Received 23January 2002 Accepted 18March 2002 Publishedonline 11 June 2002

Thesexual selection continuum Hanna Kokko1*,Robert Brooks 2,JohnM. McNamara 3 and AlasdairI. Houston 4 1Division ofEnvironmentaland EvolutionaryBiology, Institute ofBiomedical and Life Sciences, University ofGlasgow, Glasgow G12 8QQ, UK 2Schoolof Biological, Earth and EnvironmentalScience, TheUniversity ofNew SouthWales, Sydney 2052,NSW, Australia 3Schoolof Mathematics, University ofBristol, University Walk,Bristol BS8 1TW, UK 4Schoolof Biological Sciences, University ofBristol, Woodland Road, Bristol BS8 1UG, UK The evolutionof mate choicefor geneticbeneŽ ts has becomethe tale oftwo hypotheses: Fisher’ s ‘run- away’and ‘ goodgenes’ , or viability indicators.These hypotheses are oftenpitted against eachother as alternatives, with evidencethat attractive males sire more viable offspring interpretedas support for good genesand with anegative ornull relationship betweenmating successof sons and other componentsof Žtnessinterpreted as favouring theFisher process.Here, we build ageneral modelof female choicefor indirect beneŽts that capturesthe essence of both the‘ Fisherian’and ‘ good-genes’models. All versions ofour model point toa single processthat favoursfemale preferencefor males siring offspring ofhigh reproductive value. Enhancedmating successand survival are thereforeequally valid geneticbeneŽ ts of mate choice,but their relative importance varies dependingon female choicecosts. The relationship betweenmale attractivenessand survival may bepositive or negative, dependingon life-history trade- offsand mating skew.This relationship canchange sign in responseto increased costliness of choice or environmental change.Any form offemale preferenceis subjectto self-reinforcing evolution,and any relationship (or lack thereof)between male display andoffspring survival is inevitably anindicator of offspring reproductivevalues. Costly female choicecan be maintained with or withouthigher offspring survival. Keywords: Fisher process;runaway evolution;good genes; mate choice;indirect beneŽts; sexual selection

1. INTRODUCTION cessdemonstrated that in apopulation wheresome mate choiceoccurs already, andwhere preference and display Anytrait evolvesin directresponse to natural selection are heritable, thebuild-up of ageneticcorrelation between andin indirect responseto selectionon genetically corre- preferenceand display isinevitable (Lande1981) and lated traits. In thecase of mating preferencesfor exagger- leadsto runaway coevolution(Lande 1981; Kirkpatrick atedsexual displays, directŽ tnessbeneŽ ts and costs 1982). Positive preference–display geneticcorrelations imposedirect selection on mate-choice behaviour, but have beenfound in nature(Bakker 1993; Wilkinson & indirect selectiondue to geneticcovariation betweenmate Reillo 1994; Houde1994; Gray &Cade1999) andtheir choiceand other Žtnesscomponents is also thought tobe rapid build-upunder experimental conditionshas been widespread(Andersson 1994). Fisher (1930) pointedout empirically veriŽed (Blows 1999). that it wouldbe adaptive for femalesto prefer mating with In recentyears, attention has shiftedagain tothe Ž rst males that bear atrait that isfavoured by natural selection. componentof Fisher’ s theory—the evolution of prefer- Furthermore, Fisher recognizedthat asa preference encesfor male traits that indicateŽ tnessin contextsother increasesin frequency,the attractiveness of thetrait itself than mate choice.This has beenspurred by many factors, will amplify theadvantage ofmate choice.The ensuing including thearticulation by Zahavi (1975) ofthe handi- ‘runaway’process of coevolution would exaggerate the cap principle, aconsiderablebody of theoretical modelling display trait beyondits naturally selectedoptimum. Dis- (e.g.Pomiankowski 1988; Grafen1990 a,b; Iwasa et al. play exaggeration ishalted by atrade-offbetween male 1991; Houle& Kondrashov2002), theobservation that attractivenessand other componentsof Žtness,and prefer- in many speciespreferred males are more vigorous and encesfail toexaggerate whenmale survival is socurtailed long-lived than non-preferredmales (Jennions et al. 2001) that thereis nolonger any beneŽt tofemales of mating andoften produce offspring ofhigher viability (Møller & with attractive males. Alatalo 1999) andarguments that additive geneticvari- Fisher’s namehas becomeassociated in thesexual- ation in Žtnesscomponents can be maintained by pro- selectionliterature with thesecond component of his cessessuch as host– parasite coevolution(Hamilton &Zuk theory—the runaway coevolutionbetween preference and 1982), genotypeby environmentinteraction (Kotiaho et display (Fisher 1930). Early attempts tomodel this pro- al. 2001) andmutation– selection balance (Rowe& Houle 1996). Anunfortunate consequence of the success of good-genesindicators has beenan overemphasis ofthe * Authorfor correspondence ([email protected]). differencebetween these so-called ‘ goodgenes’ or ‘indi-

