Behavioral Ecology Vol. 10 No. 2: 141–148 Cooperative breeding, offspring packaging, and biased sex ratios in allodapine bees Jaco M. Greeff Arbeitsgruppe Michiels, Max-Planck-Institute for Behavioral Physiology (Seewiesen), PO Box 1564, D82305 Starnberg, Germany, and Department of Zoology and Entomology, University of Pretoria, Pretoria 0002, Republic of South Africa It is not generally appreciated that positive kin interactions do not necessarily result in an evolutionarily stable (ES) skewed sex ratio. Stability depends critically on the sex of both the helper and receiver. When help is given within one sex only, no monomorphic ES strategy exists, and local resource enhancement (LRE) between offspring of one sex does not predict a sex ratio bias toward that sex. I developed a model to clarify and examine the sex ratio biases that may be expected under cooperative breeding. I found that LRE between cooperatively breeding female allodapine bees cannot explain their female-biased sex ratios. Allodapine females feed and protect brothers, which may stabilize the female-biased sex ratio, but the model shows this is not the case because benevolence to males is likely to decrease rapidly as the number of females increases. For small broods this helping behavior causes a female bias, but bigger broods could be sufficiently male biased to compensate the population sex ratio. Considering the fact that females need to be packaged into reproductive units (multifemale colonies), of which intermediate-sized units are the most productive, it is shown that fitness returns from females are in fact a wavelike function. This results in a rugged fitness landscape, which could explain the female-biased population sex ratios of allodapine bees as an adaptation to local fitness peaks rather than a global optimum. In behaviors where organisms have to package limited resources into integer numbers of units, the possible solutions are limited, and careful analysis is required. Key words: allodapine bees, class structured groups, cooperative breeding, packaging, reproductive value, sex ratio. [Behav Ecol 10:141–148 (1999)] L loyd (1986) argued that the fact that organisms often have to package their resources into discrete functional units can have profound effects on their allocation decisions. For instance, Charnov and Downhower (1995) showed that offspring size can deviate strongly from the optimal size when limited resources have to be packaged into small integer numbers of offspring. Here I consider how sex allocation is affected by cooperative breeding and how a similar packaging principle of integer numbers is important in determining optimal clutch sex ratios and fitness landscapes. In this case the packaging occurs at a higher organizational level—namely, that of a cooperatively breeding group. I specifically consider the allodapine bees, but a similar approach could be applied to other cooperative breeders where sex ratios are skewed. In a comparative study of allodapine bees, Michener (1971) showed that all but 2 of 23 species produced female-biased population sex ratios. Michener (1971) and Trivers and Hare (1976) argued that the excess females may simply be workers. This explanation would be satisfactory if the workers are the offspring of the bee they are assisting. This is, however, not the case because females neither help nor die during their mother’s tenure, but only exhibit these behaviors in and toward the next generation. Schwarz (1988) gave an alternative explanation for the female-biased sex ratios: that positive interactions among cooperatively breeding female kin lead to local resource enhancement (LRE), which favors a femalebiased sex ratio. LRE is envisioned as the flip side of local resource competition (LRC) between females, which favors a male-biased sex ratio (Clark, 1978). LRC occurs when daughters compete among themselves or with their mother and to Address correspondence to J. M. Greef, Department of Zoological Entomology, University of Pretoria, Pretoria 0002, South Africa. Email: greeff@mpi-seewiesen.mpg.de. Received 12 February 1998; accepted 10 May 1998. q 1999 International Society for Behavioral Ecology a greater extent than sons do. By investing less resources in daughters, the mother can reduce this competition. Recent reviews of sex ratios in eusocial insects (Bourke and Franks, 1995; Crozier and Pamilo, 1996) have accepted this LRE explanation for allodapine sex ratios. Seger and Charnov (1988), however, cautioned that LRE is not merely the mirror image of LRC and suggested that additional factors have to be incorporated if an evolutionarily stable (ES), female-biased population sex ratio is to be explained. I considered the problem formally and found that the proposed explanations are insufficient. My analysis suggests that the observations could be explained as a result of the fact that females have to combine into integer numbers of groups. Positive kin interactions and sex ratios General models Trivers and Willard (1973) were the first to appreciate that positive interactions between siblings could affect the sex ratio. They stated that when the individuals of one sex help siblings of the opposite sex more than they help siblings of the same sex, then the sex ratio will be biased toward the more benevolent sex. In contrast, Speith (1974) showed that when members