Evolution of Female Choice and Male Parental Investment in Polygynous Species: The Demise of the "Sexy Son"

The University of Chicago

Evolution of Female Choice and Male Parental Investment in Polygynous Species: The Demiseof the "Sexy Son"Author(s): Mark KirkpatrickSource: The American Naturalist, Vol. 125, No. 6 (Jun., 1985), pp. 788-810Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/2461447 .

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Vol. 125, No. 6 The American Naturalist June 1985

EVOLUTION OF FEMALE CHOICE AND MALE PARENTAL INVESTMENT IN POLYGYNOUS SPECIES:

THE DEMISE OF THE "SEXY SON"

MARK KIRKPATRICK*

Department of Zoology, University of Washington, Seattle, Washington 98195

Submitted August 8, 1983; Accepted September 23, 1984

Many examples of radical sexual dimorphism in the animal kingdom are associated with polygynous mating systems in which males contribute gametes but no material resources or benefits to their mates (Darwin 1871). Following the reasoning of Fisher (1958), recent work using genetic models has shown how sexual selection in these circumstances can lead to the evolution of secondary sexual characters that seriously impair the survival of their bearers (O'Donald 1980; Lande 1981; Kirkpatrick 1982). In a large number of polygynous species, however, males interact more extensively with females and directly influence the success of their matings. These direct fitness effects most often result from male parental investment (Trivers 1972), which can take the form of a resource contributed to the female and offspring, or male parental care of the offspring. Since the earlier genetic models explicitly excluded male parental investment or any other form of direct fitness effects, it is of interest to understand how these effects alter predictions for the joint evolution of female mating behavior and male characters that affect the fecundity of their matings.

This topic recently became the focus of controversy with the introduction of the "sexy son" hypothesis of sexual selection by Weatherhead and Robertson (1979). These authors argue that under some conditions female mating preferences will evolve for "attractive" males who offer inferior resources and so yield their mates fewer offspring. Although the females mating them will suffer depressed fecundity, their male offspring (the "sexy sons") carry their father's genes and hence will be highly successful in attracting mates. The hypothesis envisions that this compensatory advantage to the sons resulting from sexual selection can offset the impact of natural selection acting against females that leave fewer offspring because of their mates' inferior contribution. In the case of the blackbirds studied

* Present address: Museum of Vertebrate Zoology, University of California, Berkeley, California 94720. Address as of September 1, 1985: Department of Zoology, University of Texas, Austin, Texas 78712.

Am. Nat. 1985. Vol. 125, pp. 788-810. ? 1985 by The University of Chicago. 0003-0147/85/2506-0002$02.00. All rights reserved.



FEMALE CHOICE AND MALE PARENTAL INVESTMENT 789

by Weatherhead and Robertson, males offer to females nesting territories that differ in the quality of food available for rearing young. The hypothesis also can be generalized to include other sources of variation between males in the productivity or fertility of their matings. The sexy son hypothesis differs from the conven- tional theory of sexual selection (Fisher 1958; O'Donald 1980; Lande 1981; Kirk- patrick 1982) by the inclusion of fecundity differences between females. In the earlier theory, a male's phenotype determines viability and mating success, but a female's mating-behavior phenotype has no effect on her survival or fecundity. By contrast, the Weatherhead and Robertson hypothesis includes fecundity differences between females that result from the type of male they mate. Their sugges- tion that sexual selection might cause females to decrease their immediate reproductive success differs from arguments made by other workers (e.g., Downhower and Armitage 1971; Downhower and Brown 1980) that females under these circumstances are expected to maximize their immediate fecundity.

In this paper I develop genetic models for the simultaneous evolution of female mating preferences and male secondary sexual traits when those male traits directly affect the average number of offspring a female will produce. Results show that when a female's mate choice affects her fecundity, an evolutionary equilibrium is reached only when females on the average mate with the male phenotype which yields them the highest immediate fecundity. Males associated with lower fecundity and the mating preferences for these males do not persist in a population at equilibrium. The only alternative evolutionary outcome to this equilibrium is an unstable runaway process in which the population follows a maladaptive evolutionary trajectory with declining population fitness, possibly leading to extinction. In either event, the results refute the contention (Weather- head and Robertson 1979; Heisler 1981) that there can develop a stable balance of evolutionary forces which maintains mate preferences for males that reduce the reproductive success of their mates.

Two classes of models are developed here. The first assumes continuous characters with polygenic inheritance, and is derived from Lande's (1981) models of sexual selection, extended to account for differences in fecundity. The second type of model assumes a two-locus haploid system of inheritance, and is based on my earlier models of sexual selection (Kirkpatrick 1982). Results from both types of models come to similar qualitative results, and so confirm the major conclusions of the study.

THE MODELS

The sexy son hypothesis concerns the evolution in a polygynous mating system of a trait expressed in males, and a mating preference in females that operates on the degree of elaboration of that trait. The description of the polygenic models will begin with the ecological and behavioral assumptions that determine the fitnesses of individuals. This section of the paper deals only with phenotypes. I will then turn to the genetic system underlying the phenotypic distributions in order to determine the evolutionary consequences of the ecology and behavior.



790 THE AMERICAN NATURALIST

Ecological and Behavioral Assumptions

Males in the population bear a secondary sexual trait whose degree of elaboration is measured as z. The character might be some morphological attribute, such as body size or a plumage characteristic. Alternatively, it might represent a behavioral quality, such as the amount of parental care or nutritional investment the male contributes, or his ability to control a high-quality territory. Three components of male fitness are affected by the trait: the probability of survival to adulthood, the success in attracting mates, and the fecundity of each mating.