Proc.R. Soc.Lond. B (2002) 269, 1331–1340 1331 Ó 2002 TheRoyal Society DOI10.1098/ rspb.2002.2020 1332H. Kokko andothers Thesexual selection continuum cator’ modelsand the Fisherian runaway process.One choiceevolution whether the viability ofattractive males couldbe forgiven for thinking, uponreading thecontem- isgreater than,less than or equal tothat ofunattractive porary literature, that Fisherian runaway andgood-genes males. indicators werealternative theoriesfor theevolution of mate choiceand that themain goal ofthe Ž eldis todis- 2. THE MODEL tinguish betweenthese as alternative hypotheses. Considerableempirical attentionhas focusedon the Weconsider a population in which births anddeaths phenotypic andgenetic relationships betweenmale sexual occurcontinuously and the primary sexratio is 1:1. Mate display andsurvival, becauseviability isone of the most choicefor indirect bene Ž tscan operate only whenmales important Ž tnesscomponents that may beindicated by vary in someheritable trait. To provide aconceptually sexual displays. Alarge numberof studies estimate the simple example, weassume the existence of a genewith phenotypic (Jennions et al. 2001) orgenetic(Reynolds & twoalternative alleles, i = 0and2, that is expressedin Gross1992; Norris 1993; Petrie 1994; Jones et al. 1998; males only.The genein  uencesthe male ’sappearance in Møller &Alatalo 1999; Wedell &Tregenza 1999; Brooks waysdescribed in thefollowing paragraphs, andit can 2000) associationbetween attractiveness, survival, fec- covary with survival positively, negatively or notat all. It undityand other Ž tnesscomponents (Petrie 1994; canthus also bethought tore  ect the ‘quality’ of a male. Møller &Alatalo 1999; Wedell& Tregenza 1999; Brooks However,to avoid any unnecessaryconnotations with ter- 2000). Apositive relationship betweenmale attractiveness minology that is biasedin favour ofone (runaway) or the andoffspring survival (or other aspectsof Ž tness) is other (indicator) direction,we simply refer tomales carry- thought tosupport the good-genes hypothesis (Norris ing thesealleles asmale ‘types’ i = 1and2, andnote that 1993; Petrie 1994; M øller &Alatalo 1999; Jennions et al. wedo not necessarily assume a correlation betweenmale 2001), whereasa negative relationship combinedwith sig- type andsurvival. niŽ cantheritability ofattractiveness is interpretedas fav- Weassumehaploid inheritance ofthe gene. Any model ouring aFisherian mechanism alone(Etges 1996; Jones et ofsexual selectionfor indirect bene Ž tswill require a al. 1998; Wedell& Tregenza 1999; Brooks2000). How- mechanism tomaintain variation in males.As our focus ever,it is conceivable within apurely good-genesframe- is noton solving thelek paradox, wesimply assumethat workthat males with thehighest geneticquality advertise sucha mechanism exists:allele 1mutatesto allele 2with sointensely that their survival becomesworse than that of a rate of m1 per generation,and allele 2toallele 1with males ofpoorer quality (Grafen1990 a;Kokko1998, a rate of m2 (seetable 1for afull list ofvariables and 2001; Eshel et al. 2000). their deŽ nitions). In dealing with therelationship betweensexual attract- The type ofa male cannotbe observed directly. How- ivenessand survival, it is crucial toremember that costly ever,a male oftype i also expressesa sexually selected sexual advertisementis alife-history trait subjectto trade- trait denotedby Di (for ‘display’; if Di = 0themale lacks offswith other componentsof reproductiveeffort and with thedisplay trait). Additionally, both males andfemales futuresurvival andfecundity (Partridge &Endler1987; carry apreferencegene P that dictatesfemale preferences

Grafen 1990a;Gustafsson et al. 1995; Kokko1997, 1998, for large values of Di.Female preferencescan vary con- 2001; Ho¨glund& Sheldon1998). Anegative relationship tinuouslyfrom absent( P = 0) tostrong (large P). A high betweenreproductive effort (including advertisement)and value of P meansthat afemale spendsmuch effort in her survival is consistentwith life-history theory (Hansen& mate choiceto Ž nda male with alarge D,butthis effort Price 1995; Kokko1998). Thus,under many conditions, also reducesher fecundity, F(P): ¶ F/¶ P , 0(sothat the it is conceivable that apreferencefor males with high lev- costof choice C(P) = (F(0) 2 F(P))/F(0); see Ž gure 1). elsof costly advertisement and poor survival prospectscan Basedon the function F(P),wedistinguish betweentwo still deliver anindirect Ž tness beneŽ ttofemales,based on conceptsof costs of choice. The environmentally determ- superior Ž tnessof offspring.It wouldfollow that adistinc- inedcostliness of choice, E,isa parameter that speci Ž es tionbetween genetic bene Ž tsdue to enhancedsurvival (in the shape of F(P),i.e.how quickly fecunditydrops with addition tomating success)and due to mating success stronger female preferences. E is high if,for example, low alone is spurious,and that thereis really aunitary process population densitycauses high energeticexpenditure in ofpreference evolution by indirect bene Ž ts. femalesthat attempt tosample several males.The fec- Here,we present a modelthat capturesthe essence of unditycost actually paid by thefemale, C, equals the sexual selectionfor indirect bene Ž ts:males vary in adis- reductionin fecunditythat thefemale sufferswhen her play trait that femalesmay Ž ndattractive (if it pays for preferenceequals P: C(P) = (F(0) 2 F(P))/F(0). Note them todoso). This trait is heritable andit may correlate that C dependsboth on E and on P,andit cantherefore with male andoffspring viability (expressedas annual besmall evenif theenvironment dictates that choiceis survival), either positively or negatively. Wealso assume costly,as a female canopt notto be particularly choosy that thereis aprocessthat maintains geneticvariation in (small P) when E is large (Ž gure 1). thetrait (andpossibly in viability) andthat female choice Wenextneed to specify male mating success.If females incursdirect costs that reducefemale fecundity.We dem- are notchoosy ( P = 0), males have equal mating success onstratethat femalesshould prefer males that display regardless oftheir display, whereashigh P values imply more intenselyif thedisplay indicatesmale breeding value that thenumber of offspring that amale siresdepends for Ž tnessand the indirect bene Ž tsof mating with males strongly onhis display, D.Wespecify that male mating of high Ž tnessexceed the direct costs of choosing rather success(the number of offspring hesires) is proportional than mating randomly. Moreover,we show that thereis toafunction M(P,D), where ¶ /¶ P(¶ M/¶ D) . 0; this means nothing qualitatively differentin theprocess of mate- that high P values describefemales who discriminate

