RISTER 442000000000
0
0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
MALE TRAPPING SURVIVORSHIP (z) D
MALE TRAPPING SURVIVORSHIP (Z) ow
Fig.6. ThisÞgureshowsforfemalemonogamywiththe n
Fig.5. Aproportion,1(cid:5)p,ofthewildmalesarenot releaseofonlyME-exposedmales,thecriticalsterilemale lo
a
attractedtotheMEsources,asaresultofpriorexposureto releaseratewhenthematingabilityofsterilemaleswas1) d
e
ME.ThisÞgureshowsforfemalemonogamywiththerelease thesameasthewildmales(c(cid:4)1.0),2)0.75timesasmuch d
ofonlyMEexposedmales,theinteractionamongthecritical asthewildmales(c(cid:4)0.75),and3)halfasmuchasthewild fro
valuesofsterilereleases,r*,andwildmalesurvivorshipfrom males(c(cid:4)0.5).Thismimicsthecaseinwhichsterilemales m
pMoErtiboanistin0.g0,,z0,.2f,oarntdhr0e.4eovfalwuieldsomfa1le(cid:5)sthpa,trearperensoetnatitntrgacptreod- htoavinecbreeeasneetxhpeoirsemdavteinrgsupsenrofotremxpanosceed.HtoeMreEFb(cid:4)ef1o0r,eare(cid:4)le5a.s0e, https
aPn(cid:4)dk1il.0le,d0a.8t,baanitdsa0n.6d)t.rMap,sm(raetpinregsebneftoedrebayttthraecctioonns;taAn,tMc(cid:4)E ands(cid:4)0.8. ://ac
attractionandcapturebeforemating.HereF(cid:4)10,a(cid:4)5.0, ad
ands(cid:4)0.8. Inthiscase,thefemalescanonlymateonthe(cid:6)thday emic
(orlater)afteremergencefromthepuparia,sothat .o
u
wildfemalesiftheyhavenotfedonMEasthosethat the equation for the virgin females will have (cid:6)(cid:5) 1 p
havehadaccesstoMEbeforerelease.Theequations termsbeforethetermformating.Asanalysisofthis .co
m
forthiscaseareshowninAppendix1Dforthecaseof modelwillonlybeintermsofthesteadystateanda /a
femalemonogamyandattractionofonlywildmalesto fulldescriptionofthemodelismessy,wewillpresent es
the ME traps. Other cases are expected to behave theequationsinsteadystateform(seeAppendix1E). a/a
similarlywithrespecttosuchMEexposure.Theequa- Ifmaleannihilationisnotbeingappliedandtheonly rtic
tions are modiÞed by the usual method of assessing controlmethodisthereleaseofsterilemales,thenthe le
competitive ability of sterile versus wild males; the criticalsterilereleaseratewithadelayinmatingis: /10
7
ainsscelusssmioennotfotfhdeifffaecrteonrtcminatitnhgesduecncoemssininatowriladlloawnds r*(cid:4)aF[a s(cid:6)(cid:5)(1(cid:5)s)]/(1(cid:5)s) [10] /1/9
7
sterilemales.Thefactorcmaytakeonvaluesbetween Noticethatif(cid:6)(cid:4)1,thenwegetthevalueofr*given /77
zero (no competitive ability) and one (being fully inequation2.With(cid:6)(cid:10)1,thevalueofr*islessthan 19
competitive).Figure6showstheeffectsofsuchre- 4
ductionincompetitiveabilityforvaluesofF(cid:4)10,a(cid:4) that in equation 2, indicating that a delay in mating by
5.0,ands(cid:4)0.8.Sterilemalesthathavenothadaccess amsosirsttasltithyeosftefreimlearleelseabseefoprreogmraamtinags.aresultofnatural gue
to ME immediately on adult emergence will be less s
successfulinmatingandtheresultsarethatthecritical kilIlefdbaotththewtirldapms,athleesnatnhderefeamrealtewsoacraeseastttroaccotendsidanerd. t on
release rate will have to be considerably greater at 1
Ifmalesareinexcessofreceptivefemales,thenweget 1
higher wild male capture survivorships than if such theequationinvolvingz,y,andr:(1(cid:5)sy)[zaF(1(cid:5) Fe
exposurehadoccurred. s)(cid:3)r(1(cid:5)sz)](cid:4)a(sy)(cid:6)zaF(1(cid:5)s),fromwhich bru
Forthecaseofmaleexcess: a
r*(cid:4)azF[as(cid:5)(1(cid:5)s)]/c(1(cid:5)sz) [9] r(cid:4)zaF(1(cid:5)s)[a(sy)(cid:6)(cid:5)(1(cid:5)sy)]/ ry 20
(1(cid:5)sz)(1(cid:5)sy) [11] 23
Thevalueofr*increaseswithadecreaseinc(lower
mating ability) and decreases with a decrease in z Thismustbesolvednumericallyfory.Ifsterilemales
(wildmalesurvivorshipwhiletrapping),andwhen are not being released and only females are being
z (cid:4) 1 and c (cid:4) 1, we recover the expression in trapped,theny*(cid:4)1/s(a(cid:3)1).Examplesofsolutions
equation2. forcriticalsterilemalereleaserateareshowninFig.