The male trait varies in a continuous manner, and its distribution in juvenile males, p(z), is assumed to be normal (Gaussian) with mean z and variance c2:

p(z) ac exp[- (z - z)2/2U2]. (1)

Natural selection acts on the juveniles, altering the distribution of z. The viability of males is determined by a Gaussian function with an optimum at O,,:

w*(z) a exp[- (OH - z)2/2W2 ]. (2)

The parameter wilm measures the width of the viability function. A large value of will corresponds to relatively weak selection, and males with a value of z falling in the range Om ? wilm have high viability relative to other males. (The possibility of viability selection on the male trait was not considered in earlier discussions of the sexy son hypothesis. Its effect can be excluded from the present models by letting

2-> oo in later equations. Directional rather than stabilizing natural selection can be mimicked by setting the optimum Om to a very large or very small value.) Following viability selection, the distribution of z among mature males, written p* (z), is normal with mean

- 2 2 2 *= Z(W7 + OmC and variance *2 = _* W_

(2 + 2 2 2 + 2 2

Secondly, sexual selection acts. A male's mating success depends on his phenotype for the trait. To determine the impact of sexual selection, a model of mating behavior that specifies the action of the female mating preference must be adopted. (The term preference here operationally refers to any mechanism by which a female's phenotype y biases the probability she will mate with a male of phenotype z. No psychological state is necessarily implied; this definition, e.g., includes cases in which the "preference" results from the mechanics of some morphological constraint of copulation [see, e.g., Licht 1976].) Lande (1981) proposed three different forms of mating-preference functions which determine the rules by which females select males. Each of the three may be appropriate for a different set of species depending upon the neurological, physiological, and morphological mechanisms involved in mating. In the body of this paper I will deal only with Lande's model of absolute mating preference because it has the simplest interpretation. The corresponding treatments of his psychophysical and relative preference models are given in the Appendix, where I show that the




qualitative conclusions for all three models of mate choice are the same. In these models I assume that female choice is responsible for variation in male mating success, and so any possible effects of male-male interactions are excluded.

The character in females that mediates mating preference is measured as y and is assumed to be normally distributed among females with mean y and variance r2:

q(y) xc exp[- (Y - /2r2] (3)

For each value y of the preference there corresponds a function, i(zjy), that describes the relative mating preference such a female will have for a male as a function of his phenotype z. Under the absolute-preference model, a female with a preference phenotype y favors mating a male with a matching value of z so that z = y. The strength of her preference for males with other values of z falls off symmetrically on either side of y as a Gaussian function with width v:

i(zjy) xc exp[- (y - z)2/2v2]. (4)

Thus, males whose z value lies within v units of y will be highly preferred over others by this type of female.

The probability that a given female mates with a male of a certain phenotype depends not only on her preference, but also on the relative abundance of that male phenotype in the population. The preference has the effect of amplifying the conspicuousness of a male or of increasing the probability of a successful mating encounter with him. The apparent frequency of a male phenotype is altered from its actual frequency by a degree that depends on the female's preference phenotype. The probability that a y female successfully mates a z male is proportional to the product of her mating preference for the z phenotype, i(zjy), and its actual frequency among mature males, p*(z). Since preferences are not exact, the average type of male that a y female will mate is a compromise between her favored type of male (with phenotype z = y) and the average male phenotype that would be chosen were there no preference at all (which is z*). The success that a z male (relative to other types of males) can expect in attracting y females is

O*(Ziy) = i(zly) p*(x)](xjy)dx

oc exp[ - z2/2V2 - y 2(*2/2(v2 + u*2) (5)

- yf*/(V2 + U*2) + zy/v2].

This calculation is made using the fact that 0x

- exp(Ax2 + Bx + C)dx ?c exp(C - B2/4A).

The expected mating fitness of a z male can now be computed as the sum over all values of y of the product of the frequency of y females and the ability of a z male to attract them:

w**(z) = { q(y)4*(zly)dy cc exp(Jlz2 + JZ) (6)




where

U2 _ *2 _ V2

2(v4 + V2cr*2 + T*2T2)

Y2(V2 + -*2) - z7T2 J2 =

V4 + V2cr*2 + c*2T2

The value of w**(z) is proportional to the number of mates that a z male obtains on the average.

The final component of male fitness to be considered is that resulting from the success of each mating, which is dependent upon the phenotype of the male. I refer to this as fecundity selection. This component is the extension to Lande's (1981) models.

A variety of biological processes affect the success of matings in polygynous species. Fecundity selection encompasses both prezygotic mechanisms, such as variation in sperm potency, and postzygotic mechanisms, such as differential probabilities of successfully raising offspring. Fecundity effects may be broadly divided into two categories. The first, which I term limited male reproductive potential, occurs when the average success of a mating is a decreasing function of the total number of females mated to the male involved. The alternative situation, which I call unlimited male reproductive potential, occurs when the success of a mating is independent of the number of females mated to a male, but does depend on some attribute of each male that varies among males in the population. With either type of reproductive potential, females may initially favor mating with a male phenotype that reduces their fecundity, as required by the sexy son hypothesis.

The redwing blackbird mating system that motivated the first discussion of the sexy son hypothesis is believed to be under limited reproductive potential. These birds exhibit resource-defense polygyny (Emlen and Oring 1977) in which males defend territories that vary in the quality of resources (such as food) needed to raise offspring. The addition of females to a male's territory is thought to mutually depress the number of young each female can raise because of local competition for resources (Verner 1964; Orians 1969). In species in which males help raise the offspring, an increase in the number of mates can decrease the quality of the male parental care contributed to each female (Alatalo et al. 1981). In other mating systems, repeated mating by males can result in sperm depletion and, consequently, a decline in the fertility of each mating (Nakatsuru and Kramer 1982; Dewsbury 1982).

The unlimited-potential assumption is more appropriate for other mating systems. In scorpion flies, for example, promiscuous males attract females with nuptial offerings of prey items. Thornhill (1981) has shown that individual males consistently differ in the size and type of prey they offer prospective mates. Nuptial-prey qualities in turn affect the number of eggs which a female will subsequently lay. Since males do not hold fixed territories and potential prey availability is high, a male's offering is not likely to be altered by the number of females he has mated and so male reproductive potential may be essentially




unlimited. Other possible examples of unlimited potential come from groups of arthropods in which the male's spermatophore contributes not only sperm but also nutrition to the female (Englemann 1970; Leopold 1976; Friedel and Gillott 1977; Boggs and Gilbert 1979; Boggs and Watt 1981; Mullins and Keil 1980; Gwynne 1982, 1983, 1984; Schal and Bell 1982; Rutowski et al. 1983; Markow and Ankney 1984). If males are not energetically limited by spermatophore production, then variation between males in the quantity or quality of their contribution would be the character measured by z in the unlimited-potential model. In some frog species, females oviposit on the territory of the male with whom they mate, and qualities of the territories (e.g., water temperature, predation rate) affect the probability that the eggs will survive (Wells 1977; Howard 1978). If females choose mates on the basis of a male attribute, the unlimited model would apply to these species.