Proc.R. Soc.Lond. B (2002) Thesexual selection continuum H.Kokko andothers 1333

Table1. Listof variablesand their de Ž nitions. i allele specifying ‘type’ of individual ( i = 1, 2) x proportion of newborn offspring thatare of type1

Di displayof male of type i P femalepreference F femalefecundity E environmentally determined costliness of choice C(P) fecundity cost paid bya femalewhose preference equals P M(P,D) function thatrelates male display D to hismating success, when femalepreference equals P m mutation rate (type1 type 2) 1 ! m mutation rate (type2 type 1) 2 ! pi relative number of malesof type i,compared withthe number of females mi(D) mortality rate of male of type i,witha displaytrait D (‘evolvable display ’ model) qi parameter (‘quality’)determining themortality function for type i males mi mortality rate of male of type i (‘constrained display ’ model) mF mortality rate of females

Ki proportion of offspring thatare type1, when father is type i ki(P) proportion of offspring thatare type1, when mother istype i and her preference equals P W(P9,P ) success of mutant strategy P9 againstresident strategy P

strongly against males with small displays, D.In all ofour 1.0 cost of preference, examples, weuse M(D,P) = exp[(1 1 D)P].Our method C(P) generalizes toother functionsthat yield similar resultsas 0.8

long asthey are able togenerate strong mating skewswith y t i high P values. d n 0.6 u

Ageneral sexual-selectionmodel mustbe able toanswer c e f thefollowing questions: e environmental costliness l

a 0.4 E = 0.1 m e

(i) assuming that apreferenceallele, P,predominates f in thepopulation, thenwill an allele P9 Þ P spread? 0.2 Which alleles P ¤ are uninvadable by alternative alleles? E = 0.15 0 2 4 6 8 10 (ii) assumingthat femalesuse preference P ¤ , what is the

distribution ofmale typesand display traits, Di, in strength of female preference, P thepopulation? Figure 1. Thetwo concepts of femalechoice costs. We assumethat stronger mate-choice preferences P in females We Ž rstoutline the general methodfor seekinguninvad- lead to lower fecundity, F(P ).All of our examplesare based able preferencealleles, whenthe preference in  uences fec- on thefunction F(P) = 1 2 {1 1 exp[2(P 2 E 2 1)]}21. The undity(direct cost) and the probability ofmating with parameter E describes whether theenvironment permits males oftypes 1 or 2(indirect bene Ž t).This method strong preferences without astrong decline in fecundity (low assumesthat themortality ofvarious typesof offspring is E),or whether fecundity declines withpreferences sooner known.We then consider two versions of the model that (high E).Theactual cost of preference C(P ) = (F(0) differin theway that male mortality is determined.In the 2 F(P))/F(0) depends both on theenvironmental costliness ‘evolvable display ’ model,males evolve toexpressthe dis- E and thestrength of preference actually used bythe female. play level, D ,tooptimize atrade-offbetween mating suc- Thus, ahigher environmental cost, E,maybe associated i witha smaller cost paid, C(P),if thefemales evolve to be cessand mortality. In the ‘constraineddisplay ’ model, lesschoosy inthissituation (inthehypotheticalexamplesmarked, type i males always expressa speci Ž c value Di, and conse- P = 2 and E = 0.15 lead to asmaller cost than P = 8 and quentlysuffer a speci Ž c,unalterable level ofmortality. E = 0.1).

(a) Findinguninvadable preference alleles To determinewhich P9 can invade P,weneed to con- types, even P = 0guaranteesmating with thepreferred trast thedirect cost that femalespay with theindirect male type.Second, the invasion prospectsof a genotype beneŽ tsof mate choice.The indirect bene Ž tsdepend both dependon the survival andattractiveness of Ž rst- onthe probability that afemale ’spreferenceactually leads generation offspring,but also, for example, onthe attract- tomating with thepreferred male andon the long-term ivenessof male grand-offspring andgrand-grand- successof offspring,either female ormale, producedwhen offspring.Hence, a female offspring that carries anallele mating with thepreferred versus non-preferred males. that makesgrandsons more attractive canbe more valu- Both componentsof the bene Ž trequire somenon-trivial able than afemale offspring withoutthis allele. The appro- calculations.First, theprobability ofmating with thepre- priate techniqueto Ž ndendpoints of evolution in such ferredmale dependsnot only on P,butalso onhowmany settingsis to calculate reproductivevalues ofoffspring males ofthe preferred type are in thepopulation. If,for basedon a residentstrategy P,andask whethermutant example, female choicehas depletedthe variation in male P9 exist that have higher Ž tnessthan theresident strategy