ForthecaseofmaledeÞcit,weobtainz*(cid:4)(1(cid:5)s)/ 7forvaluesofF(cid:4)10,a(cid:4)5.0,ands(cid:4)0.8andfortwo
[as(cid:3)s(1(cid:5)s)].Inthiscase,withthereleaseofonly valuesofmaletrappingsurvivorship(z),threeoffe-
sterilemales,itseemshighlyunlikelythatmaledeÞcit male trapping survivorship (y) and 10 values of the
wouldeverberealized. delayinmating((cid:6)).
Females Have a Maturation Period After Emer- If males are fewer than the number of receptive
genceBeforeTheyCanMate.Theequationsaregiven females,thenwegetthesolutionthatforthecritical
hereforthecaseoffemalemonogamywiththerelease valueformaletrappingisz*(cid:4)(1(cid:5)sy)/[a(cid:3)s(1(cid:5)
ofonlysterilemalesandwithtrappingbeforemating. sy)]. This indicates that the release of steriles disal-
January2014 BARCLAYETAL.:MODELOFMATANDSITFORCONTROLLINGBactrocera 105
2000 tualvaluesofz*andr*foraparticularrealpopulation
z = 0.7; y = 1.00 the models must be ßeshed out somewhat and the
ES (r*) 1500 zzzzz ===== 00000....74477;;;;; yyyyy ===== 00100....8708870577555 atwsseuemnpmtioonnosgaadmdoreusssaendd. Bpoeclyaguasmeotuhsefedmistailnecstaiopnpebaers-
S z = 0.4; y = 0.75 to be of limited importance, the development that
AA
E followswillbeonlyformonogamousfemales.Wewill
L 1000
E developthecaseforB.dorsalis,asdemographicdata
R
E areavailablethatareofgoodqualityandaresuitable
ILRI 550000 forourpurpose(Vargasetal.1984).
TE In a laboratory study of cohort development of
S three tephritid species, Vargas et al. (1984) deter-
0 minedthatthetotalpreadultmortality(herecalled(cid:7);
0 2 4 6 8 10 12 BarclayandHendrichs2014)ofeggs,larvae,andpu-
D
DELAY IN MATING (DAYS) paeofB.dorsaliswas37%(l (cid:4)(cid:7)(cid:4)0.63).Theegg, o
Fig.7. ThisÞgureshowsforfemalemonogamy,withthe larval,andpupalstageswere(cid:2)x1.6,7.8,and10.3dlong, wn
releaseofonlyME-exposedsterilemalesandwithtrapping respectively,totaling(cid:2)20d.Inaddition,theauthors loa
d
beforemating,thecriticalsterileßyreleaserate,r*,when gaveasurvivorshipcurvefromday1(oviposition)to e
mbaaittiinnggaisnddetlraaypepdinbgya(cid:6)ttrdaacytssaonfdsekxiullaslbmotahtumraatlieosn(aanpdrodpaoilry- Tabhoeugtrdeaayte1s2t6m,worhteanlitoynwlya(cid:2)si1n%thoefltahrevcaolshtoargter,efomllaoinweedd. d from
tionof1(cid:5)zeachday)andsomefemales(aproportionof1(cid:5) bysenescentadultsataboutday115.Fecundityand h
lytewveoeacllsehvoedflassyuo)rf.vsTiuvhroverisvshioxirpslihnoifepsfoerfmempaarleelessse(n(yzt(cid:4)(cid:4)the01..7s0iax,n0cd.o80m7.54b,)iananandtdio0tnh.7sr5eo)ef. ewghgehnatthcehÞabrsitlimtyacteudrvfeesmwaleere(Fa)lsoovgipivoesnitifornomoccduaryre2d6,, ttps://a
Thedelayinmating,(cid:6),takesonallintegralvaluesbetween untilday126,bywhichtimeovipositionhadvirtually ca
d
1and10inclusive.HereF(cid:4)10,a(cid:4)5.0,ands(cid:4)0.8. ceased.WeÞttedcurvestoboththesurvivorshipand e
m
fertility data and computed the mean daily fertility ic
(MDF,herecalled(cid:8)),or“meanfertileeggsperßy- .o
u