of one sex increase the viability of siblings of the same sex only, then there is no monomorphic ES sex ratio. Toro (1981, 1982) showed that in such a case a mixed evolutionarily stable strategy (ESS) exists, with some of the parents producing males only and the remainder females only. Taylor (1981) provided a clear model of how kin interactions affect the sex ratio. In his analyses terms are separated in such a fashion that one can clearly see how interactions among and between sexes would be expected to skew the sex ratio. He suggested that positive interactions between kin of cooperatively breeding groups such as Florida scrub jays and among lion brothers may select for a skew in their sex ratios. This formulation may prompt us to wrongly equate positive interactions among kin of one sex into a bias in sex ratio 142 toward that sex. This intuitive prediction stands in contrast to the studies of Speith (1974) and Toro (1981). It is indeed true that Taylor’s (1981) model, applied to single-sex interactions, correctly shows that there is a biased sex ratio at which an additional daughter gives the same fitness returns as an additional son. This is, however, only the first requirement that needs to be met for a strategy to be optimal. It is still required to show that this point is a maximum and not a minimum; in technical terms, we still need to illustrate that the second derivative is negative. In fact, Toro (1982) proved with regard to single-sexed interactions that the point where a marginal son and daughter are of equal value is the worst strategy, being invadable by all others. In contrast, when offspring of one sex helps siblings of the other sex or both sexes, then the ES sex ratio would be biased toward the helping sex (Toro, 1982). In summary, the models show that positive interactions between sibs can result in two outcomes depending on whom receives the help: type 1—when help is provided to the same sex only, then no single ratio is stable and the population may be split into two, one part producing males only, the remainder females only. LRE is thus not just the flip side of LRC; type 2—when help is provided to the opposite sex or both sexes, a population-wide ES skew can evolve. Situations to which the first type of prediction is likely to apply are cases in which offspring of one sex group together as a selective unit. On the other hand, when offspring assist their parents, the second type of prediction is likely to apply because their help will not be sex limited. A restrictive assumption of these models is that the degree of help is a function of the sex ratio, whereas it may be more realistic to assume that the number of individuals of the benevolent sex is more important. This proviso will be discussed later in the context of the allodapine bee. Male coalitions Alexander and Sherman (1977) proposed that when brothers compete as a unit to secure matings and they are more successful as a pair than singletons, then their value will be enhanced with respect to singletons. More explicitly, Taylor (1981) argued that cooperative behavior among brothers could lead to a male-biased sex ratio. Along this line, Packer and Pusey (1987) argued that coalitions between males from the same cohort can lead to what they called local mate enhancement, which biases the sex ratios of certain cohorts toward males. The argument is as follows: male coalitions of three males are more successful at securing prides of females and are able to stay in control of such prides for a longer time than smaller male coalitions. Accordingly, litters composed of three offspring are more likely to be three males than expected if sex determination was by chance. This case is a type 1 problem, and we would expect that females producing smaller litters will compensate the male bias in the sex ratio by allocating more to females. This compensation in smaller litters is not born out by empirical observations and may suggest that additional factors may be at work. Helpers at the nest, repayment, and the cheaper sex Trivers and Hare (1976) suggested that helping at the nest could explain skewed sex ratios in cooperatively breeding birds, and Taylor (1981) made a similar prediction for Florida scrub jays. Malcolm and Marten (1982), working on wild dogs where males stay in the group and help to raise their parents’ offspring, framed the problem in terms of ‘‘helper repayment.’’ They argued that the male-biased sex ratio of wild dogs can be explained by the fact that males, by helping their parents to raise subsequent offspring, become the cheaper sex to produce. Gowaty and Lennartz (1985) coined the term ‘‘local resource enhancement’’ to explain a similar skew they ob- Behavioral Ecology Vol. 10 No. 2 served in red-cockaded woodpeckers. In this case, like the wild dogs, males are more likely to stay and help their parents. Emlen et al. (1986) developed a helpers-at-the-nest model, which gives the broad predictions expected. Two assumptions of their model are important to keep in mind. First, help given by the helper has to be less than the total support he received as an offspring. When this assumption is not met, the predicted skew once again becomes unstable again. Second, the product of male and female offspring is optimized. This implicitly assumes that all offspring are considered to be of equal value, yet, a