Proc.R. Soc.Lond. B (2002) 1334H. Kokko andothers Thesexual selection continuum

(e.g.Taylor &Frank 1996; Houston& McNamara 1999; (P ¤ . 0) canevolve if femalesshow an initial preference Pen& Weissing2000). Reproductivevalues are de Ž ned P that exceedsan invasion barrier. For theESS with ascontributions of an individual tothepopulation far into P ¤ . 0, femalespay afecunditycost for their preferences, thefuture and hence take into accountall appropriate gen- but the cost C(P ¤ )is predictedto remain small (in the eticeffects. example of Ž gure 2, femalesnever evolve preferencesmore Whenderiving reproductivevalues, we assumeconstant costlythan a0.8% reductionin fecundity;the results of female mortality, mF,regardless ofher genotype,and mor- this magnitude are typical in all versionsof our model). talities m1 and m2 in males oftypes 1 and2, respectively. Figure 2a plotsthe invasion barrier measuredas the Analternative preferenceallele canspread if theallele pro- required initial preference P, whereas Ž gure 2b measures ducesa larger sumof reproductive values ofall offspring it asthe cost C(P, E)that femalespay for this initial pref- produced.The details ofthe calculation ofoffspring erence.If thecosts of being somewhatchoosy are small reproductivevalues andthe invasion criterion for P9 are (indicatedby asmall or moderate E value in Ž gure 2), the presentedin Appendix A. invasion barrier isnarrow andcan be crossed relatively easily. Asan example, evolutiontowards male displays (b) Therelationship between traitsand mortalities andcostly female preferencescan be initiated at any in the‘ evolvable-display’model E , 0.15 if it is facilitated by a ‘moderatepreference ’ allele To Ž ndtherelationship betweenmale type andits mor- that reducesfecundity by nomore than 0.01% compared tality, weneed to specify how the display D in uences with random mating ( Ž gure 2b).Suchlow, cheap levels male survival prospects.In the ‘evolvable display ’ model, ofinitial preferencecan become established exaptively, weassumea function mi(D)for eachtype, where mortality e.g.via passive processes(Wiley &Poston1996), sensory mi increaseswith thedisplay D (dmi /dD . 0). This biases(Payne &Pagel 2000) orspeciesrecognition (Veen relationship dependson male type.We then allow for the et al. 2001). Atthis initial preference P,evolutionin males evolutionof reaction normsbetween the display Di and producesdisplays Di that favour further increasesin the male type i.For eachtype i,thedisplay Di evolvesto max- preference P (and,as a consequence,in thedisplays D). imize that type ’s Ž tness.Assuming that males will mate The increaseeventually stopsat theevolutionarily stable with any female that acceptsthem, increasing display does equilibria for P ¤ and Di¤ ,dueto the increasing costsof notchange thetype distribution ofamale ’soffspring but further elaboration ofpreferences or displays. in uencestheir number.After Houston & McNamara While femalesare notpredicted to evolve very costly (1999; seealso J.M.McNamara, A.I.Houston,M. preferences,these preferences are able toselect for display Marquesdos Santos, H. Kokkoand R. Brooks,unpub- traits causinghigh mortality in males.At the evolutionarily lisheddata), an uninvadable allele for thedisplay, stable equilibria for P ¤ and Di¤ ,thepreferred males have

Di¤ ,thereforemaximizes M(P, Di )/mi(Di), where M(P, Di) larger displays Di (Ž gure 2c),butthey canhave either is proportional tothe number of offspring that hesires higher or lowermortality than thenon-preferred males (seeAppendix Afor details).Males will suffermortality (Ž gure 2d).The casewhere preferred males sufferfrom mi¤ = mi(Di¤ )at this equilibrium. Anevolutionarily stable higher mortality occurswhen the environment permits strategy (ESS)is found when the values Di¤ that are the relatively cost-freefemale choice(low E,enabling acom- bestresponses to female preferences, P ¤ , also make P ¤ bination ofhigh P with small C). uninvadable by any alternative P9. Figure 2also showsthe solutions for the ‘constrained display’ model.To aid comparison,we have calculated (c) Therelationship between traitsand mortalities theevolutionarily stable preferences P ¤ andthe invasion in the‘ constrained display’model barrier for thespread of the preference for values

Wealso considerevolution in asimpli Ž edsettingwhere Di = Di¤ and mi = mi(Di¤ )that werereached in the ‘evolvable thegenetic architecture constrainsmales oftype i to display’ model(for eachvalue of E).The preferences P ¤ expressa display trait Di,in addition toexperiencing a that wereto be found stable in the ‘evolvable display ’ mortality rate mi.In this model,we do not assume that modelremain thebest response to these display andmor- reaction normscan evolve toexploit therelationships tality values ofmales in the ‘constraineddisplay ’ model, between Di, mi andmale type.The trait, Di ,andmortality, andhence the ESSs with P ¤ . 0are thesame in thesetwo mi,cancovary positively, negatively or notat all. Weare models.The invasion barrier has adifferentshape, how- thusexploring female preferencesfor traits that either ever, as the ‘constraineddisplay ’ modelassumes that dis- increase,decrease or have noeffecton viability, butthese plays are presenteven if thepreference P is low (Ž gure correlations are Ž xedand do not evolve. 2a,b). SpeciŽ cally, theinvasion barrier candisappear if male displays covary positively with survival ( Ž gure 2a,b). The solutionsfor the ‘constraineddisplay ’ model in 3. EVOLUTIONOF PREFERENCESAND DISPLAYS Ž gure 2are special cases,as they assumea speci Ž c In the ‘evolvable display ’ model,random mating relationship (onederived from the ‘evolvable display ’ P ¤ = 0is always evolutionarily stable.If apopulation of model)between the environmental costof choice E and femalesdoes not exhibit any preferences( P = 0), males do thecovariation ofmale display with his mortality. More notevolve toshow any costlydisplays ( Di = 0). Therefore, generally, the ‘constraineddisplay ’ modelcan be solved in theabsence of any displays distinguishing between for any combination of E, Di and mi.If male display is 21 21 males,female preferencescannot invade ( Ž gure 2a). If correlated with goodsurvival ( m1 2 m2 is positive) and costsof choiceare high ( E . 0.165 in Ž gure 2), P ¤ = 0 is thecosts of choice are nottoo large ( E is small), female theonly ESSandmate choicecannot evolve at all. How- preferencesfor thedisplay caninvade withoutthe need to ever,when costs of choice are smaller, female preferences crossan invasion barrier ( Ž gure 3). If thecorrelation

