Received 28October 2002 Accepted 13March 2003 Publishedonline 25 July 2003

Quantifyingmale attractivene ss JohnM. McNamara 1* ,AlasdairI. Houston 2,Miguel Marques dosSantos 1, Hanna Kokko3† and RobBrooks 4 1Departmentof Mathematics, University ofBristol, University Walk,Bristol BS8 1TW, UK 2Schoolof Biological Sciences, University ofBristol, Woodland Road, Bristol BS8 1GU, UK 3Division ofEnvironmentaland EvolutionaryBiology, Institute ofBiomedical and Life Sciences, University ofGlasgow, GlasgowG12 8QQ, UK 4Schoolof Biological Earth and EnvironmentalSciences, TheUniversity ofNew SouthWales, Sydney, NSW 2052,Australia Geneticmodels of sexual selectionare concernedwith adynamic processin which female preferenceand male trait values coevolve.We present a rigorous methodfor characterizing evolutionary endpointsof this processin phenotypic terms.In ourphenotypic characterization themate-choice strategy offemale popu- lation members determineshow attractive femalesshould find each male, anda population is evol- utionarily stable if population members are actually behaving in this way.This provides ajustificationof phenotypic explanations ofsexual selectionand the insights intosexual selectionthat they provide. Fur- thermore, thephenotypic approach also has enormousadvantages over ageneticapproach whencomput- ing evolutionarily stable mate-choicestrategies, especially whenstrategies are allowed tobecomplex time- dependentpreference rules. For simplicity andclarity ouranalysis dealswith haploid mate-choicegenetics anda male trait that is inherited phenotypically, for example by vertical cultural transmission.The method is,however, easily extendibleto other cases.An example illustrates that thesexy sonphenomenon can occurwhen there is phenotypic inheritance ofthe male trait. Keywords: Fisher process;;sexy son;;evolutionarily stable strategy; phenotypic models

1. INTRODUCTION oftheFisher process.Subsequent work demonstrated that theFisher processcould still occurif thegenes controlling It is only adaptive for femalesto be selective about their themale trait weresubject to biased mutation choiceof mate if males differin their value toa female.A (Pomiankowski et al. 1991). male’s value asa mate may dependon direct or indirect The basisof theFisher processis that it is more valuable benefits(Kirkpatrick &Ryan 1991; Kokko et al. 2003). for afemale tomate with somemales than with others. Direct benefitsare advantages suchas food, protection or This advantage might beable tocompensatefor any direct agoodterritory that amale may provide for afemale or disadvantage associatedwith choosinga male asa mate. her young.Indirect benefits are basedon a male’s . For example, if males differin both attractivenessand the The genesthat amale passeson toits offspring may influ- care that they provide for their offspring,then it might be encethe offspring’ s ability tosurvive andreproduce. One advantageousfor afemale tomate with anattractive male way in which genesmay influenceoffspring reproductive evenif heprovides lesscare than anunattractive male. successis by determining theattractiveness of male off- This idea isknown as the sexy sonhypothesis spring tofemales. This effectwas proposed by Fisher (Weatherhead &Robertson1979) becausefemales are (1930) asthe basis for an explanation ofexaggerated compensatedfor reducedmale care by having sexy sons. male traits. Weatherhead andRobertson supported their idea with a Fisher argued that, given aninitial female preferencefor simple modelbased on counting descendants a given amale trait, both thestrength of the female preference numberof generationsinto thefuture. Kirkpatrick (1985) andthe value ofthemale trait couldbe increased by sex- pointedout that themodel of Weatherhead andRobertson ual selection.The male trait is favouredbecause of the wasincorrect. He also argued more generally that thepro- female preference,and the female trait is favouredbecause cedureof counting descendants was not valid (seealso femalesthat are choosyhave sonsthat are preferred. Arnold1983). From his geneticanalysis, Kirkpatrick con- Fisher gave averbal accountof this ‘runaway process’. cludedthat theadvantage ofsexy sonscould not compen- Lande(1981) andKirkpatrick (1982) constructedmodels satea female for adirectcost, but Pomiankowski et al. basedon population geneticsto show that Fisher’s run- (1991) showedthat thesexy soneffect can occur in some away processcould work. In thesemodels, the female does geneticmodels. notincur any costas a resultof being choosy.Pomiankow- Wecanexplain theFisher processin termsof thevalue ski (1987) showedin aparticular contextthat including toa female ofmating with differentmales. In contrastto costsof female preferenceresulted in nopermanent effect this perspective,models based on genetics deal with the change ofgene frequencies over time, rather than ideas ofvalue. Geneticmodels have donemuch to further our *Authorfor correspondence ([email protected]). † Present address: Department of Biologicaland Environmental Science, understandingof sexual selection,but it canbe argued University of Jyva¨skyla¨,POBox 35,40351 Jyva ¨skyla¨,Finland. that they lack theintuitive appeal ofphenotypic expla-