male which helps its parents cannot be counted in the same way as a male or female who starts to reproduce. This male, even though it reproduces indirectly, is neutered to some extent. The class-structured approach suggested below can be applied to this problem and will not require this assumption. Lessells and Avery (1987) made important extensions to the helpers-at-the-nest model by incorporating help between various degrees of relatives. A counterintuitive prediction of theirs is that when brothers help each other, then a 50:50 sex ratio is stable. This seems to contradict, first, the skew observed in lions and in allodapine bees and second, the results of the general models listed above. They assumed that there is a monomorphic ESS that all individuals follow. Data (Komdeur, 1996; Komdeur et al., 1997; Packer and Pusey, 1987) suggest, however, that many animals bias their sex ratios facultatively to their specific conditions, and more complex models are therefore required. On average, however, Lessells and Avery’s equations 23 and 24 show that female genes receive just as much help as male genes, and the population sex ratio as a whole is not expected to be skewed. Studies on other taxa have given more support to the connection between a biased sex ratio and help given to relatives. Stark (1992) described a female-biased sex ratio in a carpenter bee where females assist their mothers, and Lambin (1994) argued that Townsend’s voles bias their sex ratios toward females in the spring because these females are more likely to cooperate with their mothers. Social spiders Frank (1987) developed a model to explain female-biased sex ratios in communally nesting spiders. Even though this problem seems similar to the above (Cronin and Schwarz, 1997), the cause of the skew is closely linked to the multigenerational and inbred nature of these nests. Colonies with a higher female-biased sex ratio early in the nest’s development can grow faster and can reach a mature (reproductive) stage quicker than colonies with less biased skews (Vollrath, 1986). As a result of this demographical effect of sex ratios, a female-biased ratio is favored in social spiders. The generality of this model to more simple life histories as is of concern here is hence restricted. Allodapine life history and sex ratios Social allodapine bees have one generation per year (Schwarz, 1994), and colonies consists of small groups of 1–10 related females (sisters and nieces; Blows and Schwarz, 1991; Schwarz and Blows, 1991; Schwarz et al., 1996, 1997). In early autumn bees eclose from their pupae, and any remaining females from the parental generation die. During this time one or two newly eclosed females forage and feed the remainder of their nest mates. There is division of labor in colonies at this time, and males also participate in nest modification tasks (Melna and Schwarz, 1994). During autumn only one or sometimes two females within each colony mate, and these females become dominant and will be the sole egg layers toward the end of the winter. Both males and females overwinter as adults. Greeff • Cooperative breeding and sex ratios 143 domen ventrally and blocking the entrance (Schwarz, 1986) or by using their sting (Schwarz, 1994) or a pungent secretion from their mandibular glands (Cane and Michener, 1983). This means that in the absence of any females, males may starve or suffer high predation levels. Allodapine bees are outbreeding (Blows and Schwarz, 1991), and sex ratio biases that can result from inbreeding is thus absent. In many species the wet weight of individual males and females are similar, and for the sake of simplicity I assume that the numerical sex ratio is an accurate reflection of the investment ratio in the two sexes. The sex ratios of allodapines from the genus Exoneura have been reported in detail and are female biased (Figure 1). All species show a marked correlation between brood size and sex ratio, with small broods being female biased. This bias decreases as brood size increases, and in E. angophorae the largest broods are male biased (Cronin and Schwarz, 1997; Schwarz, 1988, 1994). Figure 1 Sex ratios (proportion of males) in relation to brood size in two species of allodapine bee: Exoneura robusta (circles 5 newly found nests, squares 5 overwintering nests; 10–15 and 161; data from Schwarz, 1988) and Exoneura angophorae (triangles 5 all nests; 10– 14 and 151; data from Cronin and Schwarz, 1997). Dominant females produce a clutch of eggs in late winter. In spring some additional females become mated and function as secondary reproductives within the overwintered nests. By late spring, a further group of females leave their nests to cofound new nests. When new nesting sites are in close proximity to the parental nest, sisters are able to find each other by active kin recognition, and cofounder relatedness varies between 0.49 and 0.6 (Blows and Schwarz, 1991; Schwarz, 1987; Schwarz and Blows, 1991). When dispersal distances are long, siblings rarely encounter each other while initiating nests, and single founding occurs by default. In contrast to colonies in overwintering nests, all the females in newly founded nests are mated and contribute to reproduction (O’Keefe and Schwarz, 1990; Schwarz, 1986; Schwarz et al., 1987; Schwarz and O’Keefe, 1991). Per capita reproduction of colonies increases with