Proc. Natl. Acad. Sci. USA Vol. 80, pp. 5931-5935, October 1983 Evolution

On the optimal sex ratio (optimality/evolutionary stability) SAMUEL KARLIN AND SABIN LESSARD Department of Mathematics, Stanford University, Stanford, California 94305 Contributed by Samuel Karlin, June 3, 1983

ABSTRACT The equilibrium structures for various multial- is observed that a symmetric equilibrium state corresponds to lele sex-determining genetic models of panmictic populations with an equilibrium of a classical one-sex multiallele viability model sex expression depending on the genotypes of either the zygote or with corresponding stability conditions. In Section 3 we intro- a parent are described. Even-sex-ratio equilibrium surfaces can duce a spectral radius functional that plays a fundamental role exist apart from equilibrium points having coincident male and in discernment of the existence and stability nature of the sym- female allelic frequency sets. The latter (symmetric) equilibrium metric versus even-sex ratio equilibrium types. Various opti- states entail biased sex ratios. A stable symmetric equilibrium and mality properties of the even-sex ratio equilibria are described an even-sex-ratio equilibrium segregating the same alleles cannot in 4 coexist. The tendency to evolve toward an even-sex ratio is em- Section and buttressed by numerical and simulation runs. bodied by the following optimality property: starting from an equi- The basic model for sex determination at an autosomal mul- librium having a biased sex ratio, mutant sex-determining alleles tiallele locus in diploid populations with sex expression de- can accumulate only in the case in which in the augmented system pending on the zygote genotype serves as a prototype for our all attainable equilibrium states have a sex ratio closer to one to analysis. Identical qualitative results are obtained for models one. with maternal control of brood sex ratio determined at a mul- tiallele autosomal locus. The same framework subsumes the de- 1. Introduction terminants of sex allocation to seed and pollen functions in plant populations. Environmental influences on sex expression and The evolution of the sex ratio has been considered from many fitness differences between the sexes can also be incorporated. perspectives. Although anomalous brood sex ratios, especially The corresponding models for population sex ratio evolution in in certain insect populations, have been documented (1), the haplodiploid species can be similarly dealt with. A detailed dis- predominance of the one-to-one sex ratio in diploid populations cussion of these models, complete proofs, further results, and is striking and unambiguous. The propensity toward an equal interpretations are elaborated in ref. 13. representation of males and females was understood by Fisher (2) in terms of the reproductive advantage for the rarer sex. In 2. A multiallele sex-determination model and some this vein, a number of behavioral models based on the precept preliminary results of parental expenditure that can result in an even-sex ratio in Consider a bisexual infinite, population with r possible alleles panmictic populations have been formalized (3, 4). Further Al, A2, ..., Ar at an autosomal locus primarily responsible for maximization principles founded on the idea of reproductive sex determination. We denote the frequency of genotype AiAj value have been used to predict the optimal sex ratios under in the female population by 2pyj when i $ j and by pii when i general constraints on sex allocation (5, 6). = j. The frequency of allele Ai is pi = lj=1 pt,. The quantities Sex determination governed at an autosomal locus with prob- 2qij, qii, and qi are defined analogously with respect to the male abilities of being male or female depending on zygote genotype population. We assume discrete generations, random mating, was formulated theoretically by Shaw (7) more than 20 years Mendelian segregation, and equal fertility for all mating types. ago. A two-allele model was studied by Eshel (8). Variant models Let my be the probability for an AiAj individual to be a male, (9-11) allowing for maternal (or paternal) genotype control on and