The fecundity of matings under the unlimited male potential assumption will take the form of a normal curve with an optimum at Of and a width wj. The average fecundity of mating that involves a male with phenotype z is therefore

W***(z) oc exp[-(Of - z)2/2 w]. (7a)

Matings with males whose phenotypes fall in the range Of + wf have comparatively high fecundity. The sexy son hypothesis is confirmed if there is an evolutionary equilibrium at which females on the average mate with males that differ from the phenotype with highest fertility.

Limited male potential can be modeled by making the average reproductive success of each mating of a z male be a power function of w**(z), the total number of females he mates:

w***(z) oa [w**(z)V3 a exp(3JIz2 + IJ2z). (7b)

The parameter 3, which is less than zero, measures how rapidly fecundity declines with the number of females mated to a male, with decreasing values of C

corresponding to decreasing fecundity. A value of C = 0 implies there is no effect of the number of matings on female fecundity. A value of C = - 1 implies that the loss of offspring as a result of fertility decline entirely offsets the advantage a male accrues by attracting additional matings. A value of 3 = - 0.5 means that a male that mates four females leaves twice as many offspring as a male that mates only one female, while each of the four females leaves only half the progeny she would have, had she mated a monogamous male.

The three fitness components of viability, mating success, and fecundity combine to determine the overall fitness of a male with phenotype z:

WJ(z) = w*(z)w**(z)w***(z) (8)

ac exp[(- 1/2wj2 + J1 + K1)z2 + (01?/W2 + J2 + K2)z]

where K1 - - 1/2wf for unlimited male potential and K1 = P3J1 for limited male potential; K2 = Ofl/w for unlimited male potential and K2 = 3J2 for limited male potential. The action of selection on males is illustrated in figure 1 for unlimited male potential and in figure 2 for limited male potential.




P(Z)

em V VIABILITY SELECTION

SEXUAL SELECTION - ? -- - ~~~~W q(y)

FECUNDITY SELECTION W***(z)

FIG. I.-An example of selection acting on the distribution of z, the male trait, assuming unlimited male reproductive potential. The male trait is distributed in the population as p(z), with a mean among offspring of z. The fitness of males under natural selection is given by the fitness function w*(z) with optimum 0,,. The mating preferences, measured as y, are distributed among females as q(y). The action of these mating preferences determines the fitness of males under sexual selection, given by the function w**(z). The relative success of each mating is given by the fecundity function w***(z), with optimum Of. Strength of selection has been greatly exaggerated for the purpose of this illustration.

P(Z)

w*(z) VIABILITY SELECTION

SEXUAL SELECTION fi, w**(z) , q(y)

'W***(z) FECUNDITY SELECTION

---__.1

FIG. 2.-Example of selection acting on the male trait when male reproductive potential is limited. This example differs from that of fig. 1 only in the form of the fecundity selection; viability and sexual selection are the same.

Having discussed the assumptions pertinent to male fitness, I now turn to the females. The female mating-preference character y is assumed to have no effect on a female's ability to survive or procure a mate. Its sole action is to determine the relative bias a female exhibits toward prospective mates, based on their phenotype z. Because not all matings are equally fertile, a female's fitness is affected by her choice of mate. The fitness of a female with phenotype y is simply the average fecundity of matings in which such a female will engage. This is the sum over all values of z of the probability she will mate a z male times the fertility of this type of mating:

0x

Wf(y) = J P*(Z)4,*(zly)%)***(z)dz

{L~ ~~~ C*4 K, ext(2 ? U*2)(V 2+U2 - 2v 2U*2 K1)I(9

L 2v2Klz-* + (v2 + *2)K2 ]1 + (V2 + T*2)(V?2 + U*2 - 2V2(,*2K1)JYJ




Study of this equation shows that the strength of selection imposed on female preferences can be very weak because it operates via the imperfect correlation between a female's preference phenotype and the fecundity determined by her mate's phenotype.

This completes the description of the behavioral and ecological assumptions. In order to analyze the implications these have for evolution, I will now introduce a genetic model.

A Quantitative Genetic Model

Following the standard assumptions of quantitative genetics (see Falconer 1981), I assume that the characters y and z are the sums of an additive genetic component, normally distributed in the population, and a nonadditive component, resulting from environmental effects, allelic dominance, and epistasis, that is uncorrelated with the genetic value and that is also normally distributed. The genetic loci involved in the determination of these characters are assumed to be autosomal, but an analogous model for sex-linked inheritance (Lande 1980) will give a qualitatively similar result. For the male trait, the variances of the genetic and environmental components are Gz and Ez, and for mating preferences the corresponding variances are Gq, and E,,. The overall phenotypic variances are thus

2= G + Ez for the male trait, and T2 = G + Ey for the female mating preferences.

Last, there is a genetic covariance B between the male trait and mating preference. Two sources may contribute to this covariance. In the present context, the most interesting of these is the positive correlation which spontaneously develops from the nonrandom mating resulting from variation in female mating preferences. Females with larger values of y are predisposed to mate males with larger values of z. Their offspring will tend to bear genes for large values of both y and z. The positive genetic association which so arises was first considered by Fisher (1958), and was later studied by O'Donald (1980), Lande (1981), and Kirkpatrick (1982). This cause of genetic correlation is called linkage disequilibrium. A second possible source of genetic correlation (either positive or negative) is pleiotropy, the phenomenon in which single genes influence the expression of multiple traits. If there are alleles which alter both the expression of z when carried in males and the expression of y when carried in females, then these characters are genetically coupled via pleiotropy. Pleiotropic effects are not, however, expected a priori to be important in the system being considered here.

