Proc. NatL Acad. Sci. USA Vol. 78, No. 6, pp. 3721-3725, June 1981 Evolution

Models of speciation by sexual selection on polygenic traits (mating preferences/sexual dimorphism/genetic correlation/runaway process) RUSSELL LANDE Department of Biophysics and Theoretical Biology, University of Chicago, Chicago, Illinois 60637 Communicated by James F. Crow, February 10, 1981 ABSTRACT Thejoint evolution offemale mating preferences a male trait and female preferences for it both increase geo- and secondary sexual characters of males is modeled for polyga- metrically or exponentially with time until finally checked by mous species in which males provide only genetic material to the severe counterselection. This genetic mechanism could rapidly next generation and females have many potential mates to choose create a new species by sexual isolation and phenotypic diver- among. Despite stabilizing natural selection on males, various gence of a population from its closest relatives. It also could types of mating preferences may create a runaway process in reinforce or accelerate other modes of speciation. which the outcome of phenotypic evolution depends critically on O'Donald (10) confirmed the basic of the genetic variation parameters and initial conditions of a pop- numerically operation ulation. Even in the absence ofgenetic instability, rapid evolution Fisher's runaway process, using two-locus models, in which one can result from an interaction ofnatural and sexual selection with locus with two alleles codes for variation in males and one di- random genetic drift along lines of equilibria. The models eluci- or triallelic locus influences female mating preferences. He date genetic mechanisms that can initiate or contribute to rapid found that the rate and extent of evolution is enhanced when speciation by sexual isolation and divergence of secondary sexual the most preferred genotype at the male character locus is re- characters. cessive and that linkage can influence the dynamics. However, such models greatly restrict the evolution ofa trait, which must The distinction between natural and sexual selection drawn by cease with the fixation of an allele at the corresponding locus. Darwin (1) is that natural selection arises from variance in in- For quantitative characters, it is generally more realistic to em- dividual survival (and fecundity), whereas sexual selection re- ploy a polygenic model (11, 12) and to allowfor the maintenance sults from variance in mating success. Dimorphism ofsecondary of genetic variability by mutation and recombination (13, 14). sexual characters in higher animals is caused by two major fac- Female mating preferences have been demonstrated in a tors: combat or competition between individuals of one sex variety of arthropods and vertebrates (7-9, 12), indicating that (usually males) for mates and mating preferences exerted by the there is (or was) some genetic variation for them. A complete opposite sex (females) (1). The fitness of a trait with respect to set of mating preference functions, which would specify for mating success can override its value for survival, creating a kind every female phenotype the sexual preference for each male of maladaptive evolution that may contribute to the extinction phenotype, has not been measured for any population. Al- of a population (2-6). In contrast to intermale competition, though for most species it may be difficult or impossible to ob- which entails an obvious advantage of mating success for the tain, such information is necessary to determine the course of winners, Darwin was unable to explain why in many species evolution of male secondary sexual characters and female mat- with polygamous systems of mating (where males are promis- ing preferences. To clarify Fisher's mechanism for rapid spe- cuous and invest little or nothing but gametes in their offspring) ciation by sexual selection, I analyze here the joint evolution females should prefer mates with extreme characters that are ofmale secondary sexual characters and different types offemale apparently useless or deleterious for survival, such as the ex- mating preferences which have been discussed, often incor- travagant plumage of some male birds and the exaggerated rectly, in the literature on sexual selection. horns and tusks of certain male mammals (1, 7-9). Fisher (2, 3) suggested an ingenious solution to Darwin's QUANTITATIVE GENETIC MODELS problem by outlining a genetic mechanism for the joint evo- lution offemale mating preferences and secondary sexual char- Evolution of the Mean Phenotypes. Consider for simplicity acters of males. An essential feature of this mechanism is the two