Proc.R. Soc.Lond. B (2002) Thesexual selection continuum H.Kokko andothers 1335

(a) (b) 4 8 3 – 0 1 P

3 6 , ×

e ) c E n , e P r ( e f C

e , r t

p 2 s 4

o e l c

a y t m i e d f n u c

1 e 2 f

0 0 0.10 0.12 0.14 0.16 0.10 0.12 0.14 0.16

(c) (d) 1.0 1.0

i 0.8 0.8 D

, s t n i a a p r t s

0.6 e 0.6 y f i a l l

p e s l i a d

m e

l 0.4 0.4 a m

0.2 0.2

0.0 0.0 0.10 0.12 0.14 0.16 0.10 0.12 0.14 0.16

environmental costliness of choice, E

Figure 2. Evolution of femalechoice measured as( a)preference P, and (b)fecundity cost, C(P),paid bya femalewith a preference P.Theparameter E (x-axis)measures thespeed bywhich increasing matepreferences reduce fecundity. In ( a), the arrow pairs indicate theinvasion of alleles for larger or smaller levels of femaleeffort in the ‘evolvable display ’ model (left arrows) and in the ‘constrained display ’ model (right arrows); thicksolid line, ESS;thin solid line, invasion barrier, evolvable display; dashed line, invasion barrier, constrained display. In addition totheESS indicated, P ¤ = 0isan ESSexcept between

E = 0.130 and E = 0.164 in the ‘constrained display ’ model. It is theonly ESSif E . 0.164. (c)Displaytraits, Di¤ , and

(d)expected lifespans, 1/ mi¤ , of type 1 (Ž lled circles) and 2(open circles) malesat the ESS with P ¤ . 0 (or P ¤ = 0 where this isthe only ESS).The results for the ‘evolvable display ’ model are derived assuming themortality functions mi(D) = [qi(1 2 3 21 P (D/qi) )] (we assumethat D < q), q1 = 1 and q2 = 0.5. Theother functions and parameters used are M(P,D) = exp[(1 1 D) ], m1 = 0.05, m2 = 0.01 and mF = 1. Preferred malesalways display more strongly thannon-preferred males( c),but their lifespans can belonger or shorter ( d ). betweendisplay andsurvival is zeroor negative, female dueto high mating success,low mortality or both.The preferencescan still invade butonly after crossingan modelsshow solutions where preferred type 1males have initial invasion barrier, which becomeswider with more higher mortality than type 2males ( Ž gure 2d, small E; negative correlations andwith increasing choicecosts Ž gure 3, left-handside) and vice versa ( Ž gure 2d, large (Ž gure 3). E; Ž gure 3, right-hand side).It wouldbe consistent with publishedwork on this topic toclassify theformer asan example ofa preferencefor aFisherian trait andthe latter 4. SEXYSONS AND VIABILITY INDICATORS:TWO asa preferencefor agood-genesindicator. Our modelling ENDSOFACONTINUUM producesboth kindsof outcomes from asingle process, For female choiceto be uninvadable by lesscostly ran- however,where indirect consequencesof choice dommating, it mustlead toan increasein therepro- encompassboth offspring viability andthe mating success ductivevalues ofoffspring. Males and females of the ofsonsin futuregenerations —including sonsproduced by preferredtype indeedhave higher reproductivevalues, but daughters,granddaughters etc.that have inherited alleles ourmodels do not specify a priori whetherthe increase is whoseexpression makes them more attractive.

Proc.R. Soc.Lond. B (2002) 1336H. Kokko andothers Thesexual selection continuum

0.20 1.0 P* = 0 _ (ESS at any E) 10 3 0.18 0.8 E

, e s l s a e E = 0.138 n m

i 0.6

l 0.16 1 t

P* = 1.88 s e o p

c _

4 y l 10 t

a , t n n a

e E = 0.133 p 0.4

0.14 s m

_ e n P* = 2.27

5 f i o 10 l r i v

n E = 0.12 e 0.12 0.2 P* = 2.86 0 E = 0.1 P* = 3.52

0.10 0 0.2 0.4 0.6 0.8 1.0 _0.10 _0.05 0 0.05 0.10 difference in lifespan lifespan, type 2 male type 2 male survives better type 1 male survives better Figure 4. Thecoevolution of femalechoice and male mortality in the ‘evolvable display ’ model. Thefunctions and Figure 3. Thewidth of theinvasion barrier in the parameters are asin Ž gure 2,except there isa smaller ‘constrained display ’ model, when malesof type1 develop a difference in mortality functions among males q1 = 1, displaytrait D1 = 1, whereastype 2 maleslack this trait, q2 = 0.08 and symmetrical mutation: m1 = m2 = 0.02 (zero D2 = 0.Thewidth is calculated for various values of backmutation, m2 = 0, produces avirtually indistinguishable environmental cost of choice E and differences in lifespan outcome). Therandom-mating equilibrium ( P ¤ = 0) is caused bythe trait, 1/ m1 2 1/m2 (we set 1/m2 = 1). The alwaysstable. Malemortalities are expressed asthe expected barrier width is indicated asthe fecundity cost C(P ) that the male lifespan, 1/ mi¤ .Thecoevolution of preferences and population of femaleshas to payfor its preferences P, before displaysshifts male mortalities along thecurve, provided that an increase in thestrength of preference P isfavoured by theinvasion barrier is Ž rst overcome. Some possible selection. To theright of thezero contour, C(P) = 0, i.e. endpoints (stableequilibria) are indicated withthe values of there is no invasion barrier. Theother parameters used are theenvironmental cost parameter, E,and thecorresponding m = 0.02 and m = 0.01, with M(P,D) as in Ž gure 2. 1 2 femalepreference, P ¤ ; see § 4for further explanation.