concomitantly 1 - my1 that of being a female. Clearly, 0 ' brood sex ratio led to similar qualitative conclusions in favor of = mji < 1. We refer to M = Ijmyijj i=1 as the sex-determi- an equal representation of the sexes. Eshel and Feldman (12) nation coefficient matrix. considered generalizations to multiallele cases. Two classes of The case 0 < mrj < 1 may reflect the effects of modifier genes equilibria were pointed out: symmetric equilibrium states that coupled to prenatal or neonatal interactions. Where mij = 1 or exhibit the same allelic frequencies in male and female pop- 0, the sex phenotype is determined such that the collection of ulations and phenotype equilibria characterized by an even-sex all genotypes AjAj partition into two groups, 'm and SF, where ratio. These authors dealt with the problem of the initial in- every individual of type AM and 'F iS unambiguously male and crease of a mutant allele introduced at a symmetric equilibrium female, respectively. We refer to this situation as dichotomous state. (exact) genotypic sex determination. Corresponding multitype In this report we present complete conditions circumscrib- decompositions are appropriate for studies of incompatibility ing the existence and stability nature of both types of equilib- systems and in-elucidating consequences of nonrandom mating ria. Section 2 reviews the formulation of the sex determination patterns. model and sets forth the dynamical equations connecting allele For the general model, we find that the genotype frequen- frequencies over successive generations. Direct inspection of cies over two successive generations obey the recursion rela- the equations reveals the two possible classes of equilibria. It tions my(piqj + pjqi) , (1-my)(piqj + pjqj) The publication costs of this article were defrayed in part by page charge PY = payment. This article must therefore be hereby marked "advertise- 2w 2(1-w) ment" in accordance with 18 U.S.C. §1734 solely to indicate this fact. ij = 1. r, [2.1] 5931 Downloaded by guest on October 1, 2021 5932 Evolution: Karlin and Lessard Proc. Natl. Acad. Sci. USA 80 (1983) where w = Y= m~piqj is the proportion of males, the sex These requirements guarantee that every symmetric equilib- ratio, in the total population for the given allelic frequency state. rium state is an isolated fixed point that does not yield an even- If male and female (but genotype-independent) viabilities act sex ratio and whose stability can be decided by a local linear in the ratio c to 1, the transformation equations 2.1 are unal- approximation. Such an analysis yields a correspondence of sta- tered, although the adult sex ratio differs from the juvenile sex bility properties for the symmetric equilibria in the sex-deter- ratio, being given by cw/(cw + 1 - w) compared to w. mination model and the same equilibria in the associated via- It will be convenient to employ the Schur product operation bility model. of two vectors x = (xI, x2, ..., xr) and y = (Y1, Y2, , Yr) sym- RESULT I. A symmetric equilibrium {p~p} with w- = (pMp) bolized by x o y = (x1yI, X2y2,.X. , XrYr), and the inner product <'/2 is stable in the sex-determination model (Eq. 2.2) if and denoted by (x,y) = 1:.=, xiyi. Further, if M is any matrix the only if A is stable for the one-locus viability matrix M of Eq. representation x o M stands for the matrix product D1M, where 2.3. D., designates the diagonal matrix with entries xI, x2, . , XXr down Analogously, if w > 1/2, {pp} is a stable symmetric equilib- the main diagonal. In this notation, the above system can be riumfor the sex-determination model if and only if A is stable converted into the following recurrence equations for the fre- for the viability matrix U - M. quency vectors p = (Pi, P2, Pr) and q = (qI, q2, qr): We refer to refs. 15 and 16 for some explicit conditions of poMq+qoMp stability in standard one-locus viability models. They are ex- actly opposite for M and U - M. 2w + 3. A characterization for equilibrium and stability in , po(U-.M)q qo(U-M)p [2.2] standard. one-locus multiallele viability models p-2(1 - w) where w = (p,Mq) = Ei=I pi(Mq)i = StYj=l mypiqj, and U de- This section, which is of