Assume65ddd.com

vol.
164,no.
6theamericannaturalistdecember2004E-ArticleWhenResistanceIsUseless:PolicingandtheEvolutionofReproductiveAcquiescenceinInsectSocietiesTomWenseleers,1,*AdamG.
Hart,2,andFrancisL.
W.
Ratnieks2,1.
LaboratoryofEntomology,ZoologicalInstitute,UniversityofLeuven,Naamsestraat59,B-3000Leuven,Belgium;2.
LaboratoryofApicultureandSocialInsects,DepartmentofAnimalandPlantSciences,UniversityofShefeld,ShefeldS102TN,UnitedKingdomSubmittedMarch18,2004;AcceptedAugust12,2004;ElectronicallypublishedNovember9,2004Onlineenhancements:Mathematicanotebookandcode.
abstract:Insocialgroupscomposedofkin,inclusivetnessben-etscanfavorgreatercooperation.
Alternatively,cooperationcanbeenforcedthroughthepolicingoflesscooperativeindividuals.
Here,weshowthattheeffectofpolicingcanbetwofold:notonlycanitdirectlysuppressindividualselshness,itcanalsoentirelyremovetheincentiveforindividualstoactselshlyintherstplace.
Wetermsuchindividualrestraintinresponsetosociallyimposedpo-licing"acquiescence"andillustratetheconceptusingexamplesdrawnfromthesocialHymenoptera(theants,bees,andwasps).
Inclusivetnessmodelsconrmthatwhenapolicingsystemisinplace,in-dividualsshouldbelesstemptedtoactselshly.
Thisisshowntohaveimportantconsequencesfortheresolutionofconictwithintheirsocieties.
Forexample,itcanexplainwhyinmanyspeciesveryfewworkersattempttoreproduceandwhyimmaturefemalesusuallydonotattempttodevelopasqueensratherthanworkers.
Althoughouranalysesareprimarilyfocusedonthesocialinsects,ourconclu-sionsarelikelytobegeneralandtoapplytoothersocietiesaswell.
Keywords:socialpolicing,altruism,acquiescence,socialinsects,workerreproduction,casteconict.
Nonclonalindividualswithinsocialgroupsfrequentlyhavedissimilarreproductiveinterests,leadingtoconict(Rat-*Presentaddress:WissenschaftskollegzuBerlin,InstituteforAdvancedStudy,Wallotstrasse19,14193Berlin,Germany;e-mail:tom.
wenseleers@bio.
kuleuven.
ac.
be.
E-mail:a.
hart@shefeld.
ac.
uk.
E-mail:f.
ratnieks@shefeld.
ac.
uk.
Am.
Nat.
2004.
Vol.
164,pp.
E154–E167.
2004byTheUniversityofChicago.
0003-0147/2004/16406-40349$15.
00.
Allrightsreserved.
nieksandReeve1992;Hamilton1995;Frank1998;Keller1999).
Conictsofinterestcanariseatalllevelsofsocialorganization,fromintragenomicandintercelllineagecon-ictswithinorganisms(Hurstetal.
1996;Michod1999;Pomiankowski1999)toparent-offspringandinterindi-vidualconictsinfamiliesandsocieties(Trivers1974;Triv-ersandHare1976;Hamilton1995;QuellerandStrass-mann1998).
Insocialgroups,whereallindividualsareaffectedtoagreaterorlesserextentbythebehaviorofothers,conictisusuallydetrimentalbecauseindividualswhoservetheirownselshinterestsreducegrouppro-ductivity(Hardin1968;Frank1995;Heckathorn1996).
However,conict-inducedsocialcollapseisnotinevitablebecauseconictsareoftenabsentorelseefcientlyre-solved(RatnieksandReeve1992).
Onesituationwhereconictsarereducedoccursingroupscomposedofkin.
Greatergeneticrelatednessbe-tweengroupmembersincreasestheirsimilarityofinter-ests,therebyreducingthepotentialforconict("kinse-lection";Hamilton1964,1995;MaynardSmith1964;Frank1998).
Socialpolicing,wherebygroupmembersmu-tuallysuppresseachother'sselshtendencies,isanotherimportantmechanismthatcanresolveconict(Ratnieks1988;Frank1995,1996,2003).
Policingisusuallyenvis-agedassomecoercivebehaviorthatdirectlyrepressesin-dividualselshness(Leigh1991;Frank1995,2003),forexample,individualsforcefullypreventingtheoveruseofcommonresources(Hardin1968;Leigh1991;Frank1995;Ostrom1999;Ostrometal.
1999).
Additionally,policingcanalsodescribesituationswhereselshindividualsarepreventedfrombeingsuccessful.
Forexample,ininsectsocieties,workersofteneator"police"eggslaidbyotherworkers(g.
1;Ratnieks1988;RatnieksandVisscher1989).
Eggeatingdoesnotpreventselshworkersfromlayingmoreeggs,butitdoespreventthemfromsuccess-fullyreproducing,therebymaintainingthereproductivemonopolyofthequeen(Ratnieks1988;RatnieksandVisscher1989).
Theaboveexamplesemphasizethedirectinuenceofpolicing.
However,policingcouldalsoplayanadditionalandimportantroleinresolvingconictsbyindirectlymak-Figure1:TwoformsofsocialpolicinginthesocialHymenoptera.
a,Policingofworker-laideggsinthehoneybeeApismellifera(RatnieksandVisscher1989).
b,Executionofadwarfqueen(left)thatdevelopedinaworkercellinthestinglessbeeSchwarzianaquadripunctata(RibeiroandAlves2001;Wenseleersetal.
2003,2004b).
Seetextforexplanations.
PhotosbyF.
L.
W.
Ratnieks(a)andT.
Wenseleers(b).
E156TheAmericanNaturalistingitunprotableforindividualstoactselshly.
Forex-ample,ifeggslaidbyworkersareeaten,thenthismayselectagainstworkersactivatingtheirovariesintherstplace.
Wenamethis"acquiescence,"thatis,asituationinwhichindividualsareselectedtoconformtosocialnormsandregulationsthatareenforcedagainsttransgressors.
Althoughthepossibilitythatpolicingcanindirectlyreduceselshnessandfavorgreatercooperationhasbeenalludedtoseveraltimes(e.
g.
,Ratnieks1988;Visscher1989;Rat-nieksandReeve1992;Frank2003),thisideahasneverbeenanalyzedformally.
Theaimofthisarticleistomakeaformalanalysisofacquiescenceandtodetermine,foravarietyofscenarios,whenindividualsarebetteroffiftheydonotresistpolicing.
Ouranalysesexplicitlyinvestigatethetwomaintypesofpolicinginsocialinsects,whicharealsoamongthebest-studiedexamplesofpolicingintheanimalkingdom:policingofegg-layingbyworkers,thatis,thepreventionofmaleproductionbyworkers(g.
1;Ratnieks1988;RatnieksandVisscher1989;FosterandRatnieks2000,2001c),andcaste-fatepolicing,thatis,thepreventionofimmaturefemalesfromsuccessfullydevel-opingintoqueensratherthanworkers(g.
1;BourkeandRatnieks1999;Wenseleersetal.
2003;WenseleersandRat-nieks2004).
Thedetailsofeachwillbediscussedbelow.
Althoughwefocusonsocialinsects,wearguethattheevolutionofacquiescencewithinsocialsystemsisageneralprocessthatcanreduceconictandpromotecooperationinmanyothersocialsituations.
MethodsWeusekinselectionmethodssimilartothosedevelopedbyFrank(1998)toanalyzethelikelihoodthatacquiescencecanevolveinresponsetopolicing.
However,unlikeFrank(1998)wetakeaninclusivetnessratherthanadirectorneighbor-modulatedtnessperspective.
Eachoftheanal-yseshasalsobeenconrmedusingallelefrequencymeth-ods.
(SeetheonlineeditionoftheAmericanNaturalistfortheMathematicanotebookwithkinselectionmodelsandequivalentgeneticmodels.
)Inallcases,thedifferentmod-elingmethodsgivenumericallyidenticalresults.
Itshouldalsobenotedthatinallmodels,policingisincludedasaxedparameterratherthanbeingallowedtocoevolvewiththelevelofintragroupselshness(Frank1995,1998).
Thisisbecausetheevolutionarybasisforpolicingiswellun-derstoodineachcaseandweareconcernedonlywithhowthepresenceofpolicingperseinuencestheoptimallevelofselshnessinthegroup.
Seetable1fordenitionsofmodelnotationandparameters.
ResultsWorkerSterilityasAcquiescenceInthesocialHymenoptera,maleproductionisoneofthemostimportantcontextsinwhichconictcanarise.
Inthemajorityofspecies,includingmostants,Apishon-eybees,Meliponinaestinglessbees,andVespinaewasps,workersretainovarieseventhoughtheycannotmate(Wil-son1971;Bourke1988).
ThroughthehaplodiploidsexdeterminationmechanismfoundintheHymenoptera,theseworkerscanlayunfertilized,male-destinedeggsandtherebychallengethereproductiveprimacyofthequeen.
Fromaninclusivetnessperspective,workerreproductionisbenecialattheindividuallevel(Hamilton1964;TriversandHare1976),becauseanyworkerismorerelatedtoherownsons()thantoherbrothers,thequeen'srp0.