Proc.R. Soc.Lond. B (2003) 270, 1925–1932 1925 Ó 2003 TheRoyal Society DOI10.1098/ rspb.2003.2396 1926J. M.McNamara andothers Quantifying male attractiveness nationsbased on values andrewards. The genetic is.Typically choosinessincurs costs. For example, a approach is clearly fundamental.Does this meanthat the choosyfemale might tendto take longer tofind a mate intuitive arguments basedon value are sloppy, orcanthey andhence leave fewersurviving offspring,either because bemade precise in thesense that they give thesame final sheis in poorer conditionat reproductionor because outcomesas a geneticanalysis? If phenotypic explanations young producedlate have lower survival prospects.The cannotbe made precise then the intuition affordedby ver- female’smate-choicerule may also affectthe probability bal arguments may bemisleading. Conversely,if they can that shesurvives to breed again nextyear. Weassume, bemade precise,then careful verbal arguments basedon however,that themate-choice rule usedby afemale does asuitable idea ofvalue provide considerableadvantages. notaffect her beforeher first breeding season. It canbe difficult toanalyse theevolution of a large num- A female’smate-choicerule isdetermined by asingle ber oftraits in ageneticmodel. The conversionto a haploid autosomal locus.Alleles are denotedby p, p9, etc. phenotypic modelmakes it possibleto characterize end- Weassume a one-to-onecorrespondence between poss- pointsin termsof game theory or optimality andhence ible mate-choicerules and alleles. Thusfor every mate- makesthe analysis mucheasier. More importantly, the choicerule thereis auniqueallele that codesfor this rule. phenotypic approach offersgenuine insight intosexual Giventhis assumptionwe simplify notation,allowing p to selection(Grafen 1990 a,b;Pen& Weissing2000; Kokko denotean allele andthe mate-choice rule that is codedfor et al. 2002). by theallele. Amale carries, butdoes not express, the Taylor (1990) developsa techniquethat canbe used mate-choiceallele that it inherits from its parents.Each totranslate geneticmodels into phenotypic modelsusing offspring inherits themate-choice allele ofits mother with reproductivevalue. Pen& Weissing(2000) usethis tech- probability 0.5 andthat ofits father with probability 0.5. niqueto give aphenotypic accountof mate choiceand sexual selection.In this paper wepresent an alternative (a) Invasibilityand evolutionary stability techniquethat gives thesame end result as the analysis of Wewill say that strategy p is theresident mate-choice Taylor (1990), butachieves this resultmore directly in the strategy if all population members are genetically p. Con- casewe analyse.Our phenotypic accountis similar tothat sidera population where p is theresident strategy. Let ofPen & Weissing(2000) butis applied toa different this population bedemographically stable,i.e. the popu- aspectof sexual selection.We considera simple modelin lation hasstable growth (or constantsize, if density- which femaleschoose the male that they will mate with dependenteffects are acting) andhas a stable composition onthe basis ofthe value ofa male ’strait. Afemale ’s rule over time. In particular, thesex ratio is stable,as are the for choosinga mate is genetically determinedby asingle proportions ofmales ofeach type. Suppose that amutant haploid locus.For simplicity weassume that thetrait of p9 allele arisesin this resident p population. Will the amale is determined,with error, by thetrait ofhis father, mutant invade into theresident population? If the i.e.transmission is basedon phenotype rather than geno- mutation arisesjust once, then it may becomeextinct type.This paternal effectcould be based on vertical cul- becauseof demographic stochasticity,even if themutation tural inheritance,such as inheritance ofsong in some confershigh fitness.We ignore suchchance events and speciesof bird ortheinheritance ofwealth in humans.(It focuson mutations that are nottoo rare. Theninvasion couldalso betransmitted ona chromosomethat only is concernedwith whethermean mutant numbers grow occursin males.)For this modelwe show that theend- fasterthan residentnumbers when mutant numbers are pointsof selectioncan be characterized phenotypically in still rare compared with residents. arigorous way.The heart ofour analysis involves To quantify therate ofgrowth ofmutant numbers we assigning anattractiveness to a male that quantifiesthe censusthe population annually at thestart ofeach breed- advantage toa female ofmating with themale. This value ing season.At a censustime a p9 allele canbe in oneof is notassumed in advance;instead it emergesin aself- K 1 1typesof individual; afemale or atype j male, consistentway from ouranalysis. j = 1, …, K.First considera p9 female presentat thestart Wehave deliberately chosena contextin whichthe logic ofa breeding season.The p9 descendantsof this female ofour approach canbe clearly demonstrated.Both diploid that are presentat thestart ofthe breeding seasonnext determination ofthemate-choice rule andgenetic inherit- year comprise:(i) any p9 offspring producedthis year that anceof the male trait introducecomplications that are survive until nextyear; and(ii) thefemale herself,if she describedin thediscussion. Despite these complications, survives.We enumeratesuch descendants by thefunctions ourapproach canstill beused to obtain aphenotypic accountin thesecases. ai(p9, p) = expectednumber of p9 type i male descen- dantsleft nextyear.