some independent mathematical in- notes the matrix with all unit entries. terest, is more technical, but it is the key for a deeper analysis The transformation equations 2.2 represent an important case by establishing a functional link between the even-sex-ratio of the dynamical system describing the allele frequency changes equilibria and the symmetric equilibria. The biological reader for a two-sex viability model. Formally the sex-determination can skip this material without loss in understanding the re- model where an offspring of genotype AjAj is male (female) with maining sections and their qualitative implications. probability my ((1 - mij)) is equivalent to a sex-differentiated Eq. 2.4 can be viewed in the following manner. For each p viability model with viability matrices M = I|mIlI and F = Ill - in A0 (that is, for a frequency vector interior to the allelic fre- mIlI = U - M for males and females, respectively (14). The fact quency domain A) we form the matrix that M and F = U - M are generated from a single generic matrix M allows a more.tractable analysis of the system. B(p) = DPM + DMP. [3.1] An equilibrium of the sex-determination model consists of (Dp is the diagonal matrix with the components of p down the a pair of frequency vectors {p* q*} having p' = p = p* and q' main diagonal.) If the matrix M [and therefore B(p)] is positive, = q = q* in Eq. 2.2. Two classes of equilibria were pointed we know by the Perron-Frobenius theory (16) that the eigen- out in ref. 12: symmetric equilibria exhibiting identical allelic value of largest magnitude for B(p) is positive, admitting a unique frequencies in male and female populations and even-sex-ratio (apart from a scale factor) strictly positive right eigenvector y equilibrium states entailing a sex ratio w* = (p*,Mq*) = 1/2, > 0. We denote by p(p) the spectral radius of B(p) that is the that is, with an equal.representation of males and females. This eigenvalue oflargest magnitude of B(p) and take the associated classification follows.from the equilibrium relationship (2w* - positive eigenvector y satisfying X:,=1 y, = (y,u) = 1. Then y 1)q* = (2w* - l)p*. = y(p) and p(p) vary continuously (actually analytically) as a The following facts are fundamental to our later analysis. A function of p in AO. The principal eigenvalue-eigenvector iden- symmetric equilibrium {p,i'} of Eq. 2.2 (abbreviated by A) must tity is verify To = p, where = p o Mp B(p)y(p) p(p)y(p). [3.2] Tp,= [2.3] Comparing with Eq. 2.4, we see that an even-sex-ratio equi- (p,Mp), librium represented by {,'p}, having both q and p interior to that is, T is the transformation for the classical one-locus mul- A, exists if and only if p(p) = 1, and then the corresponding tiallele viability model with M as a viability matrix (14, 15). An y(p) is 4. These conditions can be extended by continuity al- even-sex-ratio equilibrium state {q,p} is characterized by the lowing p and q on the boundary of A. relationship It is remarkable that the same spectral function characterizes q= ooM4 + oMo. [2.4] the equilibria of the viability model (Eq. 2.3). In the following 'Note that this relationship entails a one-to-one sex ratio. Sum- result, the matrix M is assumed to satisfy the genericy condi- ming over all coordinates, we have 1 = (u,qi) = 2(p,Mq), where tions. u is the vector of all unit components and therefore wb = (p,Mq) RESULT II. Let p(p) be the spectral radius of the matrix B(p) = 1/2. However, the latter equation alone is not sufficient for as defined in Eq. 3.1 for every p in A (the simplex offrequency equilibrium. vectors). To avoid unimportant technical details we assume hence- (i) p* is a polymorphic equilibrium of M (i.e., an equilibrium forth, unless stated otherwise, that M satisfies the following ge- of Eq. 2.3 with all positive components) if and only if p* is a nericy (nondegeneracy) conditions: (a) 0 < myj< 1 for all i and critical point of thefunction p(p), meaning that the derivative j; (b) every principal submatrix of M is nonsingular; (c) for every of p(p) in every direction relative to the simplex A is zero at p*. equilibrium p- of Eq. 2.3 (ii) An equilibrium p5 of M (interior or on the boundary