lmowalessth(ebuctasneotoffemmaalleesd)eÞarceit.reTlheausse,dwwheitnhevtheersutseeriolef day,”thatCarey(1989)gaveas: p.co
ME baiting and trapping (and males are in excess of (cid:8) (cid:3) (cid:12) lxhxmx/(cid:12)lx [13] m/a
receptive females), the trapped males will likely only e
inwhichthesumistakenoverallovipositingagesof s
assistthecontrolprogrambyreducingthenumberof adults,andl,h,andm are,respectively,theadult a/a
wildmalesandtherebyincreasingthesterile:fertileratio. femalesurvivxorsxhiptoagxeclassx,theegghatchability rtic
traIpfso,tnhlyerewiisldonmlyaloensearceasaet.trIafcmteadleasnadrekiinlleedxcaetssthoef fxr.oUmsinagduthlteadgaet-acolafsVsaxr,gaansd,ththeemfeeacnunddaiitlyyfoefrtaiglietyc(la(cid:8)s)s le/10
7
receptive females, then the control equation re- wascalculatedtobe8.76usingtheactualsurvivorship /1
ducesto: /9
dataand11.69usinganexponentialapproximationto 7
z*(cid:4)r/{aF[a(s)(cid:6)(cid:5)(1(cid:5)s)](cid:5)sr} [12] thesurvivorshipcurveinwhichtheendpointswere /77
1
0.59atday26and0.01atday126. 9
(seeequationsinAppendix1E).Thisdoesnotreduce Weborrowtherelevantequationstobuildonfrom 4 b
toanythingmeaningfulwhenris0,astheequations theworkofBarclayandHendrichs(2014)thatmod- y g
assumethatmalesareinexcess,andifr(cid:4)0,thenthe eled control by MAT only. Assuming that with the ue
systemisnotcontrollablewithmaleexcess.Examples release of sterile males, the males will always be in st o
ofsolutionsforthecriticalvalueofsterilereleaserate excessofreceptivefemales,thentheequationsareas n
(r)arealsoshowninthesameFig.7fortwovaluesof showninAppendix1F.Inthoseequations,F ,G ,V , 11
maletrappingsurvivorship,z,threeoffemaletrapping M ,andN arethevaluesoftheearliestagec1lass1eso1f Fe
o(sfudrevliavyorinshfiepmya(cid:4)le1m;yat(cid:4)ing0.(8(cid:6)7)5.,Iyf(cid:4)ste0r.i7l5e)m,aanleds1(0bvuatlnueost twhi1eldfemrtaillee1sf,eamnadlerse,lsetaesreildizsetderfielemmaleasle,sv,irrgeisnpefecmtivaelelys,, bruary
females) are being released, then males cannot be andthesubscriptTreferstothetotalsofthesecate- 2
fewerthanthenumberofreceptivefemales,sothat gories. These equations yield a critical rate for the 02
3
casedoesnotexist. releaseofsterilemales,r*,of
TheUseofDemographicDatainMEBaitingand
TrappingPrograms.Fromtheresultgiveninequation r* (cid:3) z(cid:7)(cid:8)FT(cid:8)(cid:12) (km(cid:5)1)zi(cid:13)(i)sj]
9, we see that the data required for estimating the {[(cid:12)(kf(cid:5)1)yi(cid:13)(i)s](ykv(cid:13)(kv)s)(cid:7)(cid:8)(cid:5)1}/
criticalbaitingandtrappingsurvivorship,z*,toelim- j j
inate an insect population, are daily fecundity and [(cid:12)(km(cid:5)1)zi(cid:13)(i)u] [14]
j
adultsurvivorship.Hiddeninthisequationisalsoall
Because we have no data from Vargas on the survi-
themortalitythatwouldbeaccumulatedbyaninsect
vorshipofsterilemales,weassignu (cid:4)s foralli,and
cohortbetweenovipositionandemergenceofadults, i i
equation14simpliÞesto
andalsobetweenßyemergenceandthestartofre-
production. The theoretical models are purposely r* (cid:3) z(cid:7)(cid:8)F (cid:14)(cid:8)(cid:12) (kf(cid:5)1)yi(cid:13)(i)s]