5sons().
However,ifthequeenmatesmultiplerp0.
25times,asoccurs,forexample,inthehoneybee(PalmerandOldroyd2000),thenworkersareonaveragemorerelatedtoqueen'ssons()thantootherworkers'rp0.
25sons(full-andhalf-nephews;).
Thisfavorswork-r!
0.
25erstoselectivelyremoveor"police"eggslaidbyotherworkers(g.
1;Ratnieks1988).
Workerpolicingcanalsobefavored,evenwhenpaternityisbelowtwo,ifitincreasescolonyefciency(Ratnieks1988)orifitallowstheworkerstobiasthesexratiotowardfemales(theworkeroptimum;TriversandHare1976)atlowcostbytheremovalofmalesintheeggstage(FosterandRatnieks2001b).
Thequeen,fromherperspective,alwaysfavorstheproductionofherownsons()overworkers'sons(hergrandsons;rp0.
5).
Hence,thequeenisalsoselectedtoeatworker-rp0.
25laideggs(queenpolicing;TriversandHare1976;Ratnieks1988)ortoassistinworkerpolicingbymarkinghereggswithaspecicsignal(Ratnieks1995).
Bothqueenandworkerpolicingarewidespread.
Workerpolicingviaeggeatingisknowninseveralspecieseachofhoneybees(g.
1;RatnieksandVisscher1989;Hallingetal.
2001;Oldroydetal.
2001),Vespinaewasps(reviewedinFosterandRatnieks2001c),andants(KikutaandTsuji1999;D'Ettorreetal.
2004;Endleretal.
2004).
Queenpolicingbyeggeatingoccursinants(MonninandPeeters1997;KikutaandTsuji1999;Monninetal.
2002),bum-blebees(Freeetal.
1969;Pomeroy1979;Cnaanietal.
2002),DolichovespulaandVespularufawasps(Greene1979;FosterandRatnieks2001c;Wenseleersetal.
2005a,2005b),Polistespaperwasps(FletcherandRoss1985;Reeve1991),andthehalictidbeeLasioglossumzephyrum(MichenerandBrothers1974).
Policingcanbehighlyeffective.
Forexample,inthehoneybee,Apismellifera,approximately7%ofallmaleeggsarelaidbyworkers(Visscher1996),yetonly0.
1%ofalladultmalesareworkerderived(Visscher1989).
Thisimpliesthatapproximately99%ofallworker-laideggsarePolicingandAcquiescenceinInsectSocietiesE157Table1:NotationandparametersusedinthemodelsSymbolDenitionModelin"WorkerSterilityasAcquiescence":nColonysize(numberofworkers)WfTotalnumberofmalesproducedbyafocalworkerinacolonyWnTotalnumberofmalesproducedbyeachofthen1nestmateworkersinthecolonyWmqTotalnumberofmalesproducedbythequeenWqTotalnumberofqueensorswarmsproducedbythecolonyxProbabilitywithwhichafocalworkerinacolonyactivatesitsovariesxProbabilitywithwhicheachofthen1nestmatesactivatesitsovariesxColonyaverageprobabilitywithwhichworkersactivatesitsovariesp(1n)x[(n1)/n]xXPopulationaverageproportionoflayingworkersXEvolutionarilystablestrategyproportionofreproductiveworkerswPopulation-wideproportionofmalesthatareworkerproducedqFecundityofthequeenintermsofmaleproductionrelativetoasinglereproductiveworkerSSurvivalofworker-laideggsrelativetoqueen-laideggs(declineswhentheyareselectivelykilled)PEffectivenessofpolicing(selectiveeatingofworkereggs)p1SRLife-for-liferelatednessModelin"AcquiescencetoCasteFate":nNumberofworkercellspresentatanyonetimeqNumberofqueencellspresentatanyonetimeWfExpectedtnessofafemaleinaworkercellWnExpectedtnessofeachoftheothern1femalesinworkercellsWqExpectedtnessofanormalqueendevelopingfromaqueencellWmTotalnumberofmalesproducedbythecolonyyProbabilitywithwhichafocalfemaleinaworkercelldevelopsasaqueenyProbabilitywithwhicheachofthen1otherfemalesinworkercellsdevelopsasaqueenyColonyaverageprobabilitywithwhichfemalelarvaeinworkercellsdevelopasqueensp(1n)y[(n1)/n]yYPopulationaverageprobabilityofdevelopingasaqueenwheninaworkercellYEvolutionarilystablestrategyprobabilityofdevelopingasaqueenwheninaworkercellSSurvivalofsmallqueensrelativetonormalqueens(declineswhentheyareselectivelykilled)PEffectivenessofpolicing(selectiveeliminationofqueensdevelopinginworkercells)p1SRfLife-for-liferelatednesstosisterqueensrearedinthecolonyRmLife-for-liferelatednesstomalesrearedinthecolonypoliced.
Doesthishighpolicingeffectivenessreducetheincentiveforworkerstolayeggs,asseveralauthorshavesuggested(e.
g.
,Ratnieks1988;Visscher1989;RatnieksandReeve1992)Forexample,coulditexplainwhysofewworkersinqueenrightcolonies(approximately0.
1%–0.
01%)haveactiveovaries(Ratnieks1993)Ifso,thiswouldbeagoodexampleofacquiescence.
Thefollowingmodelinvestigateswhethereffectivepo-licingcancauseacquiescence.
Letnbethenumberofworkersinthecolony,Sthesurvivalofworker-laidmaleeggsrelativetoqueen-laidmaleeggs(whichisreducedbybothqueenorworkerpolicingbutunaffectedbyrandomeggmortality),andqthefecundityofthequeenrelativetoasinglereproductiveworkerintermsoflayingmaleeggs.
Assumethatafocalworkerinacolonyactivatesherovariestolayeggswithprobabilityxandthateachofitsnestmatesactivatetheirovarieswithprobabilityxn1sothatthecolonycontainsegg-layingworkers,wherenxistheaverageprobabilitywithwhichworkersactivatextheirovaries,.
Wecannowwritexp(1/n)x[(n1)/n]xthetotalnumberofmalesproducedbythisfocalworkerandbyeachnestmateworkerasxSWpG(x)#,(1)fnxSqxSWpG(x)#,(2)nnxSqwhereisthecolonyproductivity(totalnumberofG(x)E158TheAmericanNaturalistmalesreared)asafunctionofhowmanylayingworkersthereareinthecolony(egg-layingworkersgenerallyper-formlessworkandsodecreasetotalcolonyproductivity;MichenerandBrothers1974;Landoltetal.
1977;vanHonkandHogeweg1981;Ward1983;Cole1986;Hil-lesheimetal.
1989;Reeve1991;MonninandPeeters1999;HartmannandHeinze2003).
Thefollowingtermrepre-sentstheproportionofallmalesthatareworkers'sons.
Thatis,thetotalnumberofsonsofthefocalandotherworkersthatsurvivepolicing(xSorxS),dividedbyallsurvivingmales,whichincludesbothworkers'sons()nxSandqueen'ssons,laidinproportiontotherelativerateqatwhichtheseareproduced.
Forsimplicity,wewillassumethatworkerreproductionlinearlyreducescolonyproduc-tivity,thatis,.
Gp1xByasimilarargument,thetotalnumberofmalespro-ducedbythequeenisqWpG(x)#.
(3)mqnxSqFinally,thetotalamountoffemalereproductionbythecolony(wingedqueens,orswarmsforswarm-foundingspeciessuchashoneybees)isalsoadecreasingfunctionof.
Forsimplicity,weassumethatworkerreproductionxreducesqueenandmaleproductionequally.
Hence,thetotalnumberofqueensorswarmsproducedisWpG(x)p1x.
(4)qTocalculatetheevolutionarilystablestrategy(ESS)pro-portionofworkerswhoactivatetheirovariesweneedtocalculatetheinclusivetnessgainedbyafocalworkerwhohasamarginallygreaterprobabilityofreproducingthanotherworkers.
ThiscanbecalculatedfollowingFrank(1998)asWWfn#R(n1)##RsonnephewFFxxx,xrXx,xrXWWmqq#R#R.
(5)brothersisterFFxxx,xrXx,xrXThisisaHamilton'srulecondition(Hamilton1964)inwhichthederivativesmeasurehowgreaterreproductionbythefocalworkeraffectstheproductionofvariousclassesofkin(sonsofthefocalworker,nephewsproducedbythenestmateworkers,andbrothersandsistersproducedn1bythemotherqueen;Frank1998).
TheRvaluesarethelife-for-liferelatednessestoeachclassofoffspring.
Thederivativesareevaluatedforthecasewheretheworkers'behaviorapproachesthatoftherestofthepopulation,(),becauseweneedtodeterminetheinvasionx,xrXconditionforamutantthatisonlymarginallydifferentinphenotypefromthecommontype(Frank1998).
Life-for-liferelatednessistheproductofregressionrelatedness,ameasureoftheproportionofgenestherecipienthasincommonwiththeactorrelativetoarandomindividualandsex-specicreproductivevalue,ameasureofthege-neticcontributionofagivensextothefuturegenepool(Pamilo1991).