2. THE MODEL b(p9, p) = expectednumber of p9 female descendantsleft next year. Weconsider a large, well mixed, sexually reproducing population. Malesare ofoneof K distinctphenotypes lab- Nowconsider a p9 type j male presentat thestart ofa elled 1, …, K.Type ispassed (with error) from father to breeding season.His p9 descendantsare enumeratedby sonphenotypically. There is anannual breeding seasonin thefunctions which eachfemale chooseswhich male or males tomate a (p) = expectednumber of p9 type i male descendants with.Offspring that resultfrom in onebreeding ij left nextyear. seasonare sexually mature by thenext breeding season.

A female’smate-choicerule determineswhat typesof b j (p) = expectednumber of p9 female descendantsleft males thefemale prefersto mate with andhow choosy she next year.

Proc.R. Soc.Lond. B (2003) Quantifying male attractiveness J.M.McNamara andothers 1927

Thesequantities depend on p but not p9 becauseof the V(p)L(p, p) =l(p, p)V(p). (2.5) following assumptions:(i) themate-choice is not Then V (p)isthe reproductive value ofatype j male rela- expressedin males;(ii) a p9 male will almost certainly j tive toa female.This is theratio ofthe expected number mate with a p female becausethe p9 allele israre; and(iii) ofdescendants left at sometime far in thefuture by the daughtersare notaffected by themate-choice allele that male relative tothe expected number left by afemale at they carry until their first breeding season;sons are also this time (e.g.Houston & McNamara 1999; Caswell notaffected. 2001). Supposethat in year t there are n (t) p9 type j males and j Wenow use reproductive value toconstructthe payoff n (t) p9 femalespresent at thestart ofthebreeding season. f in another game. For strategies p9 and p set Then,provided numbersof the p9 allele are nottoo small (sothat demographic stochasticity canbe ignored), num- K W(p9, p) = a (p9, p)V (p) 1 b(p9, p). (2.6) berspresent in year t 1 1are related tonumbers present O i i in year t by nT(t 1 1) = L(p9, p)nT(t), where nT (t) is the i = 1 transposeof the row vector n(t) = (n1 (t), n2 (t), …, nK(t), Werefer tothe game with payoff W asthe phenotypic nf(t)) and L(p9, p) is the (K 1 1) by (K 1 1) matrix game. W(p9, p)is theexpected total reproductivevalue of theoffspring left nextyear by afemale usingmate-choice a1 1 (p) a1 2(p) … a1 K(p) a1 (p9, p) strategy p9,wherereproductive value is that underthe a2 1 (p) … … a2 K(p) a2 (p9, p) residentstrategy p. L(p9, p) = . (2.1) In Appendix Ait is shownthat thegenetic and pheno- typic game payoffsare related by aK1 (p) … … aKK(p) aK(p9, p) b (p) … … b (p) b(p9, p) W(p9, p) , W(p, p) l(p9, p) , l(p, p). (2.7) 1 K , The left-handinequality saysthat amutantfemale which We refer to L(p9, p)asthe projection matrix for the p9 usesmate-choice strategy p9 this year, butwhose descend- allele whenthe resident strategy is p.Over time, the antsrevert tothe resident mate-choice strategy p in future annual proportionate increasein thetotal numberof years,will leave fewerdescendants far in thefuture than mutantalleles will settledown: aresidentfemale. As relationship (2.7) shows,this is equi- valent tothe mutant leaving fewerdescendants if all her total mutantnumbers in year t 1 1 l(p9, p), (2.2) descendantsuse strategy p9 rather than reverting to p. total mutantnumbers in year t ! Relationship (2.7) allows usto reformulate thecon- wherethe stable growth rate l(p9, p)is themaximum ditionsfor theevolutionary stability ofa strategy p¤ in eigenvalue ofthe projection matrix L(p9, p).Following terms of W rather than l.By criterion (2.4) andresult Metz et al. (1992) weassumethat theresident startegy p (2.7) asufficientcondition for p¤ tobe an ESS is that is stable against invasion by themutant allele p9 if W(p9, p¤ ) , W(p¤ , p¤ ) for all p9 Þ p¤ . (2.8) l(p9, p) , l(p, p), (2.3) In thelanguage ofgame theory, if astrategy p¤ is the that is,within theresident p population thegrowth rate uniquebest response to itself in thegenetic game it is the ofmutant allele numbersis less than that ofresidentallele uniquebest response to itself in thephenotypic game. numbers.This invasion criterion doesnot just count chil- (This doesnot mean that thetwo games have thesame drenor grandchildren. Instead,by usingstable growth bestresponse functions; in general they donot.) The ratesit countsthe number of copies of itself that anallele characterization ofevolutionary stability given by leaves far intothe future. expression(2.8) canalso beobtained using the method given by Taylor (1990). Aresidentstrategy p¤ is anevolutionarily stable strategy For thespecific mate-choice model we present below, (ESS)if nostrategy differentfrom p¤ can invade every strategy has auniquebest response. That is,for (Maynard Smith 1982). Thusa sufficientcondition for p¤ tobe an ESSis that every strategy p thereis a uniquestrategy B(p) such that ( ( ), ) = max ( , ). (2.9) l(p, p¤ ) , l(p¤ , p¤ ) for all p Þ p¤ . (2.4) W B p p W p9 p p9 In other words,if weregard l(p9, p)asthe payoff tothe Thus,in this casea strategy p¤ is anESS if andonly if strategy p9 playing against thestrategy p in agame, then asufficientcondition for theevolutionary stability ofa B(p¤ ) = p¤ . (2.10) mate-choicestrategy p¤ is that p¤ is theunique best responseto itself in this geneticgame. Wenow introduce 3. EXAMPLE:MALES ENCOUNTERED AS A aphenotypic game whoseequilibrium solutionscoincide POISSONPROCESS with thoseof this game. To illustrate theabove theoretical resultswe specify a (b) Thephenotypic game detailedmodel ofthe mate-choice process. Assume non- Supposethat all population members are genetically p, overlapping generationswith ageneration time of1 year. that is p is theresident mate-choice strategy, thenthe The annual breeding seasonstarts at time ofyear t = 0 and population projectionmatrix is L(p, p).Letthe vector ends at time t = Tseason .During abreeding seasona female