The processcan produce either outcome,regardless of 2males with display D2 = 0.74. The reproductive value of whetherwe assume that males evolve toexploit atrade- attractive offspring is high; notbecause they survive well, offbetween mating successand costs of developing the butbecause females prefer tomate with them —an erst- attractive trait ( ‘evolvable display ’),or that they simply while ‘Fisherian’ outcome. differwith respectto a genetically codedtrait that also has Considernow that thescenario with E = 0.1 has been survival consequences( ‘constraineddisplay ’). The bal- established.The high-mating skewmight nowshift the ancebetween the importance of ‘sexy-son’ or ‘good-genes’ costof choice, E,upwards.This canhappen becauseof beneŽ tscan be altered simply by varying anenvironmen- theaccumulation ofsexually antagonistic genesthat bene- tally determinedcost factor E that speciŽ eshow easily Ž tmales (Rice1992, 1996), more widespreadsexually femalesproduce a strong mating skewand, hence, how transmitted disease(Boots & Knell2002; Kokko et al. competitive amating environmentthe sons will experi- 2002) or increasedinbreeding (Farr 1983; Amos et al. ence. 2001a,b).Also,external changesin environmental factors Wefurther illustrate thearbitrariness ofdistinguishing may alter searchcosts by increasing searchtime, the thetwo hypotheses and the generality ofour modelling, energeticcosts of searching or exposureto predators by another example ofthe ‘evolvable display ’ model (Pomiankowski 1987). Anyprocess increasing E now (Ž gure 4). Asabove, random mating is evolutionarily leadsto reduced female choiceeffort P,smaller male dis- stable.Under random mating, males oftype 2suffer play traits D andincreased male lifespan 1/ mi. This higher mortality than males oftype 1. Considera case eventually restoresthe function of the display asan indi- wherethe environmental costof choice is small, cator ofviability (at E . 0.133, Ž gure 4). However,the E = 0.1. If female preferencecrosses a very small switchin relative male mortalities at E = 0.133 doesnot C(P) , 0.0005% for P = 0.1) invasion barrier, coevolution involve any qualitative change in theprocess responsible. ofthe male display andfemale choiceincreases male mor- Further environmental change canagain causecoevolu- talities until their expectedlifespan is lessthan 20% ofthe tion of P, D and m upand down the curve ( Ž gure 4). original value ( Ž gure 4). Here,the preferred males ’ display decreasestheir lifespan tofall belowthat ofthe less pre- 5. DISCUSSION ferredmales ( Ž gure 4, C = 0.1) andthe mating skewis strong: P ¤ = 3.52 predictsthat type 1males with display Wehave built ageneral modelof the evolution of female

D1 = 0.95 enjoy34.7 times ashigh-mating successas type preferencesfor traits that indicateindirect bene Ž ts. The