Hence,theRvaluescanbeexpandedto,,andRpr#np1#nRpr#nsonsonmmnephewsisterm,whereistherela-Rpr#np(1/2)#nnbrotherbrothermmmtivereproductivevalueofmalestofemales,whichis,whereisthepopulation-wideproportionof1/(2w)wmalesthatareworkers'sons(Pamilo1991).
Inourcase,itcanbeseenthat,whereXisthewpnXS/(nXSq)averageproportionoflayingworkersinanaveragecolonyinthepopulation.
Anevolutionarilystablestrategy(MaynardSmith1982)occurswhenthereisnonetinclusivetnessgainfromincreasedordecreasedworkerreproduction;thatis,whenequation.
Calculatingthepartialderivatives(see(5)p0theMathematicanotebook)andsolvingforXshowsthatthisoccurswhentheproportionoflayingworkersis2BB4ACXp,(6)2Awith22Ap2nS(1R),sisterBp2S[q(1nR4nR)sistersister(n1)n(1R)S],sisterCpq[q(14R)nS].
sisterBecauseeffectivepolicingreducesthesurvivalofworker-laideggs,wecansubstituteforSintheabove1Pequation,wherePistheeffectivenessofpolicing.
TheESSproportionoflayingworkersincreasesunderthreecon-ditions:rst,withlowerrelatednessamongsistersbecauselowsister-sisterrelatednessincreasestherelatednessgainofreplacingnephewsbysons(cf.
Bourke1988);second,withgreatercolonysizebecausecompetitionovermaleproductionwillthenprimarilybewithotherworkersratherthanwiththequeen;third,withincreasingmaleeggproductionbythequeen(thereverseargument).
Im-portantly,theESSproportionoflayingworkersdeclinesastheeffectivenessofthepolicingincreases(g.
2a).
In-tuitively,thisisbecausethebenettoaworkeroflaying()declineswhenfewerofhereggsareW/xFx,xrXfreared.
Thissupportstheideathatpolicingofworker-laidFigure2:Acquiescenceinthecontextofconictovermaleparentage.
a,Evolutionarilystablestrategy(ESS)proportionofegg-layingworkers(eq.
[6])asafunctionoftherelativeprobabilityPwithwhichworker-laideggsarekilled(policed)relativetoqueen-laideggs.
ThecurvesaredrawnforparametervaluesthatapplytoworkerpolicinginthehoneybeeApismellifera(1,multiplemating,,workers,relativequeenrp0.
3np35,000fecundity),thecommonwaspVespulavulgaris(2,doublemating,,workers,relativequeenfecundity),andtheqp25rp0.
51np3,000qp25EuropeanhornetVespacrabro(3,mostlysinglemating,,workers,relativequeenfecundity),andtoqueenpolicinginrp0.
67np400qp2.
5Dolichovespulamaculata(4,singlemating,,workers,relativequeenfecundity).
NotehowtheESSdeclinesastheefciencyrp0.
75np180qp2.
5ofpolicingapproaches100%.
Forexample,forinhoneybeesandforinthehornet,itbecomesunprotableforanyworkerP199.
8%P197.
7%toreproduce.
Thisshowsthatpolicingcanselectforworkerstocompletelyrefrainfromegglaying.
ParametervaluesarefromSeeley(1985),Ratnieks(1993),FosterandRatnieks(2001a),Fosteretal.
(2002),andT.
Wenseleers(unpublisheddata).
b,Criticalefciencyofpolicingrequiredtofavorcompleteworkersterilityasafunctionofsister-sisterrelatednessandfordifferentn:qratios(colonyqueenfecundity).
size/relativeE160TheAmericanNaturalisteggsselectsforself-policingoracquiescence(Ratnieks1988;Visscher1989).
Iftheeffectivenessofpolicingishighenough,workersareselectedtocompletelyrefrainfromegglaying.
Thatis,allworkersacquiescebyshowingcompletereproductiverestraint.
Theconditionsunderwhichthisshouldoccurcanbedeterminedbyevaluatingcostsandbenetsofworkerreproductioninapopulationwhereworkerre-productionisabsent,thatis,.
SubstitutionofXp0inequation(5)showsthatworkerlayingcanonlyXp0invadewhenS11#(RR)#R#R10.
(7)sonbrotherbrothersisterqnnThisequationhastheintuitiveinterpretationthatasin-glelayingworkerinacolonywouldreplaceafractionS/qofthequeen'smaleeggsbysonsbutwoulddiminishtotalcolonyreproductionby1/nifitdidnowork.
Fromequation(7)itisclearthatcompletereproductiverestraintoccurswhenthesurvivalofworkereggsislowerthanRRbrothersisterSpcrit(n/q)(RR)sonbrother1/4Rsisterp(8)(n/q)(1/4)14Rsisterp(n/q)orwhentheeffectivenessofpolicingP()isgreaterp1Sthan.
Thecriticalpolicingeffectivenessrequired1Scritforcompleteworkeracquiescenceunderdifferentdegreesofrelatednessamongworkersandfordifferentntoqratiosareshowningure2b.
Figure2aillustratesnumericallyhowpolicingreducestheESSproportionofegg-layingworkersforparametervaluesthatapplytothreewell-characterizedsystemsofworkerpolicing—thehoneybeeApismellifera(RatnieksandVisscher1989),thecommonwaspVespulavulgaris(FosterandRatnieks2001a),andtheEuropeanhornetVespacrabro(Fosteretal.
2002)—andonesystemofqueenpolicing,inthewaspDolichovespulamaculata(Greene1979).
Allfourspecieshaveverylowlevelsofworkerreproduction,with!
2%oftheworkershavingactiveova-ries(Ratnieks1993;FosterandRatnieks2001a;Fosteretal.
2001,2002).
Itisclearfromouranalysisthatsuchlowlevelsofworkerreproductionmustbeevolutionaryre-sponsestopolicingandnottokinshipalone.
Queensinboththehoneybeeandthecommonwasparepolyandrous(PalmerandOldroyd2000;FosterandRatnieks2001a),whichreducesrelatednessamongfemaleoffspring(Apis,,PalmerandOldroyd2000;V.
vulgaris,,rp0.
3rp0.
5FosterandRatnieks2001a).
Themodelpredictsthatwithsuchrelatednessvaluesandintheabsenceofpolicing,averyhighproportionoftheworkersshouldreproduce:56%inApisand33%inVespula.
Similarly,theopti-mumproportionofreproductiveworkers,intheabsenceofpolicing,wouldbe19%inVespacrabroand14%inDolichovespulamaculata(g.
2a).
Themuchsmallerob-servedproportionofreproductiveworkersinthesespecies,!
2%(Ratnieks1993;FosterandRatnieks2001b;Fosteretal.
2001,2002),isprobablyaresponsetotheeffectivepolicingthatoccursinthesespecies(g.
2a).
AcquiescencetoCasteFateInsocialHymenoptera,socialpolicingcanalsooccurinotherareasofreproductionbesidesmaleproduction,in-cludingtheconictoverfemalecastefate(BourkeandRatnieks1999;Ratnieks2001;Wenseleersetal.
2003).
Ad-vancedeusocialspeciesarecharacterizedbythepresenceofacastesysteminwhichimmaturefemalesirreversiblydevelopasoneoftwomorphologicallydistinctcastes,queenorworker(Wilson1971).
Becausequeenshavemuchgreaterreproductivepotentialthanworkers,anim-maturefemalegenerallybenetsfromdevelopingasaqueenratherthanasaworker(BourkeandRatnieks1999;Ratnieks2001;ReuterandKeller2001;Strassmannetal.
2002;Wenseleersetal.
2003).
However,herfateisnormallyoutofherowncontrol,beingsociallyimposedbytheamountorqualityoffoodsheisfedasalarvabytheadultworkers(Wheeler1986;BourkeandRatnieks1999;Beek-manandRatnieks2003;Beekmanetal.
2003).
Thepre-ventionofindividualsfromselshlydevelopingasqueensisreferredtoascaste-fatepolicing(Wenseleersetal.
2003).
Butwhyshoulddevelopingfemalesacceptthefateim-posedonthemanddevelopasworkersinresponsetoreducedfeedingInseveralstinglessbeegenera,femalesrearedinworkercellsovercomefeedingcontrolbyde-velopingasworker-sized"dwarfqueens"(reviewedinEn-gelsandImperatriz-Fonseca1990;Imperatriz-FonsecaandZucchi1995;Wenseleersetal.
2003,2005c).
Inthisway,theycansuccessfullycircumventcaste-fatepolicing.
Sim-ilarly,inmanyantspecies,femalescansometimesdevelopassmallintercastequeens(Heinze1998).
Giventheim-maturefemalesinsomespeciescanovercomecaste-fatepolicing,whyshouldfemalesacquiescetotheirimposedfeedingregimeandnotdevelopasdwarfqueensinallspeciesOnelikelyreasonisthatadwarfqueenmaynotgetthefullreproductivebenetofanormal-sizedqueen.
Indeed,inthestinglessbeeSchwarzianaquadripunctata,dwarfqueensarelessfecundthannormalqueens(RibeiroandAlves2001;Wenseleersetal.
2005c).