colony size, peaks at intermediate size, and decreases after a threshold is passed. Females defend the nest by bowing their ab- A CLASS-STRUCTURED MODEL In species such as polygynous mammals, one can expect males to gain more benefits from being large than females would, because larger males can secure more matings. Hence, Trivers and Willard (1973) argued that mothers with more resources should produce sons rather than daughters and vice versa. Similarly, Charnov et al. (1981) argued that because parasitoid females benefit more from being larger than males, mothers should oviposit female eggs in bigger hosts and male eggs in smaller hosts (see Charnov et al., 1981, for more examples). The model derived here has an underlying analogy to these arguments and is based on a class-structured approach developed by Taylor (1990). A complete class-structured model has been derived for this problem, but because the same qualitative predictions are reached with this more simplistic model, I present the latter. To find the optimal clutch sex ratio I optimize the total kin value obtained through daughters and sons. The total kin value of individual Y to X is equal to the product of the reproductive value of Y and its relatedness to X (Hamilton, 1972; Ratnieks and Reeve 1992; see Table 1). Although we are certain about who Y is in this case, X could be the females laying the eggs or perhaps their helpers. Fortunately, in this case the ratio of the relatedness of sons to their mother and that of daughters to their mother is the same as the ratio for respective relatednesses for nephews and Table 1 Variables used in the model, their definitions, and related variables Variable Definition vi The average reproductive value of an individual of sex i, which is the probability that an allele drawn at random from a future generation descends from a specific individual of sex i The average reproductive value of a female in a nest of size i The total number of males in the population The total number of females in the population The proportion of females in nests of size i The reproductive value of all individuals in sex i, which is defined as the product of vi and the number of individuals of that class or sex. Grafen (1986) and Bourke and Franks (1995) and Crozier and Pamilo (1996), respectively, used Vi and vi to denote this value. The regression coefficient of relatedness of i to the focal individual, which is often denoted by b (Hamilton, 1972; Ratnieks and Boomsma, 1997) or by r (Bourke and Franks 1995), and which is equivalent to Crozier and Pamilo’s (1996) pedigree coefficient of relatedness denoted by g. The sex ratio calculated as the proportion of sons [M/(M 1 F )] The kin value of i to the focal individual, calculated as Rivi, equivalent to Hamilton’s (1972) life-for-life coefficient of relatedness and Ratnieks and Reeve’s (1992) kin value in that the regression relatedness is weighted by the class reproductive value and average mating success of the class vfi M F ui ci Ri r Ki Behavioral Ecology Vol. 10 No. 2 144 Figure 2 The kin value of (a) individual females as a function of the colony size to which they belong. K*fi is the average reproductive value of a female, considering females from all types of nest. The kin value of females have this humped shape because colonies of intermediate size have a higher per capita number of brood. Based on the data presented by Schwarz (1988). (b) The kin value of males (solid line) in relation to the population sex ratio. The dotted line indicates the scale difference between the two graphs. Note that the units of measure of the y-axis is K*fi. nieces. Therefore it does not matter if X is the egg layer or a helper; the ratio that will be optimal for the one would also be optimal for the other. Define vj, the average reproductive value of an individual of sex j, as the probability that an allele drawn at random from a future generation descends from a specific individual of sex j ( j 5 m for males and f for females; variables are listed in Table 1). Because females from different nest sizes have different per capita reproductive success, let vfi be the average reproductive value of a female in a colony of size i. This formulation makes it explicit that females should be counted in the context of their nest size. If there are M males and F females in total and a proportion, ui, of the F females are in groups of size i, then we can calculate the total reproductive value of each sex as a whole as cm 5 M(vm) and cf 5 F(Sivfiui). In haplodiploid species, cf is twice as large as cm (Price, 1970), as long as there are no unmated daughters who produce males in their mother’s nest (Crozier and Pamilo, 1996). We can thus write nm 5 [ ] 1 (1 2 r) 2 r (n*f i ) (1) where, v*fi 5 Sivfiui is simply the average reproductive value of a female, and r 5 M/(M 1 F ) is the population sex ratio expressed as the proportion of sons. To obtain the kin value of individual Y to individual X, KY, we need to multiply the reproductive value of Y with its regression coefficient of relatedness to X (Rf and Rm for female and male offspring, respectively). This is in essence equivalent to Hamilton’s (1972) complete or life-for-life coefficient of relatedness in that the regression relatedness is weighted by the reproductive value and expected mating success. Because Rf 5 0.5 and Rm 5 1 in haplodiploids (Crozier, 1970), we can write: [ ] (2) 1 K *f i 5 n*f i (R f ) 5 (n*f i ). 