The genetic assumption of a bivariate normal distribution of additive genetic effects for z and y is consistent with several more-detailed models of polygenic characters that specify the number of loci and alleles involved and include the action of mutation and recombination (Kimura 1965; Lande 1977; Bulmer 1980; Turelli 1984). These models account for the evolution of genetic variances and covariances. Since the major questions concerning the sexy son hypothesis in- volve only the means of the male trait and the female mating preference, the analysis can be greatly simplified by treating the variances G, and G., and the




covariance B as fixed parameters. Since this procedure does not affect conclusions regarding the location of the evolutionary equilibria for the means, this will be the approach used in the body of the paper. In order to determine whether the evolutionary equilibria will be stable or unstable, the evolution of the variances and covariance must be considered. In the Appendix I present the derivation and results obtained from one particular model of the evolution of the variances and covariance (based on that of Lande 1977) to determine the stability of the equilibria.

The evolution of the means of z and y in this model is determined by two equations that give the per-generation change in these quantities:

2 ( 2Z s T2 y (lOa)

I 2G T SV + B2 SZ (10b)

where SZ and Sv are the selection differentials (Falconer 1981) for the male trait in males and the female mating preference in females. The selection differential is defined as the difference in phenotypic value between the selected and unselected individuals. That is,

SZ z*** - E[W P(Z)j -W (IIa)

S., = y- y = EL WY) q(Y)| - (lIb)

where WM and Wf are the mean fitnesses for males and females, and Ed] denotes the expectation.

The recursion equations (10) merit explanation. Each is the average of two terms. The first of these is the effect of selection acting directly on each trait in the sex in which it is expressed. The coefficients Gz/&f2 and G /T2 are the heritabilities (Falconer 1981) of the male trait and mating preference, respectively. The second terms, on the right-hand side of equations (10), are the correlated selection responses. Because of the genetic correlation between z and y, as selection alters the distribution of z in males, it changes the frequency of genes coding for y (which are carried but not expressed in males). A correlated response likewise alters the frequency of z genes because selection acts directly upon y in females. (For a fuller discussion of correlated responses see Lande 1979; Falconer 1981, chap. 19; Slatkin and Kirkpatrick 1985). Since the loci for the male trait and female mating preference are (by assumption) autosomal, the two sexes contribute equally to the genetic change in these characters, with the result that the genetic change is the simple arithmetic average of the change within each sex.

One important conclusion to be drawn from equations (10) is that in general the selection coefficients SZ and Sy must equal zero at an equilibrium. The only other possibility for an equilibrium is when appropriate genetic variation no longer remains, a condition indicated by the absence of genetic variance for one of the




/,-P(Z)

w*(z)> ViABIeTY SELECTION

SEXUAL SELECION - w**(z) -.7 y

W***(z) - FECUNDIT SELECTION

FIG. 3.-An evolutionary equilibrium for the male trait. Unlimited male potential is assumed. See legend of fig. 1 for explanation of the variables.

characters, or by a perfect genetic correlation between the two characters. Loss of all genetic variance for the male trait or mate preference is not likely and has not been previously discussed as a factor in the sexy son hypothesis, but will be treated below in connection with a runaway process of sexual selection. Perfect genetic correlation seems so biologically implausible that I will not discuss it further.

The selection differentials equal zero when the phenotypic means of the characters are unaltered by the overall effect of selection within each generation (eqs. [11]). The mean values that satisfy this requirement are determined by the fitness function. The fitness functions of equations (8) for males and (9) for females are of the formf(x) oc exp(Ax2 + Bx), which requires the phenotypic mean to lie at the point - B!2A for the selection differential to vanish. (A fitness function of this form represents a Gaussian curve with optimum at -B/2A when A < 0.)

The equilibria for the means of z and y can therefore be determined directly from equations (8) for W,,(z) and (9) for Wf(y). In the case of unlimited male potential, the equilibria for the means of z and y are

Z = Of (12a)

y = v2(0f - Om)!W2n ? of. (1 2b)

The equilibrium for the mean of the male trait coincides with the phenotype that has the maximum fecundity per mating. The action of sexual selection produced by female preference exactly offsets the action of viability selection, and results in males equilibrating at the fecundity optimum. This is illustrated in figure 3. The equilibrium for the male trait, however, may differ substantially from the point at which male survivorship is maximized. Consider, for example, a case in which males help feed the young, and the prey fed to the offspring differ from those on which adults feed. This model predicts that females will evolve preferences that result in their mates bearing trophic structures appropriate for the feeding of offspring. Because the male trophic apparatus now differs from that which favors male survival, the result is that males will forage for themselves less efficiently and suffer increased mortality.

If there is limited male reproductive potential, the equilibrium is simply

Z YOm. (13)




The equilibrium for both the male trait and the female preference lie at the ecological optimum which confers on males the highest survivorship. Given this equilibrium for the male trait, it can be shown that the equilibrium for the preference at y- =z is the point at which female fecundity is maximized. The location of this point is independent of A, which acts only to help determine the strength of the evolutionary force driving the preference toward this equilibrium.

The results from both limited and unlimited male potential models immediately negate the principal tenet of the sexy son hypothesis, which holds that an equilibrium can be established in which females depress their fecundity by preferentially mating with males offering inferior material rewards. The equilibria given by equations (12) and (13) show that when the joint evolution of both the preference and the male trait is considered, the direct fitness effects felt by females because of their choice of mates result in a unique equilibrium in which female fecundity is maximized. The maximization principle applies within each generation. No compensatory force from future genetic benefits arises, contrary to the arguments of Weatherhead and Robertson (1979) and Heisler (1981).

The fact that the unique evolutionary equilibrium maximizes female fecundity does not, however, necessarily imply that evolution will always lead to this state. An equilibrium for z and y can be either stable or unstable. If the equilibrium is stable, then evolution will drive the system to this point. If, on the other hand, the equilibrium is unstable, the system will rapidly evolve away from that point. This is an example of the type of runaway process of sexual selection that was first described by Fisher (1958).

A runaway process can be triggered when the genetic coupling between the male trait and the female preference becomes too strong (i.e., the genetic covariance B becomes too large). An intuitive explanation of this event is as follows. If the mean mating preference is displaced slightly from its equilibrium, then the male trait begins to evolve in response to the change in the force of sexual selection so created. Evolution of the male trait, however, feeds back to the mating preference through the correlated selection response. As z changes, it causes a change in y. This change in the mating preference can be sufficiently large that the male trait cannot catch up with the new equilibrium for z established by the preference, and the process snowballs with self-reinforcing feedback.