V(p) = (V1(p),V2 (p), ..., VK(p),1) bethe eigenvector of searchesuntil shefinds a suitable mate.While searching L(p, p)satisfying sheencounters males asa Poissonprocess of rate unity.

Proc.R. Soc.Lond. B (2003) 1928J. M.McNamara andothers Quantifying male attractiveness

Encounteredmales are drawnat random from thepopu- suchan ESS existsthere is amirror image ESSwith pref- lation. Oneach encounter the female mustdecide whether erencefor type 2males,and vice versa. tomate with that particular male, orrejecthim andcon- Afemale that followsmate-choice strategy p = (0, t2) tinueto search. On shestops searching, produces always acceptstype 1males.The larger theswitch time t2 , offspring andthen dies. If shehas notmated by theend themore costsshe is prepared toincur to obtain atype 1 ofthe breeding seasonshe dies without producing any off- male rather than atype 2male. In other words,as t2 spring. Amale may mate with many femalesduring asea- increasesso does her strengthof preference for type 1 son.After the breeding seasonhe also dies. males.An increase in thepreference of resident females for

Mating with atype i male produces,on average, Ni off- type 1males increasesthe number of matings ofthe sons, spring, ofwhich half are sonsand half are daughters.Sur- grandsonsand so on, of type 1males.Thus it increasesthe vival ofoffspring tomaturity isindependent of their sex advantage toa mutantfemale ofmating with atype 1male andtype. It isalso independentof the time at whichthe rather than atype 2male. This advantage, quantifiedas parentsmated. Thus the only costto the female ofrejecting relative attractiveness,is illustrated in figure 1 a. amale isthat shemay notencounter another beforethe In this symmetric case,when resident females prefer endof the breeding season,and hence may fail tobreed. type 1males,the best response of amutantfemale is either

Define ri(p)tobe the expected total reproductivevalue toshow no preference or toshow a preferencefor type 1 ofthe surviving offspring if afemale mateswith atype i males aswell. With aslight abuseof notation we denote male. Werefer to ri(p)asthe attractiveness of a type i thebest response to the resident mate-choice strategy male. By equations(2.6) and(2.9), afemale following the p = (0, t2 ) by B(p) = (0, B(t2 )). Figure 1b illustrates best bestresponse strategy B(p)maximizes theexpected responsefunctions. attractivenessof the male that shechooses. This strategy Aresidentstrategy p¤ = (0, t2¤ )is anESS if andonly if canthus be found by standarddynamic optimization tech- B(t2¤ ) = t¤2 .This criterion meansthat nomutant following niques(e.g. Whittle 1982). Becausean ESS is thebest another strategy caninvade intoa population wherethe responseto itself, in anevolutionarily stable population residentstrategy is already p¤ .Anotherstability criterion theattractiveness of a type i male describesthe strength is that ofcontinuousstability (Eshel1983). Apopulation ofpreference of resident females for amale ofthis type. is continuouslystable if it will evolve to p¤ wheninitially Whatever theresident mate-choice strategy, it canbe displacedslightly from p¤ .Letthe resident mate-choice shownthat thebest response is uniquely definedand has strategy be p = (0, t2 ). If B(t2 ) . t2 ,thena female follow- thefollowing form determinedby the K switchtimes ing thebest mutant strategy has astronger preferencefor t1 , t2 , …, tK .Onencountering a type i male at time t a type 1males than residentfemales. Under suitable regu- searching female rejectsthe male if t , ti andmates with larity conditions(see Appendix B)this meansthat the the male if t > ti. population will evolve towardshigher preference.Con-

versely, if B(t2 ) , t2 ,thepopulation will evolve towards (a) Twomale types lesspreference. Thus for thespecial one-dimensionalcase