ofA) and is a strict local maximum (i (*,MO) 54 1/2, is stable if only if p(Q5) of p(p) r with respect to A. maxima (ii) (Mo)i = 2 muj (p,Mp), where = 0. (iii) The localmaxima of p(p) in A are twice the local j=1 of(pMp) that are.achieved at the same points. An interior local Downloaded by guest on October 1, 2021 Evolution: Karlin and Lessard Proc. Natl. Acad. Sci. USA 80 (1983) 5933 maximum is actually a unique global maximum. Table 1. Comparisons between the occurrences of symmetric and These characterizations make possible a precise description even-sex-ratio equilibrium states of the equilibrium structure of the sex-determination model as - Number of alleles we duly elaborate. -- 2 3 4 5 6 4. Properties of even-sex-ratio equilibria of the sex- Convergence to determination model 1:1 sex ratio 0.51 0.69 0.83 0.90 0.93 The basic results dealt with in this section concern the existence Proportion of matrices leading to: of even-sex-ratio equilibrium states along with possible coex- (i) 1:1 sex ratio only 0.39 0.55 0.72 0.74 0.85 istence of stable symmetric equilibria. Note first that even-sex- (ii) symmetric ratio equilibria make up a continuum when present, contrary equilibria only 0.36 0.17 0.06 0.02 0.02 to isolated symmetric equilibrium points (in generic cases), as (iii) either 0.25 0.28 0.22 0.24 0.13 indicated in the following result. RESULT III. (i) Even-sex-ratio equilibria of the sex-deter- two-allele case has been obtained by analytical calculations.) It mination model (Eq. 2.2), when they exist, are part of pairs of should be noted that, as the number of alleles increases, even- equilibrium surfaces of dimension r - 2 in the simplex A of sex-ratio equilibrium surfaces are-more likely to occur and be dimension r - 1 for the allelicfrequencies in the male andfe- attainable. Because the existence of such surfaces precludes male populations, respectively. the existence of an interior stable symmetric equilibrium, the (ii) A stable symmetric equilibrium (interior or on the bound- domains of attraction of the symmetric equilibria are expected ary) that is globally stablefor the corresponding viability model to shrink (and do) with an increase of dimension (i.e., with more (see Result I) cannot coexist with any even-sex-ratio equilibrium allelic variants involved in sex determination). and any other stable symmetric equilibrium in A. Result IV below further amplifies the predominant endow- (iii) There may coexist one or several pairs of even-sex-ratio ment of even-sex-ratio equilibrium realizations under sex-ratio equilibrium surfaces with stable boundary symmetric equilib- genotypic determination involving extreme allelic variants -i.e., ria. when the sex phenotype is controlled at -a single multiallelic (iv) Two symmetric equilibria Pa and p3 with associated sex locus with exact genotype (zero-one) sex ascertainment. ratio Wa < 1/2 and We > 1/2, respectively, are completely sep- RESULT IV. Suppose all genotypes involving r alleles at a sin- arated by at least one pair of even-sex-ratio equilibrium sur- gle locus divide into two groupings (all genotypes in ;M become faces. males unambiguously, while all genotypes in OF become fe- An analytic formula for the surface (or surfaces) of even-sex- males). Assuming random mating in such a dioecious popula- ratio equilibria in the female population is described as the set tion, the only possible stable equilibria (or equilibrium sur- of all p in A satisfying faces) entail a one-to-one sex ratio-i.e., thefrequency ofa stable equilibrium phenotype representation must satisfyfreq(%M) = freq(8F) = 1/2. 