As a runaway process proceeds, the fitness of the population rapidly deterio- rates because of the decline of fecundity caused by the increasingly deviant male trait. Were runaway sexual selection initiated in an actual population, it would only be brought to a halt by factors not specified within this genetic model (Fisher 1958; Lande 1981). One possibility is that genetic variation for one of the characters becomes depleted, which would arrest its evolution. The other character would be left to evolve under the fitness regime determined in part by the fixed character. Thus if GZ, the genetic variance for the male trait, goes to zero, the mate preference would still evolve to maximize female fecundity to the extent possible with the available range of male phenotypes. If Gy, the genetic variance for the preference, becomes exhausted, then the male trait will equilibrate at an intermediate point between the point favored by natural selection and that favored by sexual selection. In the absence of genetic variation for one of these charac-




ters, however, an unstable runaway is not possible. Another possible end to a runaway process is that declining fecundity ultimately pushes the population beyond its reproductive capacity and extinction results.

In order to determine when the action of sexual selection alone is sufficient to generate a genetic covariance that will destabilize the equilibrium for the means Z and y, a detailed genetic model that includes the evolution of the genetic variances and covariance is necessary. In the Appendix, I use one such model (derived from Lande 1977) and find that under the assumptions of that model there are indeed situations in which a runaway process can be initiated by the action of sexual selection.

It is quite possible that none of the species that have been discussed in connection with the sexy son hypothesis have ever experienced runaway sexual selection in their evolutionary histories. The point of discussing the runaway process is that it represents the only alternative evolutionary outcome to an equilibrium at which average female fecundity is maximized.

Alternative Genetic Models

A question that naturally arises is to what extent the conclusions of this theory are dependent upon the specific genetic assumptions used in the models. I have attempted to answer this using other models of the sexy son hypothesis based on my earlier two-locus haploid models of sexual selection (Kirkpatrick 1982).

Briefly, the assumptions are as follows. The male trait locus segregates for two alleles, T, and T2. These alleles affect male viability such that a T2 male, on the average, survives to maturity with probability 1 - s,, relative to a T1 male. The alleles at this locus are also responsible for differences in fecundity per mating. Under the unlimited reproductive potential assumption, the difference is a fixed attribute of the genotype. Each mating with a T2 male results in 1 - Sf times as many offspring as a mating with a T1 male. (The parameters s, and Sf can be set to positive or negative values, putting the T1 males at a selective advantage or disadvantage, respectively.) Under the limited reproductive potential assumption, fecundity per mating is a power function of the number of mates each male obtains. The T genes are not expressed when carried by females.

Two alleles also segregate at the female preference locus. These preference alleles confer on females fixed biases toward mating with each of the two types of males in the population. A female bearing the PI allele perceives T1 males as more conspicuous or attractive than T2 males by a factor of a,. Similarly, P2 females prefer T2 males by a factor of a2. Further details about the mating assumptions of these models are given in Kirkpatrick (1982).

Results from simulations of the haploid models are consistent with the quantitative genetic analysis. Polymorphism for the male trait cannot be maintained: one of the alleles at the trait locus is always lost. In the case of unlimited male reproductive potential, as the less fecund trait allele is being lost, the female mating-preference allele for that trait also declines in frequency. A similar decline in the frequency of the preference for the less viable male trait occurs under the limited reproductive potential assumption as that male trait allele is lost from the




population. These outcomes result in the evolution of preferences that increase female fecundity, and so are analogous to the results from the quantitative genetic models.

The haploid models also exhibit runaway behavior. With certain combinations of parameters, if the mating preference for the less fecund or less viable male type begins at a sufficiently high frequency, then the force of sexual selection which that preference allele generates can drive the preferred male allele to fixation and simultaneously cause an increase in its own frequency via the genetic correlation, as described above for the quantitative models. In this case, the absolute fitness of the population declines as evolution proceeds. The runaway continues until genetic variation is exhausted with the fixation of the less fertile or less viable male trait allele. The sets of initial conditions leading to the two classes of outcomes (adaptive evolution leading to fixation of the more fertile or more viable male trait, vs. a maladaptive runaway leading to fixation of the less fertile or less viable male trait) are separated by a curve (technically known as a separatrix).

The predictions of the sexy son hypothesis are not met in any event. The balance of evolutionary forces predicted by the hypothesis should create an equilibrium in which both types of males are maintained in the population, corresponding to Weatherhead and Robertson's original observation of the pres- ence of less fertile and more fertile males in the blackbird populations. The present results indicate that the evolutionary forces considered here cannot main- tain such a polymorphism. Other factors must account for the variation in male resource quality and female preferences for males that are observed in natural populations.

DISCUSSION

Results of the genetic models presented above show that joint evolution of male secondary sexual characters and female mating preferences for those traits lead to an equilibrium at which females maximize their immediate fecundity. The models investigate evolution under two alternative assumptions about how males affect the fecundity of each of their matings. In the case of unlimited male reproductive potential, the success of a mating is a fixed attribute of the male involved. The male phenotype that maximizes the fecundity per mating may differ substantially from that which is optimal for male viability. In such a case, because the equilibrium maximizes fecundity per mating, average male survival can decline substantially. Under the alternative assumption of limited male reproductive potential, the success of each mating is a decreasing function of the total number of females that are mated to a male. Here the equilibrium not only maximizes female reproductive success, but, furthermore, is at the point that maximizes male survival.

The equilibrium in either case can be stable or unstable. If it is stable, the population will evolve toward the equilibrium. If the equilibrium is unstable, the population will evolve rapidly away from the equilibrium. Males then become increasingly "sexy," but also increasingly less able to contribute to their mates and offspring, pushing the population toward extinction. Which of these outcomes occurs depends on details of the behavioral and genetic systems that are involved.




The sexy son hypothesis posed by Weatherhead and Robertson (1979) contends that the forces of natural and sexual selection can balance in such a way that females favor mates that decrease their fecundity. The reasoning behind this idea is that sexual selection generated by the mating preferences will benefit the sons of these females and so offset in the following generation their immediate loss in fitness. Results from the models presented in this paper show that such an equilibrium cannot arise. Decreased fecundity cannot be offset by the reproductive success of progeny.