Ifthelifetimefe-PolicingandAcquiescenceinInsectSocietiesE161cundityofdwarfqueenswerelessthan75%thatofnormalqueens,thenundersinglematingandmonandry,itisbettertodevelopintoaworkerthanintoadwarfqueenbecauseproducing100%ofafullsister'soffspring(rp)wouldbeasgoodasproducing75%asmanyone's0.
375ownoffspring().
Thiscostofbecomingadwarfrp0.
5queencouldselectagainstcheatingandfavorfemalestoacquiescetotheirintendedworkerfate.
Anadditionalconditionunderwhichacquiescencemaybeexpectediswhenqueenscompetelocally,asoccursinspecieswithswarm-foundedcoloniessuchashoneybeesandstinglessbees.
Inswarm-foundingspecies,queensrearedinworkercellswouldprobablyhaveacompetitivedisadvantageovernormal-sizedqueens,particularlyifworkersprefernormalratherthanundersizedqueenstoheadnewswarmsorifdwarfqueenshaveareducedght-ingability.
WorkerdiscriminationagainstdwarfqueensappearstooccurinS.
quadripunctata.
Whereaslargequeensarepreferentiallystoredinspecial"queenprison"cells,potentiallytoheadacolonyatalaterdate(Imper-atriz-FonsecaandKleinert-Giovannini1989;Imperatriz-Fonseca1990),mostdwarfqueensareeliminatedbytheworkers(Imperatriz-FonsecaandKleinert-Giovannini1989;Imperatriz-Fonseca1990;Wenseleersetal.
2005c).
Consequently,theproportionofdwarfqueensheadingcolonies(22%)ismuchlowerthantherateatwhichtheyarereared(86%,Wenseleersetal.
2005c).
Coulddiscriminationagainstdwarfqueensbyworkersmakedevelopingasadwarfqueenlessrewarding,similartothewaythateggpolicingmakesitlessrewardingforworkerstolayeggsToinvestigatethis,considerthefol-lowingmodel,whichisanextensionofapreviousmodel(Wenseleersetal.
2003).
LetnandqbethenumberofworkerandqueencellspresentatanyonetimeinthecolonyandSbethesurvivalofdwarfqueensrearedinworkercellsrelativetonormalqueensrearedinqueencells(Sdeclineswhenworkersselectivelykillsmallqueens).
Assumethatafocalfemaleinaworkercellcheatsbydevelopingintoadwarfqueenwithprobabilityyandthateachofthefemalesinn1workercellscheatwithanaverageprobabilityofy.
Ifnormalandsmallqueensengageinascramblecompetitiontoformnewswarms,thenitcanbeshown(app.
A)thattheESSproportionoffemalesinworkercellsthatshoulddevelopasqueensis2BB4ACYp,(9)2Awith22ApnS(1R),mBpS{q[1n2nR(1n)R]mf(n1)n(1R)S},fCpq[q(RR)n(1R)S],mffwhereRmandRfarethe(life-for-life)relatednesstomalesandqueensrearedinthecolony.
Plottingequation(9)showsthatmorefemalesinworkercellsshoulddevelopasqueenswheneithertherelatednessamongsisters(Rf)ortherelatednesstothemales(Rm)rearedinthecolonyislower(g.
3).
Thepresenceofmanyworkercells(highn)orfewqueencells(lowq)alsoheight-enstheconictandfavorsmorefemalesinworkercellstodevelopasqueens.
Thisisbecauseitshiftsthequeen-queencompetitiontobeingprimarilyamongthefemalesdevelopinginworkerscells.
However,asbefore,thepres-enceofapolicingmechanism,hereintheformoftheselectiveeliminationofsmallqueens,canfavorindividualstoacquiesceanddeliberatelyaccepttheirintendedworkerfate.
Iftheprobabilitythatsmallqueensarekilledissuf-cientlyhigh(i.
e.
,ifSisloworifPishigh),thenfemalesinworkercellscanbeselectedtocompletelyacquiesceandneverattempttodevelopasaqueen.
Asbefore,thecon-ditionsunderwhichthisshouldoccurcanbedeterminedbycalculatingwhenfemalesinworkercellsareselectedtocheatandbecomequeensinasituationinwhichthestrategyisinitiallyabsent.
ThisisthecasewhenS11#(1R)#R#R10.
(10)ffmqnnInequalityequation(10)hastheintuitiveinterpretationthatasinglefemalethatwoulddevelopasaqueenratherthanaworkerwouldwinoutovernormalqueensduringcolonyfoundingwithprobabilityS/q(becauseitwouldsurvivewithprobabilitySbutcompeteswithqnormalqueens)butthatherlosstotheworkforcewouldreducetheproductivityofthecolony(maleandswarmproduc-tion)byafraction1/n.
Fromequation(10),itisclearthatfemalelarvaeinworkercellsareneverselectedtobecomequeenswhenthesurvivalofdwarfqueensislessthanRRmfSp(11)crit(n/q)(1R)forwhentheeffectivenessofpolicingP()isgreaterp1Sthan.
Thiscriticalefciencyofpolicingrequired1Scrittofavorcompletecaste-fateacquiescence,fordifferentre-latednessvaluesandfordifferentn:qratios,isshowningure3b.
Figure3:AcquiescenceinthecontextofconictovercastefateinswarmingsocialHymenopterasuchasstinglessbeesorhoneybees.
a,Evolutionarilystablestrategy(ESS)proportionoffemalesinworkercellsthatshoulddevelopasqueens(eq.
[9])reducesastheeffectivenessofpolicingincreases(intheformofdiscriminationagainstdwarfqueensrearedinworkercells).
TheESSisplottedforthesituationofsinglemating(),asisRp0.
75ftypicalforstinglessbees(Petersetal.
1999),withallmalesbeingeitherqueen'ssons(;toptwocurves)orworkers'sons(;Rp0.
25Rp0.
75mmbottomtwocurves)andwitheither1:1,000(solidlines)or1:100(dashedlines)cellsbeingspecializedqueencells(numberofworkercellsnpor1,000foreveryqueencellthatisbuilt).
b,Criticaleffectivenessofpolicingrequiredtomakeitcompletelyunprotableforfemalesinworker100cellstodevelopasqueens,plottedasafunctionofsister-sisterrelatedness,withallmalesbeingeitherqueen'ssons(;topsolidanddashedRp0.
25mcurves)orworkers'sons(;bottomsolidanddashedcurves)andwitheither1:1,000(solidlines)or1:100(dashedlines)cellsbeingRp0.
75mspecializedqueencells.
PolicingandAcquiescenceinInsectSocietiesE163Supportforthesepredictionsisprovidedbypatternsofqueenproductionindifferentgeneraofstinglessbees.
InthegenusMelipona,normal-sizedqueensrearedfromspe-cializedqueencellsareabsent().
Instead,allqueensqp0andworkers(andmales)developinidenticallysizedcellsonasimilarlysizedprovisionmass(EngelsandImperatriz-Fonseca1990;Wenseleersetal.
2004).
Furthermore,caste-fatepolicingisimpossible()asqueendevelopmentPp0isunconstrainedbynutrition(Kerr1950;Wenseleersetal.
2003,2004,2005c).
Inlinewithprediction,averylargeproportion,5%–14%,ofMeliponafemalesselshlyde-velopintoqueens(Kerr1950;WenseleersandRatnieks2004).
Bycontrast,instinglessbeeswherequeensarenor-mallyrearedfromlargerroyalcells,feworzerofemalesinworkercellsdevelopasqueens(EngelsandImperatriz-Fonseca1990;Imperatriz-FonsecaandZucchi1995;Ri-beiroetal.
2003).
Forexample,inS.
quadripunctata,just0.
6%ofallfemalesinworkercellsbecomedwarfqueens(Wenseleersetal.
2005c).
Thisisaspredictedgiventhatdwarfqueensarelessfecund(RibeiroandAlves2001;Wenseleersetal.
2005c)andthatasaresultofselectivecullingbyworkers(g.
1),theirsurvivalSisonlythatofnormal(0.
22/0.
86)[(10.
86)/(10.
22)]p4.
6%queens(cf.
RibeiroandAlves2001;Wenseleersetal.
2005c;anddatacitedabove).
Fecunditycostsareprobablyalsothereasonwhythehoneybeefemalesinworkercellsneverdevelopassmallqueens:thefecundityofsuchqueenswouldalmostcertainlybetoolowtosustaintheirlargecoloniesofover35,000bees(Seeley1985;Winston1987).
Inaddition,andunlikeinstinglessbees,larvaearerearedinopencellsinthehoneybee(Winston1987),whichwouldenableworkerstokilllarvaeattemptingtodevelopintoqueensatanearlyage.
DiscussionSocialpolicingisakeymechanismforreducingconictandpromotingcooperationwithinsocialgroups(Rat-nieks1988,2000;RatnieksandVisscher1989;RatnieksandReeve1992;Frank1995,1996;BourkeandRatnieks1999;MonninandRatnieks2001).
However,ouranalysisshowsthatpolicingmaybefarmoreeffectiveatreducingconictthanpreviouslyrealized.
Notonlycanpolicingpreventindividualsfromactingselshly,itcanalsomakeitworthlessforthemeventotrytoactselshlyintherstplace.
Thiscanfavorsocietieswheremostindividualsacquiescebyadoptingaless-favoredpositionorstrategyinsociety.