made simple to facilitate the derivation of general T j
principles;however,whenitcomestocomputingac- (ykv(cid:13)(kv)s)(cid:7)(cid:8)(cid:5)1} [15]
j
106 ANNALSOFTHEENTOMOLOGICALSOCIETYOFAMERICA Vol.107,no.1
Although equations 14 and 15 appear complicated, 1.6e+5
theyonlyinvolvethedatapresentedbyVargasetal. 11.44ee++55 BBaaccttrroocceerraa
(th1o9s8e4)daatnad, wstiatthisttihcestehxactepcatinonbeofeastseilryiledemriavleedsfurrovmi- s (r) 1.2e+5
vorship,u.Inusingequations14and15,wewillassume se 1.0e+5 z = 1.0; y = 1.0
thatachievablevaluesofyandzaredeterminedfrom aelea 8.0e+4 zzz === 000..5775;5 ;; yyy === 111.0.00
priorexperimentswithtrapsandthenwecalculatethe R 6.0e+4 z = 1.0; y = 0.8
rtheqeupirreeddevtealrumeinoefdstvearliuleesreolfeyasaensdafzteinrtsouebqstuiatutitoionns1o4f rile e 4.0e+4 zz == 00..57;5 ; yy y= = 0 0.8.8
or15.Usingtheparametervaluesaboveandassigning St 2.0e+4
y(cid:4)z(cid:4)1.0,weÞndthatifSITisusedwithouttrapping, 0.0
then r* (cid:4) 201,577. The equilibrium value of M is
M (cid:4)(cid:7)(cid:8)F [(cid:12)(km(cid:5)1)zi(cid:13)(i)s].IfwesubstituteTthis 0 2 4 6 8 10
intTo equatTion 15, we get r* (cid:4)j z M {[(cid:12) (kf (cid:5) 1)yi Mating Delay (Days) Do
(cid:13)(i)s](ykv(cid:13)(kv)s)(cid:7)(cid:8)(cid:5)1}/[(cid:12)(km(cid:5)T1)zi(cid:13)(i)s].We Fig.8. ThecriticalME-exposedsterilemalereleaserate, wn
canthjenquantifyjthecriticalsterilereleaseinjterms r*,forB.dorsaliswhencapturesurvivorshipformales(z)is loa
ofM :r*(cid:4)5.41M ,sothatthedailyreleaserateneeds 1.0, 0.75, or 0.5 and that for females (y) is 1.0 or 0.8; in de
T T addition,thedelayuntilmatingisshownforvaluesfrom1to d
tobe5.41timestheequilibriumsizeofthewildmale 10donthehorizontalaxis.Theverticalaxisshowsvalues fro
componentofthepopulation.Thevaluefory*andz* fromzeroto1,00,000(inexponentialnotation)fordailymale m
a0c.7t2in1ganaldozn*e(cid:4)we0r.0e6(5B.AarscalnayilalunsdtrHateionndroifcthhse2s0y1n4e)ryg*is(cid:4)m sthteernilethreelreealeseass.eIrfarteelseawsoeusladrenmeeaddteowbeeemkluyltriaptlhieedrtbhya7n.dWaiiltyh, https
amongthethreeparametersinthismodel,ify(cid:4)0.9 even modest wild male capture rates, required sterile ßy ://a
andz(cid:4)0.5,ther*(cid:4)6646,(cid:2)3%ofthevaluewithno releasescanbemuchlowerthanwithoutcapture.HereF(cid:4) ca
trapping.Inthiscase,r*(cid:4)12.08M . 100,a(cid:4)7.67,ands(cid:4)0.961. de
T m
WecouldalsouseasimpliÞedmodelthatassumes ic
age-independent adult survivorship and an inÞnite .o
u
numberofadultageclasses(Appendix1F).Thesur- obtain:r*(cid:4)M [(cid:8)(cid:7)(sy)kv(cid:5)(1(cid:5)sy)](cid:4)7.20M , p
T T .c
vivorship between ages 26 (the Þrst day of mated which is approximately seven times the equilibrium o
m
femaleoviposition)and126wasconvertedtoacon- malepopulation. /a
stantdailysurvivorshipbytakingthegeometricmean Ifzis0.5andy(cid:4)0.9,thenr*isreducedto773,(cid:11)1% es
perdayovertheperiodof100dfrom0.59to0.01;this ofthevaluewithz(cid:4)y(cid:4)1.0,andinthiscaser*(cid:4)3.81 a/a