This conclusion can be understood in light of earlier models of sexual selection which lack the fecundity-selection component of the sexy son models. Lande (1981) and Kirkpatrick (1982) showed that in polygynous mating systems in which males mate but do not otherwise interact with females, not one but rather an infinite number of equilibria for the male trait and female preference are possible. These equilibria fall on a line that relates the mean mate preference to the mean of the male secondary sexual character in the population. At each point along the line of equilibria, the force of natural selection driving the male trait toward its ecological optimum is exactly offset by the force of sexual selection favoring more extreme males. If the mating preferences of females are altered such that they now prefer males with a greater elaboration of the trait, the mean of the trait increases until a new balance is struck between natural and sexual selection. Hence, there is a continuum of possible equilibria. Once a population achieves a combination of the male trait and female preference which falls on this line, there is no determin- istic force causing directional evolution along the line. This is because in males there prevails the balance of natural and sexual selection just described, while in females there is no net directional selection since (by assumption) female mate choice is uncorrelated with survival ability and fertility. These assumptions therefore do not result in any evolutionary force that, on the average, favors females bearing mating preferences for more-viable males.

With the incorporation of the additional component of fertility selection, female mate choice now has a directional force acting upon it. When the male phenotype determines the number of successful progeny that a mating will produce, selection will favor those females which mate with males that on the average cause them to produce more offspring. The difference between this result and those of Lande (1981) and Kirkpatrick (1982) is that in those models the only fitness differences result from the degree of expression of the male trait. The sexy son hypothesis, however, involves a situation in which female mating behavior is under direct selection because of fecundity differences among females as a result of their mate choice. In the case of unlimited male reproductive potential, the logic can be framed in the familiar terms of fitness maximization. Assume that the mean of the male character is fixed in the population and so prevented from evolving. The fitness associated with each female preference is then constant in time, and under this condition, selection will act to maximize the population mean fitness for the character (Lande 1979). This corresponds to the development of preferences which increase female fertility.

There is a second way to understand the logical flaw of the sexy son hypothesis. At any evolutionary equilibrium, the forces acting on the genes must equilibrate within each generation. The sexy son hypothesis, however, is based on the




premise that a loss in fitness to females caused by mate choice can be recon- stituted in future generations by their male descendants. Unfortunately for this argument, there is no mechanism by which Mendelian inheritance can recoup fitness deficits from earlier generations.

Weatherhead and Robertson's (1979) original paper on the sexy son stimulated a number of discussions in the literature. Wittenberger (1981) disputed the empir- ical inference Weatherhead and Robertson drew from their data, but concluded that the suggested mechanism might operate in some circumstances. Searcy and Yakusawa (1981) also took issue with the data, and in addition argued on intuitive grounds that mating preferences that lowered fecundity would be selected against. Weatherhead and Robertson (1981) replied to these authors with a further discussion of the data supporting their hypothesis.

Two papers on the subject invoked mathematical arguments. Weatherhead and Robertson (1979) used an inclusive-fitness calculation to support their hypothesis. Heisler (1981) tallied the grandchildren of females with different mating preferences as a metric of female fitness, and concurred that the sexy son mechanism could in fact work. Neither of these is a true genetic model in the sense that they are not based on explicit Mendelian mechanisms for changes in gene frequencies, and both apparently led to erroneous conclusions because of this. Inclusive- fitness calculations such as those used in those two papers cannot be used reliably to draw inferences about evolutionary outcomes.

The models described in this paper help to clarify the logical status of the notion of "good genes" in the theory of sexual selection. A pervasive argument in evolutionary biology, promoted by many authors (Williams 1966, p. 184; Orians 1969; Trivers 1972, 1976; Mayr 1972; Halliday 1978; Borgia 1979; Searcy 1982; and others) is that selection will favor females that mate with males that bear good genes or are of high genetic quality. This intuition has not been validated when rigorous genetic models of the mechanisms have been constructed. Female choice may evolve in response to immediate, material forces that affect female survival or fecundity, but not to any force that can be attributed to the fitness of descendants.

SUMMARY

Males of many polygynous animal species contribute nutrition, nest sites, parental care, or other material benefit to their mates in addition to their gametes. Variation in female reproductive success may be caused by variation in the quantity or quality of the male contribution, and in such cases a female's mate choice affects her reproductive success.

I present here polygenic and haploid two-allele models for the joint evolution of a male secondary sexual trait and a female mating preference for that character. The male trait affects male survivorship, mating success, and the number of offspring produced per mating. A female's reproductive success is consequently affected by the type of male she mates. This selection on fecundity is assumed to operate in one of two ways. Under the assumption of limited male reproductive potential, the fecundity per mating is a declining function of the number of females




to which a male is mated. Under the alternative assumption of unlimited male reproductive potential, the fecundity per mating is independent of the number of females mated, but does depend directly on the male's trait.

The evolutionary equilibrium under both forms of fecundity selection is found to maximize immediate female reproductive success. The equilibrium, however, may or may not maximize male survivorship: when male reproductive potential is unlimited, the equilibrium for the male trait may lie far from the point that is optimal for male viability, and so average male survival can decrease under this form of sexual selection. The models also show that it is possible for the evolutionary equilibrium to become unstable, in which case the population will evolve rapidly away from the equilibrium in a maladaptive runaway process.

These models address the "sexy son" hypothesis of Weatherhead and Robert- son (1979), which contends that selection can create an equilibrium at which females on the average mate with certain attractive types of males that give them inferior material resources and therefore decrease the females' immediate reproductive success. The logic of this argument is that the females are compensated for their fecundity deficit by the mating success of their sons, which bear their fathers' genes and so are attractive to females. Results from the models developed here show that no such evolutionary balance is possible, and the sexy son hypothesis is not supported. Female fecundity is maximized at the evolutionary equilibrium, and is not offset against the fitness of descendants.

ACKNOWLEDGMENTS

I thank S. J. Arnold, R. K. Colwell, J. Felsenstein, I. L. Heisler, R. Lande, L. Mueller, M. Slatkin, M. Turelli, D. B. Wake, P. J. Weatherhead, and L. Wilson for comments and discussion. I am particularly grateful to L. Mueller for sharing with me the results from his work on this topic. This research was supported by N.S.F. grant D.E.B. 8128050 to M. Slatkin and by the Miller Institute for Basic Research in Science. This paper is dedicated to the memory of Lenny Wilson.