2 (3) K m 5 nm (R m ) 5 1 (1 2 r) (n*f i ) 2 r and K *f i denotes the average kin value of daughters. Equation 2 and 3 can now be combined to give: Km 5 [ ] (1 2 r) K *f i r (4) Note that we are not interested in the actual values of Km and K*fi, but only in the ratio of these two values to each other. To make the representation easier, we ‘‘freeze’’ the value of K*fi, in Figure 2, and only allow Km to vary with the population sex ratio. Equation 4 illustrates how the ratio of Km to K*fi is affected by the population sex ratio. More explicitly, we can say that when the sex ratio is r, then the kin value of a male is (1 2 r)/r times the average kin value of a female. At a population sex ratio of 0.5, the kin value of the average male equals that of the average female, but as the sex ratio becomes more female biased (as r decreases), the kin value of a male, as compared to that of a female, rapidly increases (compare Figure 2a and 2b). At a population sex ratio of 0.25, Km is three times as much as K*fi. This would mean that by investing in a son, a mother will on average gain three times more inclusive fitness than had she invested that energy in a daughter. Using Equation 4, we can consider the optimal sex ratio decisions of a colony. When the population sex ratio is biased to the extent that the highest value of Kfi, Kf4 in this example (see Figure 2a) is still smaller than Km, the colony should invest in males only. If the population sex ratio is less biased so that there are values of Kfi that are higher than Km (as de- Greeff • Cooperative breeding and sex ratios 145 Females help their brothers Figure 3 The per capita kin value accrued through each son (squares) or daughter (circles) as a function of the number of offspring of that sex. Notice that fitness through sons is not a function of the number of sons, whereas fitness through daughters is dependent on the number of daughters. In area A it is optimal to produce sons, in B to produce daughters, and in C to produce a mixture of both sexes (see text for explanation). picted in Figure 3), then the sex allocation decision will depend on the number of eggs being reared. First, when we consider a small number of eggs (less than seven) and assume that the females reared from this clutch will all form one colony the next year, then we can use Figure 3 to obtain the optimal decision: in area A the colony should produce sons only because one or two males are, respectively, more valuable than a colony of one or two females. In area B the colony should produce daughters only. In area C the colony will achieve the highest fitness returns per egg by producing the optimal number of daughters and producing sons with the remaining eggs. Sons should be produced with the leftover eggs because extra daughters will result in either too large or too small daughter colonies. Hence, we expect a mixture of offspring in area C. The exact location of the B/C boundary depends on the specific magnitudes of the reproductive values. The prediction of areas B and C gives a qualitatively correct answer—namely, that clutch sex ratios increase as the clutch size increases. In this base model, two predictions are incongruent with the data. First, the prediction from area A is at odds with the data because colonies producing small numbers of broods invariably produce females only (Schwarz, 1988). Schwarz (1994) suggested that one must take into account that males depend on their sisters for protection and food. Second, if colonies in areas A and B produce female-biased ratios, then the population sex ratio is female biased, and as a result the kin value of males is much higher than that of females. Therefore, colonies raising large numbers of eggs (more than 7), should optimally produce sex ratios that are so male biased that the population sex ratio will be at equality again. When we look at the data (Figure 1), it is clear that larger colonies do not compensate the population sex ratio by producing more males. Seger and Charnov (1988) suggested that the overall female bias (a result of no compensation by colonies with large broods) may be explained by the fact that females help their brothers too. A clutch of two eggs can only have a sex ratio of 0, 0.5, and 1. All male clutches (r 5 1) will suffer a reduced fitness due to the lack of protection and food from their sisters. Similarly, a clutch with only one female (r 5 0.5) will have a reduced fitness because this female will need to forage, and during her foraging excursions her brother is left unguarded. As a consequence, small clutches have the highest fitness when they are completely female biased. If we take this dependence of males on their sisters into account, the model for small brood sizes is in concordance with the data. Two reservations must be stated: (1) the observed increase in clutch sex ratio as clutch size increases is much slower than that predicted; (2) the population-wide female-biased sex ratio is still unexplained. I now consider whether Seger and Charnov’s (1988) suggestion that the overall female bias might be explained by the help sisters provide to brothers is plausible. At first sight, their explanation seems to push the problem into a type 2 problem such as Toro (1982) studied. When male fitness is