sex-limited quantitative traits: a male character, z, and a genetic correlation between the sexes; that is, the extent to female mating preference, y, each influenced by multiple au- which variations in male and female traits are influenced by the tosomal genes and subject to environmental effects. On an ap- same genes or segregating factors. Even if the genes affecting propriate scale ofmeasurement, both traits are assumed to have these characters are not mutually pleiotropic, a positive cor- normal distributions, p(z) and q(y), with means i and g and phe- relation between them will nevertheless arise in the population notypic variances o2 and A. For continuously varying charac- because of assortative mating created by genetic variance in ters, a logarithmic scale of measurement often renders the dis- mating preferences (where the more discriminating females tributions approximately normal with variances roughly mate with the more extreme males). The evolution of mating independent ofthe mean values (11). The additive genetic var- preferences may be self-reinforcing because, once started, fe- iances of the male and female traits are denoted as G and H, males are selecting not only for more extreme males but also and the additive genetic covariance between them, B, is due indirectly, through the genetic correlation, for a higher inten- to pleiotropy and nonrandom associations ofalleles at different sity of mating preferences. Fisher (3) stated that the result of loci. In a population ofautosomal genotypes, B is the covariance this positive feedback could be a "runaway process," in which of additive genetic effects when in males with those when in females. These genetic variation parameters ofa population can The publication costs ofthis article were defrayed in part by page charge be estimated from phenotypic correlations between relatives payment. This article must therefore be hereby marked "advertise- or from artificial selection experiments (11, 12). ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. It is assumed that in each generation every female is insem- 3721 Downloaded by guest on September 30, 2021 3722 Evolution: Lande Proc. Natl. Acad. Sci. USA 78 (1981) inated and that males do not help raise offspring or protect or to the fingers (15). It seems likely that in many higher animals, provision their mate(s); hence, the expected number ofprogeny as in man, both sensory perceptions and emotional reactions of from a given female is independent ofher mate choice. In any an individual often scale as power functions ofquantities related particular generation, female mating preferences do not change to secondary sexual characters and mating displays. Such a the mean fitness in a population but act only to redistribute fit- mechanism may underlie the responses of many animals to su- ness among the different male phenotypes. Thus, there is no pernormal stimuli (16). Thus, suppose that a quantitative char- selection directly on female mating preferences, which evolve acter of males, 4, produces a perception and associated sexual only as a correlated response to selection on males. The direct preference, 4i, in a given female proportional to k, where y is response to one generation of selection on males and the cor- a constant pertaining to the particular female. Ifthe male char- related response in female mating preferences are acter is analyzed on a logarithmic scale, z = lno, as is often appropriate for statistical purposes, the psychophysical pref- E= '/2GS/cr Ay= 112BSIo', [1] erences of a given female can be written as in which S is the selection differential on males, the difference oc eYZ. [8a] between selected and unselected adults, and the factors of 1/2 qzizly) account for the sex-limited expression of both traits (6, 11). Individual females are assumed to differ in the degree of dis- Natural selection on males is assumed to act through differ- crimination in mate choice, y. ential viability, followed by sexual selection through differential Animal perceptions ofsome sensory modalities, such as color mating success. Writing the viability of males with phenotype or the pitch of a sound, and matching constraints between the z as w*(z), the distribution ofmale phenotypes after natural se- sexes may result in unimodal preferences. Such preferences lection is could be either an absolute intrinsic property ofeach female or could be relative and scaled to the distribution of male phe- p*(z) = w*(z)p(z)/fw*(z)p(z)dz. [2] notypes in a population. Females who prefer most a particular value of a male char- Weak natural selection toward an optimal male phenotype, 6, acter, regardless of its distribution in the