9p(p) = det (I Pkmkj) Sji + pimyi =0 [4.1] It is interesting to contrast Results III and IV. When the ge- ~~~i~jk~~~~~~l notype-phenotype classes of sex determination are not abso- (Sij = 1 if i = j; 0 when i # j). Note that the determinant p(p) lute (that is, 0 < m# < 1), then the possibility of a stable non- is an algebraic function of degree r in the variables (pi, P2, , even sex ratio can emerge. Result IV asserts that with exact Pr). By symmetry a similar formula stands for the even-sex-ratio genotype sex determination (mu = 0 or 1 for all ij) engendering equilibrium frequencies in the male population. The exact pair- the sex dimorphism of the two phenotype classes with 'gM con- ing is defined by Eq. 2.4. sisting of all genotypes with m, = 1 and. OF9 consisting of all Result III precludes the existence of isolated even-sex-ratio genotypes with mo = 0, only an even-sex-ratio outcome can be equilibria with three alleles or more in generic cases. They stable for such a randomly mating population. compose pairs of continuous unidimensional curves of equilib- ria in the three-allele case and pairs of multidimensional hy- 5. Optimality of one-to-one sex-ratio realizations persurfaces with more alleles. Note that if there exists a stable symmetric polymorphic equilibrium p*, then no even-sex-ratio The concept of the evolutionary stable strategy (ESS) intro- equilibrium can exist and p* is the unique stable equilibrium, duced by Maynard Smith and Price (17) has been cogently ap- for the sex-determination model. It must be emphasized that plied in many studies of behavioral genetic models, including stable symmetric equilibrium states and even-sex-ratio equi- sex-ratio evolution (6), genetics of altruism (18), and optimal libria can coexist, but then the symmetric and unsymmetric dispersal rates (19, 20). In a population genetic approach, Eshel equilibrium states do not segregate the same set of alleles. and Feldman (12) conjectured that an even-sex ratio is optimal A variety of numerical runs and analytic studies of special (they call it evolutionarily genetically stable). In the framework classes in the sex-determination model provide evidence that of the sex-determination model of r alleles (Eq. 2.2), the op- each pair of even-sex-ratio equilibrium surfaces is strongly sta-- timality property is understood to mean that with the intro- ble to the extent that, for a perturbation off an even-sex-ratio duction of a new allele from r to r + 1 alleles, all possible stable equilibrium, iteration of Eq. 2.2 returns to an even-sex-ratio equilibrium states for the extended model cannot attain a sex equilibrium belonging to the same equilibrium pair of surfaces ratio farther from one to one than exists in the r allele subsys- generally geometrically fast. In order to contrast the extent and tem. This expresses an evolutionary tendency toward an even- domain of stability between symmetric versus even-sex-ratio sex ratio. equilibrium states, we constructed random sex-determination In this framework, consider a symmetric equilibrium {p*,p*}, matrices (each entry independently uniformly distributed on pt > 0, i = 1, ..., r for the r allele model (Eq. 2.2) with w* [0,1]). In each case we iterated the transformation (Eq. 2.2) from = (Mp*,p*) < '/2. With a new allele Ar+i, let mir+ I be the a sample of random starting points until an equilibrium was probability for a zygote of genotype AiAr+i to be a male and achieved. The main results are summarized in Table 1. (The (1 -mi,r+ l) to be a female, respectively. The marginal fraction Downloaded by guest on October 1, 2021 5934 Evolution: Karlin and Lessard Proc. Natl. Acad. Sci. USA 80 (1983) of male progeny carrying allele Ar+I at the equilibrium state the proportion of haploid offspring is kept constant, say a, = X r+i. p* is W*+1 I Pe',i A symmetric equilibrium {p*,p*} maintained by exogenous environmental and genetic influ- stable for the r allele sex-determination model becomes unsta- ences and the relative viability of haploid to diploid offsprings ble with the introduction of Ar+il if and only if is s to 1, then the sex ratio tw = min{O,(1 - a - 2as)/2(1 - a)} among diploid offspring [leading to an overall sex ratio (1 - a)/2(1 - a + as) at maturity when a - 1/(1 + 2s)] is optimal. w*+ l > w* provided w* < - . [5.1] Using our previous notation, the equations connecting the al- lelic frequencies over two successive generations are (The condition for local instability at p* is < w* if w* > w*+1 (1 - a)[p o Mq + q o Mp] + 2asp 1/2.) These conditions do not require that the marginal sex ratio I w*+I at p* should be closer to 1:1 than w*. However, in case q 2(1-a)w + 2as of departure, it was conjectured that the augmented system in p o (U - M)q + qo (U - M)p the long run would attain a sex ratio closer to one to one than pI = [6.1] to the previous equilibrium. 