WeexaminedtwomajorconictscenariosfoundintheeusocialHymenoptera,andinbothcases,effectivepolicingcanmakeresistancelargelyorcom-pletelyuseless.
Thisisaverysignicantndingbecauseithelpstoresolvemajorlong-standingissuesattheheartofeusociality:theevolutionandmaintenanceofrepro-ductivedivisionoflabor,inparticularwhymostworkerswhoretainovariesdonotusethemandwhymostto-tipotentfemalelarvaedevelopintoworkersandnotqueens(Darwin1859;Hamilton1964;TriversandHare1976).
Empirically,acquiescencecanexplaintheobservedlowlevelsofreproductiveworkersfoundininsectso-cietieswitheffectivepolicing,suchasthehoneybee(Rat-nieks1993)orthecommonwasp(FosterandRatnieks2001a),andtheraritybywhichfemalescheatontheirintendedworkerfate,whichonlyoccursatlowlevelsinaminorityofgenera(e.
g.
,insomestinglessbees;EngelsandImperatriz-Fonseca1990;Imperatriz-FonsecaandZucchi1995;RibeiroandAlves2001;Wenseleersetal.
2003;Wenseleersetal.
2005c)andisentirelyabsentinmost(e.
g.
,inthehoneybee;Winston1987).
Theultimateendpointofacquiescenceistheevolutionofobligateworkersterilityinsomespecies,forexample,inAtta,whereworkersareunabletolayviableeggs(M.
Dijkstra,personalcommunication)andinLinepithema,Mono-morium,Pheidole,andSolenopsisantsandFrieseomellittaandDuckeolastinglessbees,whereworkershaveevolu-tionarilylostovariesaltogether(OsterandWilson1978;Cruz-Landim2000).
However,becauseitiscurrentlyun-knownwhetherthesegeneraancestrallyhadqueenorworkerpolicing,wecannottestwhetherworkersterilityhasindeedevolvedinresponsetopolicingorwhetheralternativefactorsmayhavebeenthemaincause.
GiventhelargeandcomplexsocietiesfoundinmanyeusocialHymenoptera,ititperhapsunsurprisingthattheyshouldprovidesuchclearexamplesofacquiescence.
Nevertheless,acquiescencemayalsobeexpectedinothersocialsituationssuchasinvertebratesocieties(Emlen1991;KellerandReeve1994)orintheparent-offspringrelationship(MockandParker1998).
Forexample,ifparentsdonotrespondtoexcessivelevelsofbegging,thiswouldfavormoreobedient,acquiescentyoung(MockandParker1998).
Similarly,incooperativelybreedinganimals,manipulationbydominantscanim-posecostsonsubordinates,favoringthemtostayandhelpratherthanleave,thatis,acquiesce(CrespiandRagsdale2000).
Theseexamplesindicatethatacquies-cencecouldwellbeaverywidespreadmechanismforconictreductioninsocieties.
AcknowledgmentsThisworkwassupportedbytheIntegratedStudiesoftheEconomicsofInsectSocieties–EuropeanUnionTrainingandMobilityofResearchersNetwork,byaMarieCuriepostdoctoralfellowshiptoT.
W.
,andbyNaturalEnviron-mentResearchCouncilgrantNER/A/S/2000/01324toA.
G.
H.
WethankS.
Frankfordiscussionandadviceonpartsoftheanalyses.
WearealsoverygratefultoV.
-L.
E164TheAmericanNaturalistImperatriz-FonsecaformakingSchwarzianacoloniesavailableforphotography.
APPENDIXADerivationoftheESSProductionofCheatingQueensLetnandqbethenumberofworkerandqueencellspresentatanyonetimeinthecolonyandSbethesurvivalofdwarfqueensrearedinworkercellsrelativetonormalqueensrearedinqueencells(dwarfqueensurvivaldeclineswhenworkersdiscriminateagainstthem).
Assumethatafocalfemaleinaworkercelldevelopsasaqueenratherthanasaworkerwithprobabilityyversusprobabilityyfortheotherfemales.
Thecolonywillthen,atanyn1onetime,producenormalqueensfromqueencellsatarateproportionaltoqandsmallqueensfromworkercellsatarateproportionalto,ifistheaverageprobabilitynyywithwhichfemalesinworkercellsdevelopasqueens,.
Alsoassume,forsimplicity,yp(1/n)y[(n1)/n]ythatsmallqueensrearedfromworkercellsdonothavereducedfecundityrelativetonormalqueens(includingfecundityinthemodelgreatlycomplicatestheanalysisbecauseitwouldbenecessarytokeeptrackofthepro-portionsofcoloniesinthepopulationthatareheadedbysmallvs.
normalqueens).
TheprobabilitythatqueensrearedinworkercellswillsuccessfullyheadanewcolonyisproportionaltoySWpG(y)#(A1)fnySqandySWpG(y)#,(A2)nnySqwhereisthecolonyproductivity(relativenumberofG(y)newswarmsformed)asafunctionofhowmanyfemalesbecomequeensratherthanworkers.
Thetermfollowingthisrepresentstheproportionofallnewqueenswhowererearedinworkercells,whichisgivenbytheprobabilitythateachonedevelopsasaqueen(probabilityyory)andsurvives(probabilityS)dividedbythetotalnumberofsurvivingsmallqueens()plusthenumberofnormalnySqueens(q)produced.
Forsimplicity,wewillassumethatcolonyproductivitydeclineslinearlywiththeproportionoffemalesthatdevelopintoqueensratherthanworkers,.
NotethattheseequationshavethesameformGp1yasintheworkerreproductionmodel(see"WorkerSterilityasAcquiescence")butwiththeparameterssuitablyredened.
Byasimilarargument,theexpectedtnessofqueensrearedinqueencellsisproportionalto1WpG(y)#.
(A3)qnySqFinally,tnessthroughmaleproductionWmisalsoadecreasingfunctionof:yWpG(y)p1y.
(A4)mTocalculatetheESS,weneedtocalculatetheinclusivetnessgainedbyfemalesinworkercellswhoaltertheirprobabilityofdevelopingasaqueen,whichcanbecal-culatedas(Frank1998)WWfn#R(n1)##RselfsisterFFyyy,yrYy,yrYWWqmq##R#R.
(A5)sistermalesFFyyy,yrYy,yrYAsin"WorkerSterilityasAcquiescence,"thisisaHam-ilton'srulecondition(Hamilton1964)inwhichthede-rivativesmeasurehowthebehaviorofthefocalfemaleinaworkercellaffectsthetnessorproductionofvariousclassesofkin(self,sisterfemalesinotherworkern1cells,sisterqueensinqqueencells,andmalesproducedbyeitherthequeenortheworkers),andtheRvaluesmeasurethelife-for-liferelatednessestoeachofthem.
Cal-culatingthepartialderivativesandsolvingwhenthere-sultingequationis0yieldstheESSgiveninequation(9)(seetheMathematicanotebookandcode).
LiteratureCitedBeekman,M.
,andF.
L.
W.
Ratnieks.
2003.
PoweroverreproductioninsocialHymenoptera.
PhilosophicalTransactionsoftheRoyalSocietyofLondonB358:1741–1753.
Beekman,M.
,J.
Komdeur,andF.
L.
W.
Ratnieks.
2003.
Reproductiveconictsinsocialanimals:whohaspowerTrendsinEcology&Evolution18:277–282.
Bourke,A.
F.
G.
1988.
WorkerreproductioninthehighereusocialHymenoptera.
QuarterlyReviewofBiology63:291–311.
Bourke,A.
F.
G.
,andF.
L.
W.
Ratnieks.
1999.
KinconictovercastedeterminationinsocialHymenoptera.
Be-havioralEcologyandSociobiology46:287–297.
Cnaani,J.
,R.
Schmid-Hempel,andJ.
O.
Schmidt.
2002.
Colonydevelopment,larvaldevelopmentandworkerreproductioninBombusimpatiensCresson.
InsectesSo-ciaux49:164–170.
Cole,B.
J.
1986.
ThesocialbehaviorofLeptothoraxallar-PolicingandAcquiescenceinInsectSocietiesE165dycei(Hymenoptera,formicidae):timebudgetsandtheevolutionofworkerreproduction.
BehavioralEcologyandSociobiology18:165–173.
Crespi,B.
J.
,andJ.
E.
Ragsdale.
2000.
Askewmodelfortheevolutionofsocialityviamanipulation:whyitisbettertobefearedthanloved.
ProceedingsoftheRoyalSocietyofLondonB267:821–828.
Cruz-Landim,C.
2000.
OvariandevelopmentinMelipon-inebees(Hymenoptera:Apidae):theeffectofqueenpresenceandfoodonworkerovarydevelopmentandeggproduction.
GeneticsandMolecularBiology23:83–88.
Darwin,C.
R.
1859.
Ontheoriginofspecies.
J.
Murray,London.
D'Ettorre,P.
,J.
Heinze,andF.
L.
W.
Ratnieks.
2004.
Workerpolicingbyegg-eatingintheponerineant,Pachycondylainversa.
ProceedingsoftheRoyalSocietyofLondonB271:1427–1434.
Emlen,S.
T.
1991.
Evolutionofcooperativebreedinginbirdsandmammals.
Pages301–337inJ.
KrebsandN.