a function of the clutch’s sex ratio, rather than number of sisters, this explanation will apply. However, there is good reason to believe that the fitness of males is a function of the number of sisters they have; in allodapines only one or two females forage during autumn, suggesting that two females are sufficient to support a large number of siblings. All the protection strategies involve one female positioned at the nest entrance, and as long as one female is present, protection can be given. The support from two females should thus greatly exceed that of one, as a single mother leaves the nest unguarded when she forages. Females in addition to two, however, cannot contribute as much. On these grounds it is more likely that benevolence depends not on the clutch sex ratio, but on the actual number of females (Figure 4a). In a large clutch (upper line in Figure 4a), it will mean that males in clutches that are male biased will still have the same value as males in clutches with a strong female-biased ratio. Only the most male-biased ratios above will experience reduced fitness. In smaller clutches (lower line in Figure 4a), as discussed above, male dependence on sisters will lead to female-biased ratios. The logical expectation would thus be that the female bias created by small broods will be compensated by the clutch sex ratios of larger broods. This intrasex explanation (Seger and Charnov, 1988) can thus not account for the observed female-biased ratio of the population as a whole. Packaging effect and a role for local optima Thus far I have ignored the decisions of colonies producing larger broods. When a colony raises larger numbers of females, some females disperse and form one or more new nests. How do sisters group together to form new nests? Most important, only integer numbers of colonies can be formed. It is obvious that colonies of optimal size can only be formed when the number of females is a multiple of the optimal colony size. In all other cases some colonies of suboptimal size will have to be formed. Considering returns on investment in females, we expect the following relationship to hold true: Starting from the right-hand side of Figure 4b with no females in the brood, returns on investment first increase as a more optimal colony size is formed, then the colony becomes larger than the optimal size, and fitness returns per additional daughter decrease. With still further investment in females, the colony splits in two colonies of suboptimal size, which with further investment grows to more optimal sizes. This pattern continues with further investment to produce the steplike graph. Behavioral Ecology Vol. 10 No. 2 146 When we combine the fitness returns expected through sons (Figure 4a) and daughters (Figure 4b), we obtain the colony’s complete fitness returns (Figure 4c). When the population sex ratio is close to equality (solid line), the fitness line is on average horizontal, and local optimal peaks do not differ much from each other in fitness. When the population sex ratio is female biased (dotted line), as in allodapine bees, each male counts much more than each female. Therefore the fitness of more male-biased local optima is higher than female-biased local optima. A strongly male-biased brood is the global optimum. In addition, the valleys separating peaks become shallower, and at high female biases they disappear completely. It is conceivable that a population can produce a female-biased sex ratio which, despite not being a global ESS, is still a local ESS. In other words, a colony at peak a will have a lower fitness than a colony at peak b, but, because a valley separates the two, the population will not evolve toward peak b. The undulating fitness functions, resulting from the fact that integer numbers of groups must be formed, could thus explain the observed population-wide female-biased sex ratios. An analogous problem is the trade-off between number and size of offspring in small clutches (Charnov and Downhower, 1995), which is manifested here at a higher organizational level. DISCUSSION Figure 4 (a) Total fitness received through sons as a function of the clutch sex ratio for two clutch sizes: 16 eggs (squares) and 3 eggs (diamonds). Both the solid and dotted lines assume that the fitness of males with fewer than two sisters is less than that of males with two or more sisters and that sisters in excess of two do not contribute more to their brothers’ fitness. In the case of 16 eggs, the threshold where sons start to suffer due to a lack of sisters is reached at a much higher sex ratio than for the case of 3 eggs. (b) Fitness received through daughters as a function of the clutch sex ratio (brood of 16 eggs). Starting from the right-hand side, each additional point is an additional daughter. (c) The combined fitness through daughters and sons as a function of the clutch sex ratio for a population with a female-biased population sex ratio (squares) and a population sex ratio close to equality (circles). The exact shape of these steps depend on the variation in founding decisions and the relationship between colony size and fitness. Note that because it is not the fitness per egg, but the total accumulative fitness, the steps are not as dramatic as in Figure 2a. In contrast, males do not cooperate and can be produced in smaller energy units. Seger and