Supposethat thereare twotypes of males (i.e. K = 2) weare considering,an ESS p¤ = (0, t2¤ )is also continu- andthat among thesons of a type i male aproportion ouslystable if theslope of B satisfies B9(t2¤ ) , 1 (cf. Eshel

1 2 pi are type i anda proportion pi are theother type. 1983; Taylor 1989). Conversely,if B9(t2¤ ) . 1, then a

We refer to p1 and p2 asmutation ratesbecause they spec- population displacedfrom p¤ in either directionwill ify theerror in passing ontype from father toson. To evolve further away from p¤ in that direction,so that p¤ obtain apositive correlation betweenson ’s type and is situatedon an invasion barrier. It is reasonable tosup- father’stype weassumethat p1 1 p2 , 1. Notethat, given posethat anendpoint of the process of is both theassumptions of the model, the absolute values of N1 anESS andcontinuously stable; i.e. it isa CSSasdefined and N2 are notrelevant, only their ratio N1 /N2 affects by Eshel. results. Figure 1b illustrates thebest response to the resident

Atleast oneof the switch times t1 and t2 iszero because mate-choicestrategy for twovalues ofthe common it is neveroptimal for afemale toreject both typesof male mutation probability. For eachvalue, whenresident at time 0. Becausebest responses are ofthis form, in femaleshave aweakpreference for type 1males,the best searching for an ESSwecan restrict attentionto resident responseis tohave nopreference. A population where strategies ofthis form. In other words,we restrict attention initially femaleshad aweakpreference for type 1males tostrategies p that canbe expressed as a pair ofnon- wouldthus evolve tono preference. When the mutation negative numbers p = (t1 , t2 ),whereat least oneof these probability is high, femalesfollowing thebest response numbersis zero. strategy always have aweakerpreference for type 1males than residentfemales. Thus the only ESSisone of no (i) Symmetric case preference.When the mutation probability is low,females

Considerthe case where N 1 = N 2 and p1 = p2 . Then following thebest response strategy have agreater prefer- thetwo males are identical,except for alabel that isrecog- encefor type 1males than residentfemales for arange of nizedby females.In this casethe mate-choice strategy of residentmate-choice strategies. There are thenthree taking thefirst male encountered(i.e. the strategy ESSs,but the middle oneis not continuously stable and p¤ = (0, 0)) isalways an ESS(andalways continuously formsan invasion barrier betweenthe other two. stable;see below). We refer tothis strategy asone of no Figure 2showsthe effect of breeding seasonlength. preference.When presenting results we showthis ESSand Whenthe season is short a female cannotafford to be thosewhere females prefer type 1males,i.e. ESSs of the choosyand no preference is theonly ESS.For longer sea- form p¤ = (0, t2¤ ), where t2¤ . 0. By symmetry, whenever sonlength thereis less likelihood that afemale whorejects

Proc.R. Soc.Lond. B (2003) Quantifying male attractiveness J.M.McNamara andothers 1929

(a) (a) ,

s 1.8 e

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e continuously 1

1.6 (ii) m 1.2 e i stable t p y h t c

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mate-choice strategy e l 1.6 a m

b 1

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n ( s 2 o s r p / stable e s ) n * e

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s r

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( length of season, T season B 0.5 e

m Figure 2. Theeffect of thelength of thebreeding season i t T when there are two male types.( a)ESSmate-choice

h season c t i 0 0.5 1.0 1.5 2.0 2.5 rules of theform p¤ = (0, t2¤ ). For Tseason , Tcrit no w

s preference isthe only ESS.For T . T a second switch time t 2 under the resident season crit continuously stableESS is separated from theno-preference stable mate-choice strategy ESSbyan invasion barrier. ( b)Theattractiveness of type1 Figure 1. Theeffect of theresident mate-choice strategy malesrelative to type2 malesat the continuously stableESS when there issymmetry between thetwo typesof male. The withpreference for type1 males. p1 = p2 = 0.15, N1 = N2. resident strategyis p = (0, t2),indicating thattype 1 males are alwaysaccepted and type2 malesare rejected before the ratio N1/N2 .Ascan be seen, for arange ofvalues of N1 switch time t2. (a)Theattractiveness of type1 malesrelative and N2 with N 1 , N 2 thereis acontinuouslystable ESS to type2 males, r1(p)/r2(p). (b)Thebest response B(p) with preferencefor type 1males.In other words,at evol- = (0, B(t2))totheresident strategy. For case(ii), arrows utionary stability femalesprefer males that produceless markthe three solutions of theequation ( ¤ ) = ¤ , and it is B t2 t2 offspring.This isan example ofthe sexy sonphenomenon also indicated whether or not eachESS is continuously (Weatherhead &Robertson1979). stable. Casesare (i) highmutation probabilities p1 = p2 = 0.3; (ii) low mutation probability p = p = 0.15. 1 2 (b) Multiple male types Tseason = 2.7, N1 = N2. Figure 4presentsresults for anexample in whichthere are 50 typesof male. In this example thenumber of sur- atype 2male early in theseason will fail toencounter viving offspring that resultfrom mating with atype i male another male. Thusthe female loseslittle by being choosy, decreasesas i increases(figure 4 a).The meantype ofthe andthere are also twoother ESSs,one of which is con- sonsof atype i male lies between1 and i,sothat mutation tinuouslystable whilst theother is not.As in figure 1, the is biasedand tends to reduce type number(cf. Iwasa et continuouslystable ESSis separatedfrom theno- al. 1991; Pomiankowski et al. 1991). Full details onthe preferenceESS by theinvasion barrier formedby the numberand type ofoffspring are given in Appendix C. other ESS(figure 2 a).Astheseason length increases,and The figure showsaspects of the resident population at hencethe cost of being choosydecreases, the relative eachof two continuously stable ESSs.Computations attractivenessof type 1males at thechoosy ESS increases (examining sensitivity toinitial conditions)suggest that (figure 2b). theseare theonly continuouslystable ESSs.At oneESS femalesprefer males that provide themost offspring. At