B.
Davies,eds.
Behaviouralecology:anevolutionaryap-proach.
BlackwellScientic,Oxford.
Endler,A.
,J.
Liebig,T.
Schmitt,J.
E.
Parker,G.
R.
Jones,P.
Schreier,andB.
Ho¨lldobler.
2004.
Surfacehydrocar-bonsofqueeneggsregulateworkerreproductioninasocialinsect.
ProceedingsoftheNationalAcademyofSciencesoftheUSA101:2945–2950.
Engels,W.
,andV.
L.
Imperatriz-Fonseca.
1990.
Castede-velopment,reproductivestrategies,andcontroloffer-tilityinhoneybeesandstinglessbees.
Pages167–230inW.
Engels,ed.
Socialinsects:anevolutionaryap-proachtocastesandreproduction.
Springer,Berlin,Heidelberg.
Fletcher,D.
J.
C.
,andK.
G.
Ross.
1985.
RegulationofreproductionineusocialHymenoptera.
AnnualReviewofEntomology30:319–343.
Foster,K.
R.
,andF.
L.
W.
Ratnieks.
2000.
Facultativeworkerpolicinginawasp.
Nature407:692–693.
———.
2001a.
Convergentevolutionofworkerpolicingbyeggeatinginthehoneybeeandcommonwasp.
Pro-ceedingsoftheRoyalSocietyofLondonB268:169–174.
———.
2001b.
Theeffectofsex-allocationbiasingontheevolutionofworkerpolicinginhymenopteransocieties.
AmericanNaturalist158:615–623.
———.
2001c.
Paternity,reproductionandconictinves-pinewasps:amodelsystemfortestingkinselectionpredictions.
BehavioralEcologyandSociobiology50:1–8.
Foster,K.
R.
,F.
L.
W.
Ratnieks,N.
Gyllenstrand,andP.
A.
Thoren.
2001.
Colonykinstructureandmalepro-ductioninDolichovespulawasps.
MolecularEcology10:1003–1010.
Foster,K.
R.
,J.
Gulliver,andF.
L.
W.
Ratnieks.
2002.
WorkerpolicingintheEuropeanhornetVespacrabro.
InsectesSociaux49:41–44.
Frank,S.
A.
1995.
Mutualpolicingandrepressionofcom-petitionintheevolutionofcooperativegroups.
Nature377:520–522.
———.
1996.
Policingandgroupcohesionwhenre-sourcesvary.
AnimalBehaviour52:1163–1169.
———.
1998.
Thefoundationsofsocialevolution:mono-graphsinbehaviorandecology.
PrincetonUniversityPress,Princeton,N.
J.
———.
2003.
Repressionofcompetitionandtheevolu-tionofcooperation.
Evolution57:693–705.
Free,J.
B.
,I.
Weinberg,andA.
Whiten.
1969.
Theegg-eatingbehaviorofBombuslapidariusL.
Behaviour35:313–317.
Greene,A.
1979.
Behaviouralcharactersasindicatorsofyellowjacketphylogeny(Hymenoptera:Vespidae).
An-nalsoftheEntomologicalSocietyofAmerica72:614–619.
Halling,L.
A.
,B.
P.
Oldroyd,W.
Wattanachaiyingcharoen,A.
B.
Barron,P.
Nanork,andS.
Wongsiri.
2001.
WorkerpolicinginthebeeApisorea.
BehavioralEcologyandSociobiology49:509–513.
Hamilton,W.
D.
1964.
Thegeneticalevolutionofsocialbehavior.
I,II.
JournalofTheoreticalBiology7:1–52.
———.
1995.
Narrowroadsofgeneland.
Vol.
1.
Evolutionofsocialbehavior.
W.
H.
Freeman,NewYork.
Hardin,G.
1968.
Thetragedyofthecommons.
Science162:1243–1244.
Hartmann,A.
,andJ.
Heinze.
2003.
Layeggs,livelonger:divisionoflaborandlifespaninaclonalantspecies.
Evolution57:2424–2429.
Heckathorn,D.
D.
1996.
Thedynamicsanddilemmasofcollectiveaction.
AmericanSociologicalReview61:250–277.
Heinze,J.
1998.
Intercastes,intermorphs,andergatoidqueens:whoiswhoinantreproductionInsectesSo-ciaux45:113–124.
Hillesheim,E.
,N.
Koeniger,andR.
F.
A.
Moritz.
1989.
Colonyperformanceinhoneybees(Apismelliferaca-pensisEsch)dependsontheproportionofsubordinateanddominantworkers.
BehavioralEcologyandSocio-biology24:291–296.
Hurst,L.
D.
,A.
Atlan,andB.
O.
Bengtsson.
1996.
Geneticconicts.
QuarterlyReviewofBiology71:317–364.
Imperatriz-Fonseca,V.
L.
1990.
SwarmingactivityinSchwarzianaquadripunctata(Apidae,Meliponinae).
Pages744–745inG.
K.
Veeresh,B.
Mallik,andC.
A.
Viraktamath,eds.
Socialinsectsandtheenvironment.
Oxford&IBH,NewDelhi.
Imperatriz-Fonseca,V.
L.
,andA.
Kleinert-Giovannini.
1989.
AtividadesderainhasvirgensemPlebeia(Schwarziana)quadripunctata(Apidae,Meliponinae).
E166TheAmericanNaturalistRes.
doISimposioLatino-Americanosobre.
InsetosSo-ciaisNeotropicais(RioClaro)11.
Imperatriz-Fonseca,V.
L.
,andR.
Zucchi.
1995.
Virginqueensinstinglessbee(Apidae,Meliponinae)colonies:areview.
Apidologie26:231–244.
Keller,L.
1999.
Levelsofselectioninevolution.
PrincetonUniversityPress,Princeton,N.
J.
Keller,L.
,andH.
K.
Reeve.
1994.
Partitioningofrepro-ductioninanimalsocieties.
TrendsinEcology&Evo-lution9:98–102.
Kerr,W.
E.
1950.
GeneticdeterminationofcastesinthegenusMelipona.
Genetics35:143–152.
Kikuta,N.
,andK.
Tsuji.
1999.
Queenandworkerpolicinginthemonogynousandmonandrousant,Diacammasp.
BehavioralEcologyandSociobiology46:180–189.
Landolt,P.
J.
,R.
D.
Akre,andA.
Greene.
1977.
EffectsofcolonydivisiononVespulaatropilosa(Sladen)(Hymen-optera:Vespidae).
JournaloftheKansasEntomologicalSociety50:135–147.
Leigh,E.
G.
1991.
Genes,beesandecosystems:theevo-lutionofacommoninterestamongindividuals.
TrendsinEcology&Evolution6:257-262.
MaynardSmith,J.
1964.
Groupselectionandkinselection.
Nature201:1145–1147.
———.
1982.
Evolutionandthetheoryofgames.
Cam-bridgeUniversityPress,NewYork.
Michener,C.
D.
,andD.
J.
Brothers.
1974.
WereworkersofeusocialHymenopterainitiallyaltruisticorop-pressedProceedingsoftheNationalAcademyofSci-encesoftheUSA71:671–674.
Michod,R.
E.
1999.
Darwiniandynamics:evolutionarytransitionsintnessandindividuality.
PrincetonUni-versityPress,Princeton,N.
J.
Mock,D.
W.
,andG.
Parker.
1998.
Theevolutionofsiblingrivalry.
OxfordUniversityPress,Oxford.
Monnin,T.
,andC.
Peeters.
1997.
Cannibalismofsub-ordinates'eggsinthemonogynousqueenlessantDi-noponeraquadriceps.
Naturwissenschaften84:499–502.
———.
1999.
Dominancehierarchyandreproductiveconictsamongsubordinatesinamonogynousqueen-lessant.
BehavioralEcology10:323–332.
Monnin,T.
,andF.
L.
W.
Ratnieks.
2001.
Policinginqueen-lessponerineants.
BehavioralEcologyandSociobiology50:97–108.
Monnin,T.
,F.
L.
W.
Ratnieks,G.
R.
Jones,andR.
Beard.
2002.
Pretenderpunishmentinducedbychemicalsig-nallinginaqueenlessant.
Nature419:61–65.
Oldroyd,B.
P.
,L.
A.
Halling,G.
Good,W.
Wattanachai-yingcharoen,A.
B.
Barron,P.
Nanork,S.
Wongsiri,etal.
2001.
WorkerpolicingandworkerreproductioninApiscerana.
BehavioralEcologyandSociobiology50:371–377.
Oster,G.
F.
,andE.
O.
Wilson.
1978.
Casteandecologyinthesocialinsects.
PrincetonUniversityPress,Prince-ton,N.
J.
Ostrom,E.
1999.
Governingthecommons.
CambridgeUniversityPress,Cambridge.
Ostrom,E.
,J.
Burger,C.
B.
Field,R.
B.
Norgaard,andD.
Policansky.
1999.
Revisitingthecommons:locallessons,globalchallenges.
Science284:278–282.
Palmer,K.
A.
,andB.
P.
Oldroyd.
2000.
Evolutionofmul-tiplematinginthegenusApis.
Apidologie31:235–248.
Pamilo,P.
1991.
Evolutionofcolonycharacteristicsinso-cialinsects.
I.