Proc.R. Soc.Lond. B (2002) Thesexual selection continuum H.Kokko andothers 1337 sameprocess is responsiblefor theevolution of female (c) Counterargument 3.Preferences for Fisherian preferencesfor males whoseoffspring survive either better traitsare arbitrary; preferences for viability or lesswell than thoseof unattractive males.Moreover, indicatorsare not thecorrelation betweenmale attractivenessand survival In polarizing Fisher andgood-genes indicators, some canswitch from positive tonegative (or vice versa) with authorscharacterize theFisherian runaway processas the changesin theenvironmentally determinedcostliness of ‘arbitrary traits ’ model(e.g. M øller 1991; Møller & female choice,and thus the intensity of sexual selection Ho¨glund1991). The ‘arbitrariness ’ ofa preferenceor a that femalesimpose on males. At equilibrium, female pref- trait seemsto lack aconsistentde Ž nition in theliterature, erencesare maintained despitedirect costs that wehave however.Neither Fisher (1930) norLande (1981) used modelledas reduced fecundity. the word ‘arbitrary’.Fisher (1930) didnot consider male Basedon the results of our model, viewing Fisherian sexual displays tobe arbitrary in any sense,and the run- runaway andgood-genes indicators ascompeting alterna- away process(that is halted by decidedlynon-arbitrary tives isa hindranceto understanding the evolution of mate costs)is merely thesecond component of his theory ofthe choicefor indirect bene Ž ts.Instead, a single processfav- evolutionof mate choice.There is nothing in theresults oursfemale choicefor males whosire offspring ofhigh ofour model that isinconsistent with thetheory setout reproductive values.If female choiceis cheap,and thus in Fisher (1930). leadsto strong mating skewsamong males,male attract- To ourknowledge, the Ž rstusage of the expression ivenessmay becomenegatively correlated with survival or ‘arbitrary’ traits wasby Heisler(1985). Sheexplicitly used other Ž tnesscomponents. If thecosts of strong preferences it asshorthand for ‘characters that may notthemselves be are prohibitive, preferredmales are also expectedto pro- thecause of variation in Ž tness,but merely re  ect that vide geneticbene Ž tsin other componentsof Ž tness.There variation owing togeneticcorrelations with other,directly is noqualitative differencebetween these outcomes; selectedcharacters ’ (p.188). In this sense,Heisler (1985) rather, they are endpointsof a continuum( Ž gure 4). uses ‘arbitrary’ tomean male traits that are notdirectly Asadeviceto illustrate that it is notuseful to distinguish selectedthemselves, but indicate breeding value for other Fisherian andgood-genes models of sexual selection,we Ž tnesscomponents —i.e.a form ofindicator trait. respond to Ž ve important counterargumentsthat we Kirkpatrick &Ryan (1991) describedthe Fisher process anticipate might beraised. as ‘arunaway [that] will generally establish preferences that are arbitrary with respectto male survival ’ (p. 34). (a) Counterargument 1.Only Fisherian runaway Accordingto our results, females can evolve tofavour evolutioncan account for female preferences traits that covary with survival ofoffspring positively, for traitsthat lead to lowered survival rates in negatively or notat all. The processof evolving prefer- theiroffspring encesdoes not establish itself equally easily in all ofthese This view issimply wrong,as has been repeatedly cases.If survival is negatively correlated with thedisplay, pointedout (Grafen 1990 a;Kokko1998, 2001; Eshel et the beneŽ tsthrough attractivenessmust override this cost, al. 2000; Jennions et al. 2001). If males vary in quality and in addition toany directcost of choosiness. If onede Ž nes optimize their display in acondition-dependentmanner — arbitrary as ‘having any correlation with male survival ’, standardassumptions in any ‘good-genes’ model—high- oneloses the distinction that survival will have an in  u- quality males may bene Ž tfrom signalling with suchinten- enceon how easily preferenceswill evolve.Signals are all sity that their survival falls belowthat oflow-quality males. arbitrary underthis meaning. If oneinstead considers Thus,negative correlations betweenattractiveness and ‘arbitrary traits ’ tobethose that donot improve offspring survival are consistentwith ‘good-genes’ signalling. survival, oneis left with theresult that preferencesfor ‘arbitrary traits ’ canfavour males whoare ofgenetically (b) Counterargument 2.Only ‘good-genes’ models high quality andsignal this sointenselythat their survival withcondition-dependence can account for remains meagre (Grafen1990 a;Kokko2001). Wethere- female preferences for traitsthat indicate foreconclude that distinguishing betweenpreferences for highersurvival rates in theiroffspring ‘arbitrary’ and ‘non-arbitrary ’ male traits is,in itself,arbi- Our modelling showsthat displays that indicatethe trary. ‘type’ ofa male via areaction norm canevolve tocovary with high survival rates oftheir offspring.Here, ‘type’ can (d) Counterargument 4.Even if both ‘good-genes’ beinterpreted as a male ’sgeneticquality, becauseit deter- and ‘Fisherian’ modelsmake similar minesthe mortality costof adisplay ofcertain magnitude. predictions, traitsevolve in theformer asa In this sense,the ‘evolvable display ’ modelcorresponds to result ofcondition-dependent signalling of standardgood-genes modelling. However,the ‘con- overall genetic quality; thisis absent in straineddisplay ’ model also easily yields preferencesfor Fisherian theory displays that are correlated with higher survival ratesof Modelsdesigned to explore Fisherian trait evolution offspring.This modelsimply assumesthat agenehas have typically assumeda Ž xedrelationship betweenmale pleiotropic effects:it is fully conceivable that avisible (or traits andviability, in this respectresembling our ‘con- otherwisedetectable) trait canalso confersurvival advan- straineddisplay ’ model. ‘Good-genes’ models have tages.It is easyto see that femalescan bene Ž t by using assumedthis relationship tovary dependingon male qual- sucha trait asa mate-choicecue. The preferencecan be ity. They thereforecorrespond to our ‘evolvable display ’ maintained evenif it is costly,as long asthere is amech- model,if male ‘type’—i.e.the trait that determinesthe anism (in ourmodels, mutation) that maintains additive display–mortality trade-off —isinterpreted as his genetic geneticvariation in themale trait. quality.