Charnov’s (1988) model where female fitness increases exponentially with the degree of female bias is too simplistic because female fitness alternates between an increasing and decreasing function as the clutch sex ratio increases. The rugged landscapes thus produced do not specify a unique optimal strategy. Rather, many sets of strategies could be trapped at locally stable points, even if it leads to a population-wide male bias. The point to which such a system evolves depends on (1) the starting point and (2) the possible mutational steps. A direct confirmation of theory-generated predictions will therefore be difficult, but by testing the assumptions the model can direct future research in this field. Because the starting point is important in determining which locally optimal positions are reached first, differences between species may reflect history rather than current ecological difference. If the ancestral condition is an equal clutch sex ratio, then selection for female-biased sex ratio must have been strong enough to overcome the local peaks in the first place. A possible evolutionary history could be as follows: Initially the brood size–sex ratio correlation may have been absent, with a monomorphic female-biased sex ratio due to males’ dependence on their sisters. Subsequently, the strategies of larger colonies evolved away from this female bias. As they evolved along the curve in Figure 4c (and as a result of this movement), the local peaks rose relative to the valleys and eventually reached a relative altitude where the valleys were so deep that they resisted further evolution to the global optimum. The possible mutant strategies that can arise are of major importance. If long jumps from one peak to the next are possible (that is, without going through the valley of lowered fitness), then a biased population sex ratio will not be stable, and colonies producing larger broods will evolve to compensate for the female bias created by smaller broods. If the sex ratio strategy for producing a brood size of x is correlated to the sex ratio strategy for a brood of size y, strong selective forces, such as male defenselessness in small brood sizes, will also affect the sex ratio of larger broods. A step toward more male-biased ratios in larger clutches would automatically mean that smaller clutches have too few females to defend and support males. Orzack and Gladstone (1994) Greeff • Cooperative breeding and sex ratios showed that the strategy employed by a female Nasonia vitripennis, when she is the first to oviposit, is correlated to the strategy she uses when she is the second to oviposit. This suggests that similar correlations could explain the observed sex ratios. Chance factors such as predation reduce mothers’ accuracy in predicting what types of colonies their daughters will form. Under such variation the types of colonies will be a distribution along a stretch of the wave (Figure 4c). Depending on the distribution’s magnitude in relation to the wave length, the relative height of local peaks will be reduced and the valleys raised, making stability less likely. Female Allodape mucronata are already mated in autumn (Michener, 1974), and this means that protection and feeding of males during the winter should not be as important in this species. Following the expectation, a small data set of 73 A. mucronata pupae show a slight male bias (Michener, 1971). This suggests that the help received by males may be important. Frank (1990) discussed a similar packaging problem for mammals where males and females have different optimal sizes. He found that individuals need to skew their sex ratio from equality to cope with the optimal packaging of offspring. Crozier and Pamilo (1996) identified a similar problem when colonies fission to form new daughter colonies. Daughter colonies, having a large optimal size, should be produced in stepwise increments, whereas males, being smaller energy units, can be used to soak up the remaining resources. A difference is that Crozier and Pamilo consider the problem from the viewpoint of LRC between daughter queens for a worker force. A class-structured approach, as is used here, could be used to investigate cooperative breeding in vertebrates and would allow the simultaneous consideration of breeding groups at different stages of development. Helpers that join the colony can be counted in the form of ‘‘growth’’ of the parental group, whereas offspring that form new groups can be counted as reproduction. Leimar (1996) employed such a model to investigate the Trivers-Willard problem. This manuscript is dedicated to one of the allodapine pioneers, Dr. S. H. Skaife. I am grateful to the University of Pretoria for a travel grant and to Michael Schwarz for introducing me to and encouraging me to model this problem. I benefited greatly from discussions with researchers in the laboratories of Mike Schwarz and Ross Crozier. I thank Mike Schwarz, Jon Seger, Steve Orzack, Stuart West, and two anonymous referees for their comments on drafts of this paper. I am very grateful to Martin Storhas for his painstaking comments on the manuscript and for pointing out an error. REFERENCES Alexander RD, Sherman PW, 1977. Local mate competition and parental investment in social insects. Science 196:494–500. Blows MW, Schwarz MP, 1991. Spatial distribution of a primitively social