Sexallocation.
AmericanNaturalist137:83–107.
Peters,J.
M.
,D.
C.
Queller,V.
L.
Imperatriz-Fonseca,D.
W.
Roubik,andJ.
E.
Strassmann.
1999.
Matenumber,kinselectionandsocialconictsinstinglessbeesandhoneybees.
ProceedingsoftheRoyalSocietyofLondonB266:379–384.
Pomeroy,N.
1979.
BroodbionomicsofBombusruderatusinNewZealand(Hymenoptera:Apidae).
CanadianEn-tomologist111:865–874.
Pomiankowski,A.
1999.
Intragenomicconict.
Pages121–152inL.
Keller,ed.
Levelsofselectioninevolution.
PrincetonUniversityPress,Princeton,N.
J.
Queller,D.
C.
,andJ.
E.
Strassmann.
1998.
Kinselectionandsocialinsects.
BioScience48:165–175.
Ratnieks,F.
L.
W.
1988.
ReproductiveharmonyviamutualpolicingbyworkersineusocialHymenoptera.
AmericanNaturalist132:217–236.
———.
1993.
Egg-laying,egg-removal,andovarydevel-opmentbyworkersinqueenrighthoney-beecolonies.
BehavioralEcologyandSociobiology32:191–198.
———.
1995.
Evidenceforaqueen-producedegg-markingpheromoneanditsuseinworkerpolicinginthehoney-bee.
JournalofApiculturalResearch34:31–37.
———.
2000.
Workerpolicinginthehoneybee:basicfactsandideas.
InsectSocialLife3:3–10.
———.
2001.
Heirsandspares:casteconictandexcessqueenproductioninMeliponabees.
BehavioralEcologyandSociobiology50:467–473.
Ratnieks,F.
L.
W.
,andH.
K.
Reeve.
1992.
Conictinsingle-queenhymenopteransocieties:thestructureofconictandprocessesthatreduceconictinadvancedeusocialspecies.
JournalofTheoreticalBiology158:33–65.
Ratnieks,F.
L.
W.
,andP.
K.
Visscher.
1989.
Workerpo-licinginthehoneybee.
Nature342:796–797.
Reeve,H.
K.
1991.
Polistes.
Pages99–148inK.
G.
RossandR.
W.
Matthews,eds.
Thesocialbiologyofwasps.
CornellUniversityPress,Ithaca,N.
Y.
Reuter,M.
,andL.
Keller.
2001.
SexratioconictandworkerproductionineusocialHymenoptera.
AmericanNaturalist158:166–177.
PolicingandAcquiescenceinInsectSocietiesE167Ribeiro,M.
F.
,andD.
A.
Alves.
2001.
SizevariationinSchwarzianaquadripunctataqueens(Hymenoptera,Ap-idae,Meliponini).
RevistadeEtologia3:59–65.
Ribeiro,M.
F.
,V.
L.
Imperatriz-Fonseca,andP.
D.
Santos.
2003.
ExceptionalhighqueenproductionintheBra-zilianstinglessbeePlebeiaremota.
StudiesonNeotrop-icalFaunaandEnvironment38:111–114.
Seeley,T.
D.
1985.
Honeybeeecology:astudyofadaptationinsociallife.
MonographsinBehaviorandEcology.
PrincetonUniversityPress,Princeton,N.
J.
Strassmann,J.
E.
,B.
W.
Sullender,andD.
C.
Queller.
2002.
Castetotipotencyandconictinalarge-colonysocialinsect.
ProceedingsoftheRoyalSocietyofLondonB269:263–270.
Trivers,R.
L.
1974.
Parent-offspringconict.
AmericanZoologist14:249–264.
Trivers,R.
L.
,andH.
Hare.
1976.
Haplodiploidyandtheevolutionofthesocialinsects.
Science191:249–263.
vanHonk,C.
,andP.
Hogeweg.
1981.
TheontogenyofthesocialstructureinacaptiveBombusterrestriscolony.
BehavioralEcologyandSociobiology9:111–119.
Visscher,P.
K.
1989.
Aquantitativestudyofworkerre-productioninhoneybeecolonies.
BehavioralEcologyandSociobiology25:247–254.
———.
1996.
Reproductiveconictinhoneybees:astale-mateofworkeregg-layingandpolicing.
BehavioralEcologyandSociobiology39:237–244.
Ward,P.
S.
1983.
Geneticrelatednessandcolonyorgani-zationinaspeciescomplexofponerineants.
I.
Phe-notypicandgenotypiccompositionofcolonies.
Behav-ioralEcologyandSociobiology12:285–299.
Wenseleers,T.
,andF.
L.
W.
Ratnieks.
2004.
TragedyofthecommonsinMeliponabees.
ProceedingsoftheRoyalSocietyofLondonB271(suppl.
):310-312.
Wenseleers,T.
,F.
L.
W.
Ratnieks,andJ.
Billen.
2003.
Castefateconictinswarm-foundingsocialHymenoptera:aninclusivetnessanalysis.
JournalofEvolutionaryBi-ology16:647–658.
Wenseleers,T.
,A.
G.
Hart,F.
L.
W.
Ratnieks,andJ.
J.
G.
Quezada-Euan.
2004.
Queenexecutionandcastecon-ictinthestinglessbeeMeliponabeecheii.
Ethology110:725–736.
Wenseleers,T.
,A.
Tolski,andF.
L.
W.
Ratnieks.
2005a.
QueenandworkerpolicinginthetreewaspDolicho-vespulasylvestris.
BehavioralEcologyandSociobiology(inpress).
Wenseleers,T.
,N.
S.
Badcock,K.
Erven,A.
Tolski,F.
S.
Nascimento,A.
G.
Hart,T.
A.
Burke,M.
E.
Archer,andF.
L.
W.
Ratnieks.
2005b.
Atestofworkerpolicingtheoryinanadvancedeusopcialwasp,Vespularufa.
Evolution(inpress).
Wenseleers,T.
,F.
L.
W.
Ratnieks,M.
F.
Ribeiro,D.
A.
Alves,andV.
-L.
Imperatriz-Fonseca.
2005c.
Working-classroy-alty:beesbeatthecastesystem.
BiologyLetters(inpress).
Wheeler,D.
E.
1986.
Developmentalandphysiologicalde-terminantsofcasteinsocialHymenoptera:evolutionaryimplications.
AmericanNaturalist128:13–34.
Wilson,E.
O.
1971Theinsectsocieties.
HarvardUniversityPress,Cambridge,Mass.
Winston,M.
L.
1987.
Thebiologyofthehoneybee.
Har-vardUniversityPress,Cambridge,Mass.
AssociateEditor:EldridgeS.
AdamsOnlineappendixB:Whenresistanceisuseless:policingandtheevolutionofreproductiveacquiescenceininsectsocietiesT.
Wenseleers,A.
G.
HartandF.
L.
WRatnieksThefollowingMathematicanotebookshowshowallkinselectionmodellingresultsinthemanuscriptcanbederivedinanequivalentwayusingpopulationgenetic(allelefrequency)models.
Theresultsofbothmodellingapproacheswillbeshownsidebyside,andthenumericalpredictionswillshowntobeidenticalinallcases.
Thegeneticmodelsareanalysedforanexamplecolonykinstructureofsinglemating,butproofsforothermatingfrequenciescaneasilybeconstructed.
Thekinselectionmodelsarederivedbothfollowinganinclusivefitnesslogic(asgiveninthemanuscript)andfollow-ingadirector"neighbour-modulated"fitnesslogic,asfavouredbyFrank(1998).
Theresultsofboththesemodellingapproachesareshowntobeidentical.
1.
WorkersterilityasacquiescenceNotationandparameters:n=colonysize(numberofworkers)y=individualprobabilityforafocalworkertobecomeareproductiveworkerY=probabilitywithwhicheachofthe(n-1)nestmatesbecomereproductiveworkersz=averageprobabilitywithwhichworkersinthecolonybecomereproductiveworkers=(1/n)y+(n-1)Y/nQ=fecundityofthequeen(intermsofmaleproduction)relativetoasinglereproductiveworkerS=survivalofworkerlaidegg(relativetoqueeneggs)-lowerwhenpolicingismoreeffectiveP=1-S=effectivenessofpolicingvm=relativereproductivevalueofmalesü1.
1Kinselectionmodel(inclusivefitnessapproach)Thenumberofsonsproducedbythefocalworkeriswfoc=H1zLHySêHnzS+QLLê.
8z→HH1ênLy+Hn1LYênL0&&rsisterthisvalue:online_appendix_B.
nb3criticalP@nQratio_,Rsister_D=HFullSimplify@Solve@Hchangeinclfitnê.
8S→1PColonyfitness1-W-d/2(forlargecolonies)Ptypecolonies:aaxA->Colonyfitness1-(W+d)IfthefrequencyoftheAgeneinfemalesandmalesispfandpm,thenwhenrareitsfrequencyinwildtype,MtypeandPtypecoloniesareapprox.