2(1 - w) We can prove the following facts. RESULT V. Let p* be a stable symmetric equilibrium of Eq. The model of Eq. 6.1 is based on the assumption that the 2.2 with sex ratio w* = (p*,Mp*), which becomes. unstable after gametic output contributed by a diploid mating and that con- the introduction of a new allele Ar+ 1 Then, for the augmented tributed by haploid fertilization are equal. An alternative model r + 1 allele system with nondegeneracy conditions inforce either considers that the gametic output from diploid matings is twice (i) there exists a unique stable symmetric equilibrium whose sex that produced by haploid fertilization. In this perspective the ratio is closer to 1:1 compared to w* and that does not coexist equations of 6.1 are modified, replacing 2a with a as the coef- with any even-sex-ratio equilibrium or (ii) {p*,p*} is enclosed ficient of sp. In this latter model the optimal sex ratio is again by a pair of even-sex-ratio equilibrium surfaces containing no 1:1 at maturity independent of a as long as a s 1/(1 + s). stable symmetric equilibria. (v) Multiallelic sex-ratio-determination models under mater- In the conditions of Result V, the only stable equilibrium nal genotypic control. (See ref. 12.) With maternal control of points attainable from p* entail a sex ratio closer (compared to brood sex ratio (in haplodiploid as well as diplodiploid popu- w*) to 1:1 or equal to 1:1. Our numerical iterations (Section 4) lations), the progeny sex ratio of an AjAj mother is assumed to have shown convergence to either a symmetric equilibrium or be mi,:(1 - miy) for males to females. In our notation, and in- an even-sex-ratio equilibrium, in agreement with this theme. troducing z = (zl, . Zr) with zi = jr=1 m#py, the recurrence The paths of the sex-ratio values over successive generations equations are may exhibit small oscillations, but in the long run they always =2 + p, _2 p-z tend (increasingly or decreasingly) in the direction toward 1:1. 2 2w' 2 2(1 -w)' 6. Variants and extensions (p - z)oMq + qoM(p - z) z = [6.2] For completeness, we list a series of models analytically close 2(1 - w) to the basic sex-determination model (Eq. 2.2) that lead to sim- for diplodiploid populations, and they differ only by q' = z/w ilar qualitative results. for haplodiploid populations. These transformations share the (i) Sex allocation of resources to female and malefunctions dynamical properties described for Eq. 2.2. determined at an autosomal locus. (See ref. 6.) A sex allocation With relative viabilities c to 1 for males and females, re- formulation more appropriate for simultaneous hermaphro- spectively, from conception to maturity and replacement of lost dites, as occurs in plants or fish, requires only a reinterpre- offspring resulting in equal brood size at maturity normalized tation of the sex-determination parameters my as the relative to one for convenience, the progeny of an AjAj mother must proportions of fixed individual resources allocated to male as satisfy (ref. 9) against female function. (ii) Sex determination at an X-linked locus with multiple al- nucmij + nu(1 - my) = 1, [6.3] leles. A dominant male- (or female)-determining factor (as oc- where ny, is proportional to the number of offspring produced. curs in standard XX/XY systems) can be accommodated by Eq. Our results can be applied at maturity with the matrix M = 2.2 if we set mu = 1 for allj. In such a case, it can be shown 11fit'1 where m = n-cm I By extrapolation the 1:1 sex ratio at that the strong sex determiner represented by allele A1 can be maturity corresponds to a 1: c sex ratio at conception. The pa- maintained at equilibrium only with an even-sex ratio. rameter c can also be interpreted as a relative cost of a male (iii) Sex ratio evolution when male versusfemale expression compared to a female offspring, and then Eq. 6.3 connotes a varies spatially. (See ref. 21.) Assume that a population is sub- constant total expenditure per family. divided into N demes %, v = 1, ..., N, corresponding to N different environments where sex determination is genotypi- 