APPENDIX This appendix serves three purposes: (1) the description and analysis of two additional

mating-preference functions; (2) the analysis of the stability of the equilibrium for the means z and y, and (3) discussion of a model of the evolution of the genetic variances and covariance.

Mating-preference functions. -Lande (1981) proposed two mating-preference models in addition to the absolute-preference model treated in the main body of this paper. These are the psychophysical and relative models of mate choice. Under the assumption of psychophysical mate choice, the sensitivity of a female to differences in the male phenotype depends upon her value of y, and preference is a monotonic function of z:

1(z y) oc exp(yyz) (A l) where the parameter y is a constant. This then is a model of open-ended preferences in which a female's strength of preference increases with more extreme development of the male trait. In the relative mate-preference model, females prefer males differing from the average male by an amount y:

4(z y) a exp{ - [(Z* + y) - z]2/2V2}. (A2)




Males falling in the range (Z* + y) + v are highly preferred by y females over other males that have either greater or lesser development of the male trait.

Proceeding as before, the fitness functions for the male trait and female preference are:

Wm(z) oc exp I- 2 ? J1 + ?) ( ? J2 ? (A3a)

Wf (y) oc expF ckr*2K LL(_ - 2a ) - *2 + 2KJ1 (A3b)

? Ca*2 2(do*2 + Z*)K1 + (1 - 2aa*2)K2 L(I - 2aa* )(I 2av* + 2K~f*2r

where

= a ? Y2(1 - 2(a *2 + C2a*2T2)

J2 d + cy(l - 2au*2) _ C2T2(d*2 + Z*) 2(1 - 2au*2 + C2?*2T2)

and K1 and K2 are as defined earlier. In psychophysical mate choice, a = d = 0 and c = for relative mate choice a = - 1/2vA, C = 1/V2, and d = Z*/V2; and under absolute mate choice a = - 1/2v2, C = 1/V2, and d = 0. All other variables are as defined earlier. If male reproductive potential is limited, then the equilibrium is

Z = onl (A4a)

y = 0 (A4b)

for both psychophysical and relative mate preference. When male potential is unlimited, the equilibria are

z = Of (A5a)

= (of- 0t?)1(Yw2) (A5b)

for psychophysical mate choice, and

z= Of (A6a)

Y = (Of - Om)(2n + V22 ?+ a W (A6b)

for relative mate choice. Just as was found for absolute preference in the main body of this paper, these two additional mate-preference models show that the evolutionary equilibrium maximizes the fecundity of females.

Stability of the means.-To determine the conditions under which the equilibrium of the means will be unstable, I performed a standard stability analysis of eqs. (10). This approach treats the genetic variances and covariances as fixed parameters, which vastly simplifies the analysis. Because of this simplification, the resulting conditions for stability are necessary but not sufficient. Conversely, the conditions for instability are sufficient but not necessary; if they are met, the equilibrium is guaranteed to be unstable.

The equilibrium of eqs. (10) will be stable (both locally and globally) if and only if IXo + 11




< 1, where Xo is the largest eigenvalue of that system of equations. The eigenvalues of eqs. (10) are given by

I z -XA A z Ay AY - a

(A7) =2 - (Gzx1 + Bx2 + Bx3 + G,,x4)/2

? [(Gx I + Bx2)(Bx3 + Gyx4) - (BxI + G,,x2)(Gzx3 + Bx4)]/4 = 0,

where

XI= (S) = 2a(? + 1) - I/wtt, X2= a = 2ac

X3= (S) = C( + 1), and X4= ( = C_

The selection differentials are conveniently computed using the fact that S, = a2x(B + 2AX)/(1 - 2A-2r), where the character under selection has mean x and variance a , and the fitness function is of the form W(x) a exp(Ax2 + Bx). The solutions for X are then

X = f(B) - (GG - B2)(XX4- X2X34 (A8)

where

f(B) - 1 (GzxI + Bx2 + Bx3 + Gyx4). 2

The larger of the two values is XO, the eigenvalue of interest. This may be either a real or a complex number. If XO is real, which occurs when [f(B)]2 - (GzGt, - B2)(x x4 - X2X3), then the conditions for instability are:

B> - GzxI + ?5,X4 (A9a) X2 + X3

xIx4 - x2x3 < 0. (A9b)

When either condition is met, a runaway process will be initiated. If the eigenvalue Xo is complex, which occurs when [f(B)]2 < (GG,, - B2)(x x4 - x2x3), then the condition for instability is

f(B) + ? (GzGA - B2)(x x4 - x2x3) > 0. (A9c) 4

The conditions for instability (A9a) and (A9c) are met when the genetic covariance B becomes sufficiently large. Condition (A9b) shows that instability can result even without involvement of the covariance, but this condition appears to be unattainable under the assumptions of weak selection used in these models.

Since the analysis leading to inequalities (A9) for the instability of the means treats the genetic variance/covariance structure as constant, it is expected that the equilibrium




actually is unstable under less stringent conditions because of the possibility that the variances and covariances themselves may become destabilized.

A model of the genetic variances and covariance. -Detailed models of the evolution of the genetic variance structure of the male trait and mate preference can be used to investigate if and when the nonrandom mating of sexual selection alone will be sufficient to generate a genetic covariance B that will destabilize the means and start a runaway. The model I use here is derived from that of Lande (1977). Recent work by Turelli (1984) calls into question the validity of some assumptions on which Lande's model for the evolution of the genetic variances and covariances is based. At present, however, it is the only model on which the appropriate calculations for the problem at hand can be performed. Until this controversy is resolved, the following model should be considered a qualitative heuristic guide.

Lande's basic model is of an arbitrary number of additive autosomal loci whose genetic effects are at the outset of each generation assumed to be distributed as multivariate normal. This model can be extended (Lande 1981) to situations involving two sex-limited characters evolving in a dioecious population, which is the case of interest here.

The loci involved are numbered consecutively, with loci 1 to m coding for the male trait, and loci m + 1 to n coding for the mating preference. The genetic variance at locus i is incremented in each generation by an amount ui which results from mutation. The recombination rate between loci i and j is denoted rij.