1,2pfandpm:Wfreq=1;Mfreq=2pf;H2H1pfLpfH1pmL=2pfforpmandpfsmallLPfreq=pm;HH1pfL^2.
pm=pmforpfsmallLWild-typeandP-typecolonieswilleachhaveaproductivityofColSW=1W;ColSP=1HW+dL;ForM-typecoloniestheproductivitywilldependonhowmanyAaworkersthecolonycontains;inlargecolonieshalfofthemwillbeAabutinsmallcoloniesthereissomebinomialvariationontheproportionthatwillbeaaorAa.
TheproductivityofanM-typecolonywhentherearekAaworkersandn-kaaworkersisColSMfofk@k_,n_D=1HHkênLHW+dL+HHnkLênLWL;TheproportionofmalesthatwillbeproducedbyeitheranAaworkerortheAamotherqueeninM-typecolonieswillagaindependonhowmanyAaandaaworkersthereare.
IftherearekAaworkersandn-kaaworkers,theproportionofthemalesthatwillbesonsofAamothersisgivenbypropAaMCols@k_,n_D=HkHW+dLS+QLêHkHW+dLS+HnkLWS+QL;Becausedissmallwecanuseafirstorderapproximation:propAaMCols@k_,n_D=FullSimplify@HpropAaMCols@k,nDê.
8d→00&&pf0&&pm0&&S0&&W0&&d0EH4npmSH1+d+WLHd+WLHQ+nSWL2+pfHQ+nSHd+WLLH2QSHd+3dn+6nH1+WLLW+4Q2H2+d+2WL+nS2WHH2+dLdH1+nL+4H1+dLnW+4nW2LLLêH8H1+WLHQ+nSWL2HQ+nSHd+WLLLForfemales:numberofAafemalesproduced*0.
5pfng=FullSimplifyAH1ê2LikjjjjjPfreqColSP1+MfreqH1ê2Lk=0nBinomial@n,kDH1ê2LkH1ê2LnkColSMfofk@k,nDy{zzzzzìColSW,pf>0&&pf0&&pm0&&W0&&d0E14J2Hpf+pmL+dHpf+2pmL1+WNInmatrixformthiscanbewrittenas:A=genetransitionmatrixikjjjfem.
parenttofem.
offspringfem.
parenttomaleoffspringmaleparenttofemaleoffspringmaleparenttomaleoffspringy{zzzA=FullSimplifyAJpfngê.
8pf→1,pm→01.
TheeigenvaluesoftheAmatrixareHbecausedissmallwecanuseafirstorderapproximationaroundd=0Lonline_appendix_B.
nb7evalues=FullSimplify@HHEigenvalues@ADê.
8d→00&&S0&&W>0&&W0&&d>0&&d0&&W0&&S>0&&S0D@@2DD@@1DD@@2DD114n2SJH1+16nLQ+H1+nLnS+#H1+32nH1+nLLQ2+2nH1+3nH5+4nLLQS+H1+nL2n2S2NThisistheESSproportionoflayingworkers.
Theoptimumcalculatedusingakinselectionmethodologywaskinselopt=FullSimplify@ExpandAll@ESSreprwDF@n,Q,S,3ê4DD,n>0&&S>0&&S0D114n2SJH1+16nLQ+H1+nLnS+#H1+32nH1+nLLQ2+2nH1+3nH5+4nLLQS+H1+nL2n2S2NBothequationsarethesame:genoptkinseloptTrueQ.
E.
D.
online_appendix_B.
nb82.
AcquiescencetocastefateNotationandparameters:y=individualprobabilityforafocalfemaleinaworkercelltodevelopasaqueenY=probabilitywithwhicheachofthe(n-1)nestmatefemalesinworkercellsdevelopasqueensz=colonyaverageprobabilitywithwhichfemalesinworkercellsdevelopassmallqueens=(1/n)y+(n-1)Y/nQ=fecundityofthequeen(intermsofmaleproduction)relativetoasinglereproductiveworkerS=survivalofsmallqueensrelativetonormalqueens(lowerthehighertheprobabilitythattheyareselectivelykilledbytheworkers)P=effectivenessofpolicing=1-Svm=relativereproductivevalueofmalesn=numberofworkercellsQ=numberofqueencellsü2.
1Kinselectionmodel(inclusivefitnessapproach)Theexpectedfitnessofafocalfemaleinaworkercelliswfoc=HH1zLHySêHnzS+QLLLê.
8z→HH1ênLy+Hn1LYênL0&&Rsister0&&Rmales0&&Q>0&&S>0&&S∞D,Rsister>0&&Rsister0&&Rmales0&&S0&&Q>0D1Rsister1+RmalesProductionofsmallqueenswillbefavouredifthestrategycaninvadefromasituationinwhichitisinitiallyabsent(z=0):changeinclfitn=Expand@Simplify@HD@wfoc,yD1+Hn1LD@wnestm,yDRsister+QD@wq,yDRsister+D@wm,yDRmalesLê.
8y→Z,Y→Zthisvalue:criticalP@nQratio_,Rsister_,Rmales_D=HFullSimplify@Solve@Hchangeinclfitnê.
8S→1P0&&Rsister0&&n>0&&S>0&&SColonyfitness1-W-d/2(forlargecolonies)Ptypecolonies:aaxA->Colonyfitness1-(W+d)IfthefrequencyoftheAgeneinfemalesandmalesispfandpm,thenwhenrareitsfrequencyinwildtype,MtypeandPtypecoloniesareapprox.
1,2pfandpm:Wfreq=1;Mfreq=2pf;H2H1pfLpfH1pmL=2pfforpmandpfsmallLPfreq=pm;HH1pfL^2.
pm=pmforpfsmallLonline_appendix_B.
nb13Wild-typeandP-typecolonieswilleachhaveaproductivityofColSW=1W;ColSP=1HW+dL;ForM-typecoloniestheproductivitywilldependonhowmanyworker-intendedfemalesdevelopasqueens;inlargecoloniestheproductivitywouldbe1-(W+d/2)butinsmallcoloniesonehastocalculatethebinomialvariationonthenumberofAaandaaindividualsthatthecolonywouldcontain.
TheaverageproductivityofanM-typecolonywhenthecolonyproduceskAafemalesandn-kaafemalesisColSMfofk@k_,n_D=1HHkênLHW+dL+HHnkLênLWL;TheexpectedproportionofnewcoloniesthatwouldbefoundedbyAaqueenswillagaindependonhowmanyAaandaafemalesthatareproduced.
IftherearekAafemalesandn-kaafemales,theproportionofthenewqueensthatwillbeAaisgivenbypropAaMCols@k_,n_D=HkHW+dLS+QH1ê2LLêHkHW+dLS+HnkLWS+QL;Becausedissmallwecanuseafirstorderapproximation:propAaMCols@k_,n_D=FullSimplify@HpropAaMCols@k,nDê.
8d→00&&pf0&&pm0&&W0&&d0E18H1+WLHQ+nSWL2H4pmH1+d+WLHQ+nSWL2+pfH2Q2H2+d+2WL+nS2WHH2+dLdH1+nL+4H1+dLnW+4nW2L+QSHd2H1+nL+8nH1+WLW+2dHn+W+3nWLLLLonline_appendix_B.
nb14Inmatrixformthiscanbewrittenas:A=genetransitionmatrixikjjjfem.
parenttofem.
offspringfem.
parenttomaleoffspringmaleparenttofemaleoffspringmaleparenttomaleoffspringy{zzzA=FullSimplifyAJpfngê.
8pf→1,pm→01.
TheeigenvaluesoftheAmatrixareHbecausedissmallwecanuseafirstorderapproximationaroundd=0Levalues=FullSimplify@HHEigenvalues@ADê.
8d→00&&S0&&W>0&&W0&&d>0&&d0&&F0.
12844287190833065`0&&S>0&&S0D12H10+3FLn2SJH2+F18n+5FnLQH2+FLH1+nLnS+######H2+FLHQ+nSLHH2+F+2H18+5FLn+H2+FLn2LQ+H2+FLH1+nL2nSLNThisistheESSqueenproductionfromworkercellsThisistheESSforwhenallmalesareworkerproducedgenoptallmaleswprod=FullSimplify@genoptê.
8F→10&&S>0&&S0DH1+13nLQ+H1+nLnS+!
HQ+nSLHH1+nH26+nLLQ+H1+nL2nSL14n2SThisistheESSforwhenallmalesarequeenproducedgenoptallmalesqprod=FullSimplify@genoptê.
8F→00&&S>0&&S0DH1+9nLQ+H1+nLnS+!
HQ+nSLHH1+nH18+nLLQ+H1+nL2nSL10n2STheoptimacalculatedusingakinselectionmethodologywerekinseloptallmaleswprod=FullSimplify@ExpandAll@ESSqueenprodIF@n,Q,S,3ê4,3ê4DDDH1+13nLQ+H1+nLnS+!
HQ+nSLHH1+nH26+nLLQ+H1+nL2nSL14n2Skinseloptallmalesqprod=FullSimplify@ExpandAll@ESSqueenprodIF@n,Q,S,3ê4,1ê4DDDH1+9nLQ+H1+nLnS+!
HQ+nSLHH1+nH18+nLLQ+H1+nL2nSL10n2STheoptimacalculatedfromthegeneticmodelandthekinselectionmodelarethesame:genoptallmalesqprodkinseloptallmalesqprodTruegenoptallmaleswprodkinseloptallmaleswprodTrueQ.
E.
D.
online_appendix_B.
nb16