7. Discussion cally governed by M(^" =-Im(tII, v = 1, ..., N. If random mating occurs in a common area and subsequently, a fraction c, of the The variety and complexity of sex-determining systems and population settles in deme 26, (c,, > 0, X= cv = 1), then the controls are manifold. These can be classified into seven modes: transformation given by Eq. 2.2 is in force with M = N= (i) one-locus multiallele autosomal or sex linked; (ii) multigene Such a reduction is usually impossible when male and female (and polygenic) with modifier gene effects; (iii) chromosomal fitnesses vary among demes, which leads to a general two-sex heteromorphism, including distinguished XY/XX, ZZ/ZW, or viability model. balanced XO/XX forms; (iv) hermaphroditism (simultaneous or (iv) Sex determination in haplodiploid populations. (See ref. sequential); (v) mixed parthenogenesis (e.g.,- haplodiploid sys- 22.) In a haplodiploid population suppose all haploid zygotes tems); (vi) environmental sex determination (e.g., influenced develop into males while diploid zygotes are subject to prob- by cytoplasmic milleux or endogenous conditions at birth); (vii) abilistic sex determination according to a matrix M = I1myII. When extrachromosomal factors (e.g., viral particles, contagion, and Downloaded by guest on October 1, 2021 Evolution: Karlin and Lessard Proc. Natl. Acad. Sci. USA 80 (1983) 5935 conditions fostering meiotic drive). In broad terms, there are reduction of linkage associated with homozygote lethals (dys- distinguished genotypic and environmental determinants sub- function)] there could evolve chromosomal sex determination ject to zygotic, parental, or population controls. Even com- entailing a 1:1 sex ratio. Although the Y chromosome is pri- pletely genetic controls of sex expression can be manipulated marily heterochromatin, it is conceivable that with respect to by hormonal and physiological covariates. sex determination it behaves as a large multiallele system, im- Examples of sex controls based on multiple alleles or loci in- plicating a 1:1 sex ratio in line with Result V. Alternatively, the clude the wasp Habrobracon (9-11 alleles), some poecilid fishes general model of exact multiallele multilocus sex determination (polygenic), platyfish (3-4 alleles), toad (Bufo bufo) (with both systems can serve as a transition stage from one sex-determin- male and female heterogamety), common housefly (Musca do- ing mechanism to mixed heterogamety. From chromosomal sex mestica) (involving chromosomal and autosomal loci determi- determination on many occasions adaptations of sex expression nation), the mosquito Aedes aegypti (involving multiple alleles to environmental conditions arose. This includes the advan- and modifiers), and the rodent wood lemming (3-4 alleles). tages of the haplodiploid state, in which maternal facultative or There are three principal perspectives on sex ratio evolution latent responses to environmental cues are more easily man- centering on (i) optimization and adaptive criteria, (ii) sex-de- aged. This sequence of events is consistent with the prevalent termining systems and controls, and (iii) physiological and be- view that haplodiploidy evolved from diploidy. The "optimal" havioral covariates. sex ratio as proposed in the models of this paper tends to be Fisher (2) proposed that the aggregate genetic contribution 1:1 for all these systems. Moreover, an even-sex ratio enjoys to the next generation of all females is equal to that of all males, the greater population stability against sudden changes in en- implying that members of the rarer sex individually contribute vironment, which can obliterate one or other of the sexes when relatively more than members of the more abundant sex. Many represented in small numbers. authors discuss aspects of parental strategies, short- versus long- Why is the sex ratio not 1:1 for many of the invertebrates? term fitness, individual versus population emoluments, and in- This can be partly explained in terms of the following factors fluences of patchy versus fine environmental conditions di- and patterns: (i) the effects of population subdivision accom- rected toward sex ratio