(ii) Effect of N1 /N2 theother, females prefer males ofintermediate type to Wenowallow thetwo types of male toproduce differ- males that producethe most offspring andto males that entnumbers of offspring.Figure 3showshow the continu- producevery few.This is afurther example ofthe sexy ouslystable ESSsand invasion barriers dependon the sonphenomenon.

Proc.R. Soc.Lond. B (2003) 1930J. M.McNamara andothers Quantifying male attractiveness

3.0 (a)

2.5 s s e

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2.0 v i m t i t c

a h r

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s 8

1.0 n i r p s

0.5 f f o

f

o 4 0 r

0.8 0.9 1 1.1 e b

–0.5 m

N 1 /N 2 u n Figure 3. Theeffect of thenumber of surviving offspring on thenumber and stabilityof ESSsthat are present. There are (b) two typesof male, so thatresults depend on theratio N1/N2. 4 Continuously stableESSs lie on thesolid lines; invasion barriers lie on dashed lines. In showing ESSswe adopt the following convention: apositive switchtime t corresponds to 3 the ESS (0, t)withpreference for type1 males; anegative e m i switch time 2t corresponds totheESS ( t,0) withpreference t

h

for type2 males. Only asmallrange of negative values is c

t 2 shown, but thefull curve can beconstructed bysymmetry i w s withthe positive values. p1 = p2 = 0.15, Tseason = 4. 1

4. DISCUSSION Wehave shownthat theFisher processand the sexy soneffect can occur when the female trait is genetically (c) determinedand the male trait is culturally determinedand 0.075 y t i

inherited by vertical transmission.Although therehave l i b

beenprevious modelsof cultural transmissionand sexual a t s selection(e.g. Laland 1994 a,b; Aoki et al. 2001; Naka- y r jima &Aoki 2002), this particular issuehas notbeen a 0.050 n o i addressed. t u l

Anymodel that analysesthe Fisher process,or, more o v e generally, mate choicefor indirect benefits,cannot just t 0.025 a

countoffspring numberbecause the beneficial effectsof n o i choosinga sexy sondo not begin toappear until grand- t r o children are produced.In contrastto approximations that p o r

look nofurther intothe future than grandchildren p (Weatherhead &Robertson1979; Heisler1981), our 0 10 20 30 40 50 approach looksat theasymptotic rate ofspread of a mate- male type choiceallele. Wethentranslate resultson spreadrates into Figure 4. Examplewith 50 typesof male. TwoESSs, resultsabout thetotal reproductivevalue ofdescendants referred to asESS1 and ESS2,are illustrated. ( a) For each left in 1year ’stime. In this way,structuring thepopulation male type: thenumber of surviving offspring left bya female andallowing reproductivevalue todepend on type,we are if shemates with a male of thistype (the Ni of thetext), and able tolook just1 year intothe future in assessingwhether theattractiveness of amale of thistype at each ESS. Circles, amutantallele canultimately invade.This equivalenceof offspring number; squares, attractiveness ESS2;triangles, thelong-term andshort-term fitnessadvantage has been attractiveness ESS1.( b)Thetwo ESSmate-choice strategies, pioneeredby Taylor (1990) in thecontext of translating eachspecified bya sequence of switchtimes t1, t2, …, t50. geneticmodels into phenotypic models.Our analysis in Squares, switchtimes ESS2;triangles, switchtimes ESS1. (c)Theproportion of malesof eachtype at each of the this specificcase is more directthan that ofTaylor, but ESSs.Squares, proportion ESS2;triangles, proportion reachesthe same conclusion. ESS1. T season = 4. Afurther benefitof ourmethod is that wedonotneed tomake theassumption of fixed correlations between traits andpreferences. Genetic approaches needto con- siderthe correlation betweenthe female mate-choiceallele justtotal numbers,automatically takescorrelation into carried by amale andmale ’strait value, butare unable account.In other words,a female with apreferenceis togenerate this correlation from first principles exceptin more likely tohave her offspring siredby an attractive casesof very fewloci (e.g.Kirkpatrick 1982). Our male, which automatically establishesa correlation approach obviates theneed to look at correlations because betweenfemale preferencesand offspring traits whenboth countingnumbers of offspring ofeach type, rather than thetrait andthe preference is passedon to offspring.