Proc.R. Soc.Lond. B (2002) 1338H. Kokko andothers Thesexual selection continuum

Aswehave shown,both the ‘constraineddisplay ’ model thussome Fisherian runaway,is an inevitable conse- and the ‘evolvable display ’ modelhave female preferences quenceof choice. Eshel et al. (2000) showedthat this that evolve asa responseto the genetic correlation sexy-sonbene Ž tis inherentin all good-genesmodels. betweenmale display andbreeding value for total Ž tness. If self-reinforcingrunaways are takento mean open- Both modelsalso require amechanism that maintains endedevolution of male traits, it istrue that thosehave variation among males.Consequently, the preferred males beenfound in modelsof the runaway processonly. How- will, at equilibrium, be Ž tter,either through their attract- ever,as Fisher (1930) pointedout, the runaway process ivenessalone or through acombination ofimproved is likely very rapidly toreach an equilibrium whereit is viability andattractiveness. These results do not depend halted by ever-increasing costs.Existing runaway models onwhether the display iscondition dependent. Therefore, mentionthis buthave simply notincorporated thesecosts. thepreferred males canalso besaid tobeof higher genetic In ourmodel, self-reinforcing evolution of preferences quality in the ‘constraineddisplay ’ model. doeseventually halt andthere are nodifferences between Despitethis equivalencein termsof the ESS level of thedifferent versions of our model in this respect.We did choicein thetwo models that wepresent, we agree that not,however, Ž ndcasesof cyclic evolutionof preferences thegenetic architecture ofsignalling will have anin  uence in ourmodel (cf. Iwasa &Pomiankowski 1995; Houle& onthe ease with whichpreferences become established. Kondrashov2002), butwe have noreason to believe that The ‘evolvable display ’ model and ‘constraineddisplay ’ theconditions for this tooccurshould differ between the modelyield differentpredictions in onerespect: the width evolvable- versusconstrained-display versions of the ofthe initial invasion barrier. In the ‘evolvable display ’ model. model,the evolution of costly traits always requiressome In conclusion,ourmodel replaces theFisherian – initial (albeit minisculeand conceivably exaptive) mate good-genesdichotomy with ageneral processof indirect choicein females,regardless ofthe costliness of choice selectionin which Ž tnesscomponents trade-off along a andthe sign ofthe correlation betweenmale advertise- continuum.The mostimportant prediction totestis that mentand survival at theevolutionary endpoint.By con- femalesprefer males with high breeding values for total trast, a Ž xedlevel ofsexual advertising (andconcomitant Ž tness(i.e. high reproductivevalue). It isneither valid nor relationship with survival), suchas in our ‘constraineddis- usefulto measure a single Ž tnesscomponent such as play’ model,generates some cases where the preference attractivenessor survival asa testof the theory that mate canspread without an invasion barrier. This canhappen choicecan evolve by indirect selection.Nevertheless, it if thedisplay trait is positively correlated with male sur- remains worthwhile for empiricists toask whetherfemales vival. This Ž ndingechoes Fisher ’soriginal suggestionthat seekingindirect bene Ž tsusually gain merely by producing female choicespreads initially becausepreferred males are ‘sexy sons’,or if offspring survival, fecundityor other Ž t- favouredby natural selection. nesscomponents are also improved. Oneend of the The underlying geneticswill determinehow easily addi- sexual-selectioncontinuum arises when cheap choice tive variation is maintained. Mostprobably, thegeneral allows astrong mating skew,so that variation in offspring conditionof a male, i.e.his ‘type’ in thesense of the ‘evol- reproductivevalue is almost completely driven by ‘sexi- vable display ’ model,is subject to a larger mutation rate ness’ ofsons.Empirical examples consistentwith this are than asingle or afewgenes that determinedisplay trait scarce(Etges 1996; Wedell &Tregenza 1999; Brooks expressionin theconstrained display model(see Rowe & 2000), possibly becausevery cheap female choiceand Houle1996; Kotiaho et al. 2001). The conditionsfor the extreme mating skewsare rare. Atestableprediction from maintenanceof female choiceare thusperhaps more likely ourmodel is that thetrade-off between mating success tobe met in thescenario where male traits evolve asa andother componentsof Ž tnesswill covary with thecosts reaction norm totheir overall condition.While weagree ofmate choiceand the mating skewthat results. that this is abiologically meaningful distinction,the logic It also remains valid toask whether indirect bene Ž ts dif- ofthe preference evolution process is not fundamentally fer betweenoffspring sexes(Rice 1992, 1996). To retain differentin thecondition-dependent versus simple plei- conceptualsimplicity in ourmodels, we have ignored any otropic scenarios.Both require aprocessmaintaining vari- sire effectson daughters apart from theattractiveness ation in male Ž tnessand a detectabletrait that separates genesthat they carry andpass on to their male descend- Ž tmales from un Ž tones.Both also rely heavily onthe ants.It is clear that whenmale genesalso in  uence the beneŽ tsof producing ‘sexy sons’ andrequire that sexiness survival or fecundityof daughters, these factors will have bea componentof a male ’s Ž tness. tobe incorporated intothe reproductive value equation. Again, wewould expect no qualitative differencein the (e) Counterargument 5.Self-reinforcing runaways process:female choiceis selectedfor whenit leadsto a or positivepreference –display genetic higher total ofoffspring reproductivevalues. For example, correlations are foundonly in Fisherian if sexually antagonistic genes(Rice 1992, 1996) cause modelsof sexual selection daughterssired by attractive males tosurvive poorly, the Female choiceis self-reinforcing in both variants ofour invasion barrier for sucha preferenceis simply expected modelling: thebene Ž tofproducing attractive sons tobe wider than if daughterssurvive well. increaseswith female preference P,andstronger prefer- Several authorshave preacheda pluralistic understand- ences P are favouredby selectionas the importance of ing ofindirect selectionon female mate choice(e.g. sons’ attractivenessas a determinantof their mating suc- Andersson1994; Houde1997). Others have recognized cessincreases. Lande (1981) demonstratedthat where that thegood-genes process can result in positive ornega- heritable variation in mate choiceand male display exist, tive correlations (both phenotypic andgenetic) between thebuild-up of a geneticcorrelation betweenthem, and attractive advertisementand survival (Ho¨glund& Sheldon

Proc.R. Soc.Lond. B (2002) Thesexual selection continuum H.Kokko andothers 1339

W(P ,P) = F(P )m21{[xk (P ) (1 x)k (P )](n n ) 1998; Eshel et al. 2000; Kokko2001) andthat ‘sexy-son’ 9 9 F 1 9 1 2 2 9 M1 1 F1 beneŽ tsare also inherentin thegood-genes process (Eshel 1 [x(1 2 k1(P9)) 1 (1 2 x)(1 2 k2(P9))] et al. 2000; Jennions et al. 2001; Kokko2001). Since ´ (n n )}. M2 1 F2 Fisher (1930), however,we have mostly missedthe point that self-reinforcementof preferencesand the evolution of indicator traits are merely differentcomponents of one Here,the reproductive values nM1, nF1, nM2 and nF2 andthe same process of indirect selection. are calculatedusing the resident strategy P. Where (¶ W(P9,P)/¶ P9)|P9 = P . 0, selectionwill lead toan increase

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