can be obtained from the equation: 0.59 sn (cid:4) 0.01, MT.TheseinteractionsareshowninFig.8forthree rtic
wheren(cid:4)100inthiscase.Usingnaturallogarithms, valuesofzandtwovaluesofyaswellas10valuesof le
ln [(0.59)(sn) (cid:4) ln (0.01), so that ln (0.59) (cid:3) 100 kv(1(cid:5)10).ItisapparentinFig.8thatfortheextreme /10
ln(s)(cid:4)ln(0.01)(cid:4)(cid:5)4.6052.Collectingterms,ln(s)(cid:4) values,controlwillbedifÞcultbySITalonewithout 7/1
(cid:5)0.0408,sothats(cid:4)0.960.Thus,theadultsurvivorship integrationwithanyothersuppressionmethod,while /97
from 1 d to the next is 0.960 using an exponential evenmodestamountsofcaptureatMEtrapsandbaits /7
7
survivorshipcurvefortheovipositingadults.Also,we greatly reduce the required sterile release rate for 19
assigntheparameter(cid:8)thevalueof11.69.Themean eradication.TheseshowthevalueofintegratingMAT 4 b
daily fertility ((cid:8)) and the preadult survivorship ((cid:7)) and SIT when ME attracts males and some females. y g
willbeinsertedintotheequationsforthesimpliÞed ThegaininefÞciencywouldnotbenearlyasgreatif u
e
modelinAppendix1F.Anadditionalcorrectionthat MEattractedonlymales. st o
must be included is the mortality accrued by virgin It is obvious that if we take the full model as the n
adultfemalesduringtheperiodbetweenemergence morecorrectone,thenthesimpliÞedmodelseriously 11
fromthepupariaandwhentheybecomereceptivefor underestimates the required sterile release rate for F
e
mating. From the graph presented by Vargas et al. control.Thisislikelyaresultofitsuseoftheexpo- bru
(1984) this period is taken as six days, and thus the nential survivorship curve, which reduces both the a
factorof0.9606(cid:4)0.783isalsoincluded. meanageofthepopulationandthusalsothelifetime ry 2
Solving the equations in Appendix 1F for steady eggproductionpermatedfemale.Thegainineaseof 02
3
state,weobtain: calculationinthiscasecomesataconsiderablecostin
accuracy,incontrasttothecasewhenusingMATalone,
r*(cid:4)z (cid:8)(cid:7)FT(1(cid:5)s)[(cid:8)(cid:7)(sy)kv wherethesimpliÞedmodelyieldedanacceptableap-
(cid:5)(1(cid:5)sy)]/(1(cid:5)sy)(1(cid:5)sz) [16] proximation to the full model if females were being
trappedaswellasmales(BarclayandHendrichs2014).
Thisreducestoequation2whenz(cid:4)y(cid:4)1.0andkv(cid:4)
1. With the parameter values listed above ((cid:8)(cid:7)(cid:4)
(11.69)(0.63)(cid:4)7.36,s(cid:4)0.960,kv(cid:4)6,andF (cid:4)100) Discussion
T
forB.dorsalis,thevaluesofz*,y*andr*whenacting Thepreviousarticle(BarclayandHendrichs2014)
aloneare:z*(cid:4)0.005;y*(cid:4)0.721(BarclayandHen- dealtwiththeuseofMATalone,andwefoundthat
drichs2014)andr*(cid:4)132,550perday.Ifwesubstitute withoutthetrappingoffemalesaswellasmales,the
M (cid:4)M /(1(cid:5)s)(cid:4)(cid:8)(cid:7)F /(1(cid:5)s)for(cid:8)(cid:7)F /(1(cid:5)s) chancesofsuccessappeartobeslim.Inthisarticle,we
T 1 T T
in the equation for r* above with y (cid:4) z (cid:4) 1.0, we includethesimultaneousimplementationoftheSIT