Proc.R. Soc.Lond. B (2003) Quantifying male attractiveness J.M.McNamara andothers 1931

Pen& Weissing(2000) analyse aspecificmodel of sex- Wehave chosento present our phenotypic approach in ual selectionusing an approach basedon that ofTaylor thesimplest possiblecase of haploid geneticsand pheno- (1990). In their model,males inherit thepropensity y to typic inheritance ofthemale trait. This has allowed usto developinto aparticular type,and any correlation exposethe logic ofthe situation in theclearest way, but betweenthe type ofa father andthat ofhis sonis solely thebasic logic still holdsin other cases.When female dueto inheritance ofthis y value. Thusif population choiceis determinedby diploid genetics,the analysis goes members all have thesame y values (for example, at evol- through asbefore for mutantsthat are notcompletely utionary stability) thenthere is nocorrelation between recessive.Completely recessivemutants must be treated father’stype andson ’stype.Pen & Weissing(2000) con- differently (e.g.using the approach ofTaylor (1990)), as cludethat whenfemale preferenceis costlythere is no wecannot assume that individuals that are heterozygous stable equilibrium with female preferencefor sexy sons. for this mutantare rare in determining theprojection They thusinfer that modelsare most matrix. Whenthe female choicerule andthe male type relevant whenfemale preferencepre-exists for somerea- are both genetically determined,a rare mate-choiceallele sonnot included in themodel. By contrast,our analysis, canbe found in 2 K typesof individual: asbefore there are whichincorporates apositive correlation betweenthe type K typesof male butthere are nowalso K typesof female ofa father andthat ofhis sons(even at evolutionary becausethe female carries themale type allele. The pro- stability), showsthat simple modelscan predict stable jectionmatrix is thusa 2 K by 2K matrix, butour approach equilibria with female preferencefor sexy sons.Generally, canstill beapplied. Resultsdepend on whether the for asexy sonbenefit to cause the evolution of female expressionof the female mate-choiceallele candepend choicedespite direct costs that it incurs,it appears neces- onthe gene for male type that sheis carrying. If sucha sary that thereis aprocessthat maintains variation in off- dependenceis possible then models predict that females spring sexiness,despite the directional selectionprovided shouldput less effort into finding apreferredmale if they by female preference(Kokko et al. (2002); ‘handicap’ ver- are already carrying an allele for this male type (J.M. sionof themodel of Pomiankowski et al. (1991), Eshel et McNamara, M.Marquesdos Santos and A. I.Houston, al. (2000)). Modelsthat lack this assumptiondo not pre- unpublisheddata). Such complications are easyto deal dictstable female choice(Kirkpatrick (1985), Pomian- with in aphenotypic modelbased on reproductive value kowski(1987); ‘large-effect mutation ’ versionof the calculations. modelof Eshel et al. (2000)). In ourmodel, this variation is provided by errors in transmissionfrom father toson. WethankInnes Cuthill, Andrew Pomiankowski, Franjo Weiss- Our approach characterizes thestable endpointsof the ing and three anonymous referees for comments on previous evolutionary processin phenotypic terms(cf. Grafen versions of thispaper. M.M.d.S. wassupported byFundac ¸a˜o para a Cieˆncia eTecnologia. 1990a;Taylor 1990; Eshel1996; Hammerstein 1996; Weissing1996; Pen& Weissing2000). This approach gives usa consistentway tomeasure the ‘costs’ and ‘bene- APPENDIXA: THERELATIONSHIPBETWEEN THE fits’ ofmating with aparticular male, whenfitness conse- GENETICAND PHENOTYPICGAMES quencesinclude direct as well asindirect effects.We have To analyse therelationship betweenthe games, we formally definedthe attractiveness of a male asthe adapt atechniqueused by Caswell(2001) tolook at eigen- expectedvalue ofthe surviving offspring producedon value sensitivity. Considera resident p population in mating with themale. The residentmate-choice strategy whichthe mutant allele p9 is rare. At thedemographic in apopulation determinesthe attractiveness of each type steadystate the ratio ofnumbers of mutant type i males ofmale. Amutantfemale is following thebest response tomutant females, P9 is stable andsatisfies strategy if her behaviour maximizes theexpected attract- i ivenessof themale that shechooses. In this senseattract- L(p9, p)P9T = l(p9, p)P9T , (A 1) ivenessspecifies the mate-choice costs that afemale shouldbe prepared topay tobe choosy. The population where P9 denotesthe row vector is evolutionarily stable if residentfemales are following this P9 = (P91 , P92 , ..., PK9, 1). (A 2) bestresponse strategy. Thusan ESSis characterized by theself-consistency of attractiveness: the resident strategy The matrices L(p9, p) and L(p, p)have identical first K specifieshow attractive femalesshould find each male; this columns.Thus by equation(2.6) strategy is anESS if andonly if residentfemales are behav- V(p) [L(p, p) 2 L(p9, p)] = (0, ..., 0, W(p, p) ing in this way.This phenotypic characterization ofevol- 2 W(p9, p)). (A 3) utionary stability facilitates intuition about sexual selection.It also has enormousadvantages over agenetic Multiplying both sidesof equation (A 3) from theright approach whencomputing ESS.For example, wehave by P9T gives presenteda modelin which thereare 50 typesof male. In W(p, p) 2 W(p9, p) = V(p)[L(p, p) this modela female ’smate-choicestrategy specifiesa 2 L(p9, p)] P9T . (A 4) switchtime for eachtype ofmale, sothat it is determined by 50 traits. Geneticmodels would find it cumbersome Thusby equations(2.5) and(A 1) toanalyse selectionin 50 dimensions.By contrast,our W(p, p) 2 W(p9, p) = [l(p, p) approach has noproblem in identifying bestresponses 2 l(p9, p)] V(p) P9T . (A 5) usingthe power of dynamic optimization techniquessuch asdynamic programming (applied toa 50-state problem, Because V(p) P9T is positive wededucerelationship (2.7) nota 50-dimension problem). ofthe main text.