population, are de- can be approximated by a Gaussian function, scribed by absolute preferences. An important class ofabsolute w*(z) = e-(z - 0)2/2w2 [3a] "preference" occurs where homologous or complementary characters of the sexes are under a matching constraint that in which w indicates the range of male phenotypes around the determines the probability of successful mating. In many spe- optimum with high viability. After natural selection alone, the cies, male and female morphology and sexual behaviors are distribution of male phenotypes is normal with mean and mutually constrained in some way. A simple form of absolute variance preference is that for a given female the most preferred male ± z* = (zFW2 + 6o2)/(W2 + a,) [3b] phenotype is y with a tolerance of v or where the characters ofmates are somehow constrained to be within about ± vofeach (72* = W2a,2/(W2 + ag). [3c] other, Denoting the relative preference offemales with phenotype q(zIY) oc e-(z - y)2/2v2 [8b] y for mating with males ofphenotype z as i(zly), the frequency Alternatively, if an individual female surveys the population ofmatings ofthe different males with females ofthe given phe- ofsurviving males and chooses a mate from among them relative notype y is assumed to be proportional to this preference. The to the mean, with the highest preference for males ofphenotype mating success of males with phenotype z relative to that of all z* + y but with a high preference for males having any phe- surviving males in the population encountering females with notype within ± v of this value, she shows relative preferences phenotype y is then that can be described by the Gaussian function 0*(zlY) = O(zly)/fp*(z)O(zly)dz. [4] '(zty) X e-[Z - (j* + y)]2/2v2 [SC] The net relative fitness ofmales with phenotype z is the product Because natural and sexual selection act independently on of their viability and mating success averaged over the entire males, the total selection differential can be calculated as the female population, sum of two corresponding parts. The change in the mean male phenotype due to natural selection within a generation is from W(z) = w*(z)fq(y)qi*(zjy)dy. [5] Eq. 3b:

Utilizing Eqs. 2, 4, and 5, the mean fitness in the population zS - z = (6- z)O2/(W2 + cr2). is In the psychophysical model females of a given type y then W = fp(z)W(z)dz = fp(z)w*(z)dz, [6] choose mates with a mean phenotype that deviates from z* by which is the same as under natural selection alone, as required an amount o'2*y whereas with unimodal preferences the anal- by the assumptions. The selection differential on males can be ogous mean deviation from i* is computed as (Y - Ei*)of2*/(V2 + O'2*), S = WV'fzp(z)W(z)dz - z [7] in which E = 1 for absolute preferences and E = 0 for relative female = fq(y)fzp (z)0 (zjy)dzdy - z. preferences. Averaging these changes over the entire population and adding the previous contribution from natural One model of mating preferences is suggested by Stevens' selection gives the general form ofthe total selection differential (15) psychophysical law. Over a wide range of stimulus inten- on males. The assumption that selection on the variance ofmales sities in nearly every sensory modality in man, the perceived is weak (w2, v-2>> o2) yields the approximation intensity is proportional to a power ofthe actual intensity. Mea- S !i/a -(1 + E/a)i + 6 sured values ofhuman psychophysical exponents range from 0.5 = ,9i for brightness of a point source oflight to 3.5 for electric shock Cr2 W2~~~~C' Downloaded by guest on September 30, 2021 Evolution: Lande Proc. Natl. Acad. Sci. USA 78 (1981) 3723 in which, for the psychophysical model, a = 1102 and £ = 0, encing the two characters are not expected to be mutually pleio- whereas with relative or absolute unimodal preferences, a tropic. This case is ofspecial interest in assessing the magnitude 1v2/oW2 of genetic covariance between the characters that can be pro- The equilibria in each model ofmate choice, from Eqs. 1 and duced purely by the assortative mating that necessarily results 9 are all points on the line from variance in mating preferences. Enumerating the loci as 1, ... m for the male trait and m + 1, ... n for the female pref- y = (a + e)i- aO. [10] erence, any linkage map with positive recombination rates be- For any mean value of the male character there is an average tween loci is allowed, 0 < ri s 1/2 for i # j, and rij = 0. With intensity of female sexual preference that will cancel the force the assumption that all of the genetic variation is additive, the ofnatural selection tending to restore the mean male phenotype covariance between the effects ofalleles at loci i andj from