The variance/covariance structure is specified by 2n(n - 1) covariance variables. These are the covariances of allelic effects at loci i and j sampled from the same individual, and are written Cij when the alleles originated from the same parent (maternal or paternal), and C!j when the alleles came from different parents.

These quantities in turn determine the genetic variances G_ and G, and the covariance B according to the relations

In

G, = 2 Ciz (AlOa)

GI, = 2 Civ (AlOb) i=m+ I

tZ n

B = 2 Ciz =2 Ci, (AlOc) i=o in ?

where Ci, and Civ are the coefficients of regression of allelic effects of locus i on the phenotypic variables z and y:

iz (Ci + C) (AI I a) j=1

,= (Ci + C). (AlIb) j= +l

The overall phenotypic variances are & = G, + Ez for z and 72 = GI, + E), for y. When gametes combine to form zygotes, the distribution of genetic effects in the zygotes

is the sum of the distributions of effects in the sperm and in the ova. Because selection acts differently on the two sexes, the distributions from these two types of gametes are not identical, and so the resultant admixture is not multivariate normal. The new distribution can, however, be approximated by a normal with the same variances and covariances as the admixture. The approximation will be valid when selection is weak, i.e., when W.,, 4r V >> a, and y& ? 1.

To determine the appropriate variances and covariances we will need a general formula for the equal admixture of two bivariate normals. Let one of these distributions have means




2t4' and J7', variances c, and Cq'i, and covariance cT.j. Corresponding terms for the other contributing distribution are written by replacing mn with The sum of the two distributions can be shown directly to have variances and covariances

C,1 - 1 (cZ + c$) ? - - -I )(1 J - f) (A12) 2 4 ~

where the variances are the special cases Cii and Cjj. This result is used to modify eq. (1) of Lande (1977) to give the basic recursion equation

for the genetic covariances

cij(t + 1) = (1 -ri)L+ (c{T + c!/) + + (Xi - if)(jf -

(A13) + r (cg.' + cj') + (]jo1 - 1f) xfi) jf)I Ili

2 ~ ~ ~~4 -J -U

where aij is defined as 1 if i = j, and 0 otherwise. The terms Xi? and serf are the means of genetic effects at locus i in successful gametes coming from males and females, and are given by

xli?, = 1C S5, -Jff = Ci8, S...2(A14)

The terms cT. and cf are the covariance between alleles at loci i and j in gametes coming from males and females, respectively, where the allele involved came from the same parent in the previous generation. The terms cij' and c{1. have a similar definition, except the alleles come from different parents. These variables have the values

cT= Cij- k- CizCjz ct - Cij- k, ci2cjy (A15a)

cT'= Cij- k- CZ~C1. uf] = (Al5b)

where k, and k. are the proportional changes in the phenotypic variances of y and z caused by selection:

(T***2 T***2

kz = 1I- 2 9 kY_ = 1 _, (A 16)

The triple-star superscripts denote values in adults following fecundity selection. These definitions can be inserted into eq. (A13) to find that the per-generation change in Cij is

ACij = ij(Cij - C () -CizCj? + Cj,,Cj, Dij (A17)

where

=4 ( ;

T2 - 2 Ci)( ,

- -; cT).

This equation specifies half of the required recursion relations; we now need a similar expression for Ci.

Again making use of the generic result of eq. (A12) we have

Ciy(t + 1) = - r[c!(t) + cj'j(t)] + -r(1*" - X- - (A18) 2 1j4 1 - jfl)] A18




Here c!i is defined to be the actual covariance between locus i in a sperm and locus j in an ovum that unite. (Note that by this definition cf, $A cj'.) The covariance can only be due to the correlation between the male's phenotypic value z and the female's phenotypic value y. This implies (Kendall and Stuart 1979, eq. [27.15]) that

CizciY Cf= 2 P (A19)

where P* is the phenotypic covariance between z and y in mated pairs that successfully raise offspring. This is not equal to P, the phenotypic covariance between all mated pairs, because of fertility selection. A regression argument can be used to show the relation between the two is

-s = a (A20)

where the double-star superscript denotes variances following sexual selection but before fecundity selection. To determine the covariance P we examine the distribution M(z,y) of the phenotypes of mated pairs:

M(z,y) = p*(z)q(y)+*(zjy)

c exp[(a - 1/2(&2)Z2 - + (1/T2 ? 1 -2U*2 Y (A21)

/ - 1 -2a

?1y C(d&F*2 + z,) I ~ + ( *2 d)z + 2 1 - 2 Y + CZY]

This is a bivariate normal distribution. When written in the general form f(z,y) = k exp(Az2 + By2 + Cz + Dy + Ezy + F), such a distribution has means z = (DE - 2BC)/ (4AB - E2) and y = (CE - 2AD)/(4AB - E2), variances ci) = - 2B/(4AB - E2) and (J. = -2A/(4AB - E2), and covariance o>Z = MI - (E2/4AB)]. The desired covariance is therefore

P C. (A22) Equation (A18) can now be written

P* Cbj(t + 1) = (CizCjC + Cj>Civ) + Dij. (A23)

This completes the derivation of the genetic model for the variances and covariances. The stimulus for constructing this genetic model was a desire to determine sufficient

conditions for the genetic covariance to become sufficiently large to destabilize the mean male trait and mate preference, and thereby initiate a runaway process. I have been unable to determine analytically the equilibrium for the recursion equations in order to do this. The equations, however, can be iterated by computer in order to learn something about their behavior. I did this using a symmetrized version (see Felsenstein 1979) of eqs. (10) for the means and eqs. (AlO), (Al 1), (A17), and (A23) for the variances and covariance. These simulations revealed that if the mutation rate of loci coding for the female preference is sufficiently great relative to that for the male trait, then the equilibrium for the means can become unstable and a runaway process will result. Lande (1981) came to the same conclusion for his model of sexual selection, which is the same model treated here except that his lacks the fertility selection component. He was able to show analytically that relatively high mutability of the preference will be destabilizing, and so the present results are in accord with findings from a similar model. The simulations also show that the variances and covariance themselves can become unstable. The conditions for instability of the full model are therefore somewhat broader than those for instability under the assumption of constant variances and covariance, as was stated earlier.




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Evolution of Female Choice and Male Parental Investment in Polygynous Species: The Demise of the "Sexy Son"