controls. Their verbal arguments por- panied by differential gender selection over space and time, (ii) tend that many of these opposing considerations are best the consequences of inbreeding and life history strategies, (iii) reconciled by a 1:1 sex ratio. In most of these treatments the the superimposition of differential viability and fertility forces actual sex determination mechanisms (i.e., the underlying mul- on genotypic sex determination, (iv) extrachromosomal factors tiallelic or multiloci factors) are not explicitly considered. and meiotic drive, and (v) behavioral manipulations. An alternative point of view holds that sex ratios arise as an evolutionary concomitant of the sex determination system- This work was supported in part by National Institutes of Health Grant simply the consequence of the cytological machinery. For ex- GM10452-20 and National Science Foundation Grant MCS79-24310. the XY/XX system and the formal Mendelian rules help ample, 1. Hamilton, W D. (1967) Science 156, 477-488. produce and maintain a 1:1 sex ratio. This does not answer the 2. Fisher, R. A. (1930) The Genetical Theory of Natural Selection question: How did it evolve and for what reasons? Moreover, (Oxford Univ. Press, London). sex-dependent fitness components, variants of parthenogene- 3. Bodmer, W. F. & Edwards, A. W F. (1960) Ann. Hum. Genet. 24, sis, forms of hermaphroditism, and sex conversion may all af- 239-244. fect the sex ratio. 4. MacArthur, R. H. (1965) in Theoretical and Mathematical Biol- A 1:1 sex ratio seems to predominate in most animal species ogy, eds. Waterman, T. H. & Morowitz, H. (Blaisdell, New York), pp. 388-397. in the early stages of development, although a smaller number 5. Williams, G. C. (1979) Proc. R. Soc. London Ser. B 205, 567-580. of males would seem a priori more advantageous. Sex chro- 6. Charnov, E. L. (1982) The Theory of Sex Allocation (Princeton mosomes overwhelmingly induce a 1:1 sex ratio at conception, Univ. Press, Princeton, NJ). but heed the caveat that there are counterexamples (1, 6). Sex 7. Shaw, R. F. (1958) Genetics 93, 149-163. determination under two blocks of genes often shows a 1:1 sex 8. Eshel, I. (1975) Heredity 34, 351-361. ratio even with organisms. Also, selection of X- 9. Speith, P. T. (1974) Am. Nat. 87, 337-342. haplodiploid 10. Nur, U. (1974) Theor. Popul. Biol. 5, 143-147. linked genes that determine or effect the sex of offspring can 11. Uyenoyama, M. K. & Bengtsson, B. 0. (1979) Genetics 93, 721- produce an even stable sex ratio. 736. The foregoing observations are in accord with the theoretical 12. Eshel, I. & Feldman, M. (1982) Theor. Popul. Biol. 21, 430-439. results of this paper. Actual sex ratio, which is evolutionarily 13. Karlin, S. & Lessard, S. (1983) Theor. Popul. Biol., in press. "optimal" when modified by successive mutant genes, leads 14. Karlin, S. (1978) in Studies in Mathematical Biology, MAA Stud- to a sex ratio closer to this optimum. The 1:1 pop- ies in Mathematics, ed. Levin, S. A. (Math. Assn. of America, ultimately Washington, DC), Vol. 16, pp. 503-587. ulation sex ratio is optimal in this sense according to Result V. 15. Kingman, J. F. C. (1961) Proc. Cambridge Philos. Soc. 57, 574- This finding applies for exact or probabilistic genotypic sex de- 582. termination under zygote, paternal, or haplodiploid control. 16. Gantmacher, F. R. (1959) The Theory of Matrices (Chelsea, New There can be a multiplicity of male versus female genotype York), Vol. 2. determinations, but still an overall 1:1 phenotypic sex ratio tends 17. Maynard Smith, J. & Price, G. R. (1973) Nature (London) 246, 15- to result. The following evolutionary scenario may be sug- 18. 18. Charlesworth, B. (1978)J. Theor. Biol. 72, 297-319. gested. The earliest organisms (fishes and invertebrates) were 19. Motro, U. (1982) Theor. Popul. Biol. 21, 394-411. hermaphrodites that later specialized to genotypic controls with 20. Motro, U. (1982) Theor. Popul. Biol. 21, 412-429. sex expression. From single gene (or sets of genes) determi- 21. Bull, J. J. (1981) Heredity 46, 9-26. nations with the cumulation of mutations [multiple alleles and 22. 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