the to its optimum, 8. The slope of the line of equilibria, a + E, same gamete is denoted as C,, whereas the covariance ofallelic depends only on the selective forces impinging on the popu- effects within individuals but from different gametes (due to lation, if these are weak. nonrandom mating) is written as Ci,. By defining Provided the genetic variances and covariance of the char- m n acters remain constant, deterministic evolution of the mean Ci. = E (Co, + CV) cim= E1 (co + Coj), [13a] phenotypes occurs along lines of constant slope, Aq/Ai = B! j=l j=m+ 1 translation ofcoordinates G. The the genetic variances and covariance of the characters are i = E- 0/(I + E/a), g = q [Ila] m n allows Eqs. 1 and 9 to be written in matrix form as G = 2 > Ciz, H = 2 2Ci>, i=l i=m+ 1 [13b] n m ( iz (-(aa+1 e)G GB ki [lib] B = 2 , Ci3 = 2E CjY. i=m+ 1 i= 1 The eigenvalues of the matrix in Eq. lib are AO = 0, corre- Male and female traits are assumed to have normally distributed sponding to lack of deterministic movement along the line of environmental effects with variances E and F and no geno- equilibria, and type-environment interaction or correlation, so that the pheno- typic variances are respectively A = [B - (a + E)G]/(2aw2), [12] o2=G+E r2=H+F. [14] associated with the lines ofmotion. Thus, the mean phenotypes In a model of mutation with a wide range of possible allelic change geometrically with time at the rate of(1 + A)t. The gen- effects at each locus and the same rate and distribution of mu- eral criterion for stability of the line of equilibria is -2 < A tational changes for all alleles at a given locus i, mutation creates < 0. Although it is possible with discrete generations for os- a constant input of new genetic variance each generation, u2, cillations of period two generations to occur if A < -1, under but does not alter the covariances between loci (13, 14). By as- weak selection JAI << 1, so that (1 + A)t eAt. Then, if A < 0 suming that mutation produces no net directional force on the or equivalently BIG < a + E, the line of equilibria is stable; however, perturbations away from a certain point on the line mean phenotypes, the dynamics of the variances and covari- will not generally return to the same point. If A > 0 or BIG ances of allelic effects can be derived (17) as > a + 8, the system is unstable and evolves away from the line ACY = -rij(Cy-CJ) - /2kCizCjo + ijui + Dij, [15] ofequilibria at a geometrically increasing rate, in a direction that may either exaggerate or diminish the male trait (Fig. 1). Under in which S8 = 1 ifi j, and zero otherwise. k is the proportional unimodal preferences, the condition for instability is more strin- reduction in the phenotypic variance of males caused by selec- gent with absolute (E = 1) than with relative (E = 0) mate choice. tion within a particular generation, and the factor of1/2 accounts Maintenance of Genetic Variance and Covariance. When for sex-limited expression of the selected trait. Dig represents the mating preferences offemales are mediated by sensory and the disruptive influence ofsex-limited selection (due to different other nervous processes rather than by physical constraints and intensities of directional selection on the sexes); this vanishes the male sexual trait is morphological, the sets of genes influ- when the mean phenotypes are at an equilibrium (S = 0) and is negligible under sufficiently weak directional selection, near the line of equilibria, hence Dii is ignored in subsequent calculations. The diagonal Eqs. 15 with i = j have a unique admissable equilibrium solution, provided k > 0, Ciz = V2uir2/k. [16a] Although this does not completely solve the system, in con- junction with the definitions in Eq. 13b it does determine the Male character, 2 critical genetic parameter that governs the stability ofthe mean FIG. 1. Thejoint evolution offemale mating preferences and a sec- phenotypes near the line of equilibria, ondary sexual character of males. The male trait is under stabilizing n m natural selection and sexual selection by females. There is no selection directly on female mating preferences, which evolve as a genetically B/G= E vui E vui. [16b] correlated response to selection on males, along lines ofconstant slope i=m+1 i=l (with arrowheads). A (heavy) line ofequilibria exists for the mean phe- notypes that may be stable (Left) or unstable (Right), depending on the This parameter does not depend on the linkage map ofthe genes genetic variation parameters of the population. or on the form ofnatural or sexual selection on males; it depends Downloaded by guest on September 30, 2021 3724 Evolution: Lande Proc. Natl. Acad. Sci. USA 78 (1981) only on the relative mutability ofgenes affecting female mating in which y = B/\/GH is the additive genetic correlation be- preferences and the male trait. tween the sexes. In spite of stabilizing natural selection on Variance in female mating preferences exerts a disruptive males, random genetic drift in female mating preferences pro- effect on the male trait, which, if sufficiently large, may over- duces a diversification of male phenotypes among populations come the stabilizing influence of natural selection, so that k that may be quite rapid. < 0. Under weak selection on the variance of males (w2, ,2 >> an), it can be shown that existence ofthe equilibrium in Eq. DISCUSSION 16 a and b requires in the psychophysical model that T2 < w2 and, with relative or absolute unimodal preferences, that r2 In polygamous species where males are promiscuous, providing < 2(l + v2/02). Numerical analysis ofthe complete dynamical only genetic material to the next generation, and females have system (not given here) confirmed that these equations specify many potential mates, there is no selection directly on female a unique locally stable equilibrium of genetic variability. mating preferences. Nevertheless, female sexual preferences Random Genetic Drift in Mating Preferences. In a finite can evolve in response to selection on genetically correlated population, random genetic drift in female mating preferences traits, such as secondary sexual characters ofmales. A male char- produces random selective forces on males, which in turn affect acter under stabilizing natural selection toward a phenotype mating preferences through the genetic correlation between the that is optimal with respect to survival may evolve to a markedly traits. When the line of equilibria created by genetic variance suboptimal phenotype by sexual selection acting through mat- in mating preferences is unstable, random genetic drift could ing success. For a population with additive genetic variance in trigger a runaway process of sexual selection. Even when the both female mating preference and a male sexual trait, a selec- line of equilibria is stable, evolution along it can occur rapidly tive constraint on the male trait alone implies a line ofpossible through the interaction ofrandom genetic drift with natural and equilibria for the mean phenotypes. Regardless of how much sexual selection because populations starting from the same the average male deviates from the optimum phenotype under point may drift to different sides of the line ofequilibria and be natural selection, there is an intensity of sexual selection that selected in opposing directions (Fig. 1). will bring the system into balance (Fig. 1). In the models of If the effective size of a population, Ne (5), is not very small psychophysical, absolute, and relative mating preferences, the and remains nearly constant through time, the probability dis- slope of the line of equilibria depends essentially on the selec- tribution of genetic variation parameters will have negligible tive forces acting on the population. The evolutionary stability dispersion around its expected value (18); then G, B and H can of the line of equilibria is determined by the ratio BIG, the be approximated as constants. Although the effective population additive genetic covariance between male and female traits di- size changes with the degree of polygamy (5) as the intensity vided by the additive genetic variance in the male character. of sexual selection evolves, for weak selection, Ne should be If this genetic regression slope exceeds the slope of the line of nearly constant over a wide range of mean phenotypes. Under equilibria, the line is unstable (Eq. 12). these conditions, phenotypic evolution can be analyzed by using Polygenic mutation, recombination, and assortative mating the theory of Gaussian diffusion processes (19). The vector of can maintain the additive genetic variance and covariance ofthe mean phenotypes in a population of effective size Ne has a vari- traits nearly constant in spite of selection tending to deplete ance-covariance matrix due to genetic sampling (each genera- genetic variation. Genetic covariance between characters in a tion) of population is attributable to pleiotropy and nonrandom asso- ciations of alleles at loci affecting the traits. Homologous char- acters of the two sexes, such as body size, usually show similar Ne (B H)[7 patterns ofvariation and a high genetic correlation probably due which is unaltered by the translation of coordinates in Eq. lla. to pleiotropy (6). Female mating preferences operating through Starting from a specified point, the joint probability distribution