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Quantifying Male Attractiveness Nationsbased on Values Andrewards 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 into sexual 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;sexual selection;sexy son;mate choice;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 basis oftheFisher 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 benefitsare basedon a male’s genes. 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 an attractive male way in which genesmay influenceoffspring reproductive evenif heprovides lesscare than anunattractive male. successis by determining theattractiveness of male off- This idea is knownas 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 an initial female preferencefor simple model basedon counting descendants a given amale trait, both thestrength ofthe 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. Arnold 1983). 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 mean that the choosyfemale might tendto take longer tofind a mate intuitive arguments basedon value are sloppy, orcanthey andhence leave fewersurviving offspring,either because bemade precisein 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 is determinedby 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- makes theanalysis mucheasier. More importantly, the choicerule thereis auniqueallele that codesfor this rule. phenotypic approach offersgenuine insight into sexual 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 reproductive value. 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 model in lation has stable growth (or constantsize, if density- which femaleschoose the male that they will mate with dependenteffects are acting) andhas astable 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
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