bee: does genetic population structure facilitate altruism? Evolution 45:680–693. Bourke AFG, Franks NR, 1995. Social evolution in ants. Princeton, New Jersey: Princeton University Press. Cane JH, Michener CD, 1983. Chemistry and function of mandibular gland products of bees of the genus Exoneura (Hymenoptera, Anthoporidae). J Chem Ecol 9:1525–1531. Charnov EL, Downhower JF, 1995. A trade-off-invariant life-history rule for optimal offspring size. Nature 376:418–419. Charnov EL, Los-den Hartogh RL, van den Assem J, 1981. Sex ratio evolution in a variable environment. Nature 289:27–33. Clark AB, 1978. Sex ratio and local resource competition in a prosimian primate. Science 201:163–165. Cronin AL, Schwarz MP, 1997. Sex ratios, local fitness enhancement 147 and eusociality in the allodapine bee Exoneura richardsoni. Evol Ecol 11:567–577. Crozier RH, 1970. Coefficients of relationship and the identity of genes by descent in the Hymenoptera. Am Nat 104:217–217. Crozier RH, Pamilo P, 1996. Evolution of social insect colonies: sex allocation and kin selection. Oxford: Oxford University Press. Emlen ST, Emlen JM, Levin SA, 1986. Sex-ratio selection in species with helpers-at-the-nest. Am Nat 127:1–8. Frank SA, 1987. Demography and sex ratio in social spiders. Evolution 41:1267–1281. Frank SA, 1990. Sex allocation theory for birds and mammals. Annu Rev Ecol Syst 21:13–56. Gowatay PA, Lennartz MR, 1985. Sex ratio of nestling and fledgling red-cockaded woodpeckers (Picoides borealis) favor males. Am Nat 126:347–353. Grafen A, 1986. Split sex ratios and evolutionary origins of eusociality. J Theor Biol 122:95–121. Hamilton WD, 1972. Altruism and related phenomena, mainly in social insects. Annu Rev Ecol Syst 3:193–232. Komdeur J, 1996. Facultative sex ratio bias in the offspring of Seychelles warblers. Proc R Soc Lond B 263:661–666. Komdeur J, Daan S, Tinbergen J, Mateman C, 1997. Extreme adaptive modification in sex ratio of the Seychelles warbler’s eggs. Nature 385:522–525. Lambin X, 1994. Sex ratio variation in relation to female philopatry in Townsend’s voles. J Anim Ecol 63:945–953. Leimar O, 1996. Life-history analysis of the Trivers and Willard sexratio problem. Behav Ecol 7:316–325. Lessells CM, Avery MI, 1987. Sex-ratio selection in species with helpers at the nest: some extensions of the repayment model. Am Nat 129:610–620. Lloyd DG, 1986. Selection of offspring size at independence and other size-versus-number strategies. Am Nat 129:800–817. Malcolm JR, Marten K, 1982. Natural selection and the communal rearing of pups in African wild dogs (Lycaon pictus). Behav Ecol Sociobiol 10:1–13. Melna PA, Schwarz MP, 1994. Behavioural specialization in pre-reproductive colonies of the allodapine bee Exoneura bicolor (Hymenoptera: Anthoporidae). Insect Soc 41:1–18. Michener CD, 1971. Biologies of African allodapine bees (Hymenoptera, Xylocopinae) Bull Am Mus Nat Hist 145:220–301. Michener CD, 1974. The social behavior of the bees. Cambridge, Massachusetts: Harvard University Press. O’Keefe KJ, Schwarz MP, 1990. Pheromones are implicated in reproductive differentiation in a primitively social bee. Naturwiss 77:83– 86. Orzack SH, Gladstone J, 1994. Quantitative genetics of sex ratio traits in the parasitic wasp, Nasonia vitripennis. Genetics 137:211–220. Packer C, Pusey AE, 1987. Intrasexual cooperation and the sex ratio in African lions. Am Nat 130:636–642. Price GR, 1970. Selection and covariance. Nature 227:520–521. Ratnieks FLW, Reeve HK, 1992. Conflict in single queen hymenopteran societies: the structure of conflict and processes that reduce conflict in advanced eusocial species. J Theor Biol 158:33–65. Ratnieks FLW, Boomsma JJ, 1997. On the robustness of split sex ratio predictions in social hymenoptera. J Theor Biol 185:423–439. Schwarz MP, 1986. Persistent multi-female nests in an Australian allodapine bee, Exoneura bicolor (Hymenoptera, Anthophoridae). Insect Soc 33:258–277. Schwarz MP, 1987. Intra-colony relatedness and sociality in the allodapine bee Exoneura bicolor. Behav Ecol Sociobiol 21:387–392. Schwarz MP, 1988. Local resource enhancement and sex ratios in a primitively social bee. Nature 331:346–347. Schwarz MP, 1994. Female-biased sex ratios in a facultatively social bee and their implications for social evolution. Evolution 48:1684– 1697. Schwarz MP, Blows MW, 1991. Kin association during nest founding in the bee Exoneura bicolor : active discrimination, philopatry and familiar landmarks. Psyche 98:241–249. Schwarz MP, O’Keefe KJ, 1991. Order of eclosion and reproductive differentiation in a social allodapine bee. Ethol Ecol Evol 3:233– 245. Schwarz MP, Lowe RM, Lefevre KS, 1996. Kin association in the allodapine bee Exoneura richardsoni Rayment (Hymenoptera: Apidae). Austr J Entomol 35:65–71. 148 Schwarz MP, Scholz O, Jensen G, 1987. Ovarian inhibition among nestmates of Exoneura bicolor Smith (Hymenoptera: Xylocopidae). J Austr Entomol Soc 26:355–359. Schwarz MP, Silberbauer LX, Hurst PS, 1997. Intrinsic and extrinsic factors associated with social evolution in allodapine bees. In: The evolution of social behaviour in insects and arachnids (Choe JC, Crespi BJ, eds). Cambridge: Cambridge University Press; 333–346. Seger J, Charnov EL, 1988. Benevolent sisterhood. Nature 331:303. Speith PT, 1974. Theoretical considerations of unequal sex ratios. Am Nat 108:837–849. Stark RE, 1