Proc.R. Soc.Lond. B (2003) 1932J. M.McNamara andothers Quantifying male attractiveness

APPENDIXB: BEST RESPONSESAND in thequantitative theory of evolution. J.Math. Biol. 34, CONTINUOUSSTABILITY 485–510. Eshel, E.,Volovik, I.&Sansone, E.2000 OnFisher –Zahavi’s For simplicity ofexposition wedeal with thecase of handicapped sexyson. Evol. Ecol. Res. 2, 509–523. twosymmetric male types.In thenotation of the text we Fisher, R.A.1930 The genetical theory ofnatural selection . considermate-choice strategies ofthe form p = (0, t2 ), Oxford University Press. where t2 is theswitch time at whichtype 2males are first Grafen, A.1990 a Biological signals ashandicaps. J. Theor. 144 accepted.Suppose the resident strategy is p = (0, t2). Biol. , 517–546. Thenthe payoff toa mutantfemale usingmate-choice Grafen, A.1990 b Sexual selection unhandicapped bythe Fisher process. J.Theor.Biol. 144, 473–516. strategy (0, s2 )in this residentpopulation is f(s ) = W((0, s ), (0t )).The function f is maximized when Hammerstein, P. 1996 Darwinian adaptation, population genetics 2 2 2 and thestreetcar theory of evolution. J.Math. Biol. 34, 511–532. themutant employs thebest response rule; i.e. when Heisler, I.L.1981 Offspring quality and thepolygyny thresh- s2 = B(t2 ).The analysis presentedbelow is based on the old: anew model for the ‘sexy son’ hypothesis. Am. Nat. plausible assumptionthat f(s2 )isa unimodal functionof 117, 316–328. s2 with auniquemaximum at s 2 = B(t2 ). Houston, A.I.&McNamara,J. M.1999 Models ofadaptive Supposethat B(t2 ) . t2 .By unimodality behaviour.Cambridge University Press. Iwasa, Y., Pomiankowski, A.&Nee, S.1991 Theevolution of 0 , s , t W((0, s ), (0, t )) 2 2 ) 2 2 costly matepreferences. II. The ‘handicap’ principle. Evol- , W((0, t2 ), (0, t2 )) (B 1) ution 45, 1431–1442. and Kirkpatrick, M.1982 Sexual selection and theevolution of femalechoice. Evolution 36, 1–12. t2 , s2 , B(t2) W((0, s2 ), (0, t2 )) Kirkpatrick, M.1985 Evolution of femalechoice and male par- ) . W((0, t2 ), (0, t2 )). (B 2) ental investment in polygynous : thedemise of the ‘sexy son’. Am. Nat. 125, 788–810. Thusby relationship (2.7) ofthe main text andits ana- Kirkpatrick, M.&Ryan, M.J.1991 Theevolution of mating logue with thereverse inequality preferences and theparadox of thelek. Nature 350, 33–38. Kokko, H., Brooks, R., McNamara,J. M.&Houston, A.I. 0 , s2 , t2 l((0, s2 ), (0, t2 )) , l((0, t2 ), (0, t2)) (B 3) ) 2002 Thesexual selection continuum. Proc.R. Soc. Lond. and B 269, 1331–1340. (DOI 10.1098/rspb.2002.2020.) Kokko, H., Brooks, R., Jennions, M.&Morley, J.2003 The t , s , B(t ) l((0, s ), (0, t )) 2 2 2 ) 2 2 evolution of matechoice and mating biases. Proc.R. Soc. . l((0, t ), (0, t )). (B 4) 2 2 Lond. B 270, 653–664. (DOI 10.1098/rspb.2002.2235.)

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