nervous and other sensory processes are not expected to be of the mean phenotypes is approximately Gaussian, with ex- mutually pleiotropic with male morphological characters. How- pectation obeying a continuous time version of Eq. lib. The ever, even in the absence ofpleiotropy, a positive genetic cor- dispersion matrix ofthe mean phenotypes (in either the original relation between female sexual preferences and the secondary or translated coordinates) is initially null, D(O) = 0, and satisfies sexual characters of males inevitably results from assortative mating due to genetic variance in mating preferences. (19) In the present models, if there is no pleiotropy of genes in- fluencing female mating preferences and male sexual traits, the dt critical genetic parameter governing the evolutionary stability ofthe mean phenotypes, BIG, is completely determined by the in which M is the matrix in Eq. lib and MT is its transpose. The relative mutability of genes affecting the two traits. For the general solution mutation process studied here (Eq. 15; refs. 13 and 14), this ratio rt is independent of the linkage map of the genes and the forms D(t) = feM(t- VeMT(t de [19] of natural and sexual selection (Eq. 16b). The expectation ofa genetic correlation between female mat- ing preferences and the male characters on which mate choice can be evaluated by expanding the matrix exponentials in power is based can be experimentally tested. Breeding and selection series. Noting from Eqs. 11 and 12 that M2 = AM, summing experiments can be used to estimate the additive genetic vari- the series and integrating shows that when the line of equilibria ances and covariances ofquantitative characters in a population. is asymptotically stable (- 1 < A < 0), the ultimate rate of dis- But these techniques generally cannot distinguish pleiotropy persion along the line for t >>-A-' is from nonrandom association of alleles at tightly linked loci (11). D(t) constant matrix [20] As stated by Fisher, during the unstable phase, the rates of ( evolution of a male trait and female mating preferences for it H(1- y2)t 1 a+ both increase with time or ' geometrically approximately expo- N+~a~e~B/G)N,(a+e E- BIG) a~ea + E (a+ E)2 nentially; at least near the line of equilibria. Thus, the rates of Downloaded by guest on September 30, 2021 Evolution: Lande Proc. Natl. Acad. Sci. USA 78 (1981) 3725 evolution may become quite rapid, especially when the preding Although in the present models selection is limited to males statement applies tocharacters measured for statistical purposes and mate choice is restricted to females, genetic instability of on a logarithmic scale. These results contrast with the conclu- mating preferences and sexual dimorphism may occur in awider sions from simple two-locus models (10). set of circumstances. With the addition of a constraint by sta- In Fisher's account, a runaway process must eventually be bilizing natural selection directly on female matingpreferences, stopped by severe counterselection against extreme males or caused by variation in male parental behavior or mating delays against the most discriminating females because of their diffi- incurred by the most discriminating females, there would be culty in finding a suitable mate. The model ofstabilizing natural a qualitative change in the dynamics described here. Instead selection with a Gaussian fitness function (Eq. 3a) produces a ofa line ofequilibria for the mean phenotypes there would exist linear restoring force on the mean male phenotype toward its and equilibrium point where the mean fitness of females is optimum. But some intensities ofnatural selection can be over- maximized but that ofmales is not (6). However, by continuity come by the evolution of sexual preference, which in the pres- with the present limiting case, this equilibrium point could still ent models produces a linear perturbing force (Eq. 9). This im- be genetically unstable. Natural selection on mating prefer- plies that a nonlinear force from natural selection is necessary ences also creates the possibility of evolutionary oscillations. to finally stabilize runaway sexual selection or, equivalently, These models help to explain the classical observations of that the viability of individual males must decrease faster than Darwin (1) and others (7-9) that closely related species ofhigher a Gaussian curve as the mean male phenotype departs from its animals often differ most in the characters of adult males, in a optimum. substantially nonadaptive or random pattern, whereas females Despite his generally gradualistic view of evolution, Fisher resemble one another more strongly. Sexual isolation is fre- (3) believed that striking secondary sexual characters often quently a major reproductive barrier between closely related evolved in sudden bursts, followed by long periods ofcompar- animal species (20). Studies of natural and experimental pop- ative stability. He discussed runaway sexual selection as a mech- ulations ofHawaiian Drosophila (21) suggest that, in small geo- anism for rapid speciation by premating reproductive isolation graphically isolated populations, random genetic drift in female and divergence ofquantitative characters. The exponential na- mating preferences may initiate rapid speciation by sexual iso- ture ofthe instability means that the process could be explosive lation and evolution of sexual dimorphism. or virtually instantaneous on a geological time scale. Fisher (3) alluded to the indeterminancy involved in unstable I thank S. J. Arnold, J. J. Bull, J. F. Crow, M. J. West-Eberhard, systems, where small perturbations can produce large effects, E. G. Leigh, Jr., M. J. Ryan, and the reviewers for helpful criticisms. but suggested that the initial direction ofrunaway evolution in This work was supported by National Science Foundation Grant DEB- female mating preferences and male traits is determined by 7909804 and the Andrew W. Mellon Foundation. natural selection or intermale combat. Instability also implies that fluctuating selection and random genetic drift. may be im- 1. Darwin, C. (1871) The Descent ofMan and Selection in Relation portant sources of nonadaptive diversity in taxa undergoing to Sex (Murray, London). 2. Fisher, R. A. (1915) Eugen. Rev. 7, 184-192. rapid speciation. Even in the absence of genetic instability, 3. Fisher, R. A. (1930) The Genetical Theory of Natural Selection there may be rapid indeterminate evolution by random genetic (Clarendon, Oxford); (1958) (Dover, New York), Rev. Ed., pp. drift interacting with natural and sexual selection along lines of 135-162. equilibria (Eq. 20). In species with complex morphology and 4. Haldane, J. B. S. (1932) The Causes of Evolution (Harper and behavior, the diversity of possible outcomes could be enor- Row, New York), pp. 119-124. 5. Wright, S. (1969) Evolution and the Genetics ofPopulations. Vol. mous, with a hyperplane rather than a line of equilibria. The 2. The Theory ofGene Frequencies (Univ. ofChicago, Chicago). male traits most likely to become exaggerated by such mech- 6. Lande, R. (1980) Evolution 34, 292-305. anisms are those under weak natural selection and subject to 7. Gilliard, E. T. (1969) Birds ofParadise and Bower Birds (Natural relatively large variance in female sexual preferences, such as History, New York). 8. Geist, V. (1971) Mountain Sheep (Univ. of Chicago, Chicago). some behavioral and morphological elements of courtship and 9. Brown, J. L. (1975) The Evolution of Behavior (Norton, New mating. York), pp. 151-185. Male characters can be diminished as well as enhanced by 10. O'Donald, P. (1980) Genetic Models of Sexual Selection (Cam- female sexual preferences. A bias in the direction of evolution bridge Univ. Press, Cambridge). toward conspicuous male traits is inevitable during the origin 11. Falconer, D. S. (1960) Introduction to Quantitative Genetics (Ronald, New York). ofa new character. Furthermore, unlike the highly polygamous 12. Ehrman, L. & Parsons, P. A. (1976) The Genetics of Behavior mating systems often associated with exaggerated male traits, (Sinauer, Sunderland, MA). evolution toward diminished development tends to restore ran- 13. Kimura, M. (1965) Proc. Natl. Acad. Sci. USA 54, 731-736. dom mating as the population passes a lower threshold for the 14. Lande, R. (1975) Genet. Res. 26, 221-235. or its 15. Stevens, S. S. (1975) Psychophysics (Wiley, New York). development of the trait in males detection by females. 16. Alcock, J. (1975) Animal Behavior (Sinauer, Sunderland, MA), Fisher (2) noted that, for species in which individuals have lim- pp. 158-160. ited time and ability to assess potential mates, the expression 17. Lande, R. (1977) Genetics 86, 485-498. ofstrong mating preferences for one trait may decrease the op- 18. Avery, P. J. & Hill, W. G. (1977) Genet. Res. 29, 193-213. portunity for sexual selection on other traits; thus, the evolution 19. Lande, R. (1980) Am. Nat. 116, 463-479. 20. Dobzhansky, Th. (1970) Genetics of the Evolutionary Process of new secondary sexual.characters and associated mating pref- (Columbia, New York), pp. 319-325. erences may contribute to the decline ofold ones (see also ref. 21. Carson, H. L. (1978) in Ecological Genetics: The Interface, ed. 7). Brussard, P. F. (Springer, New York), pp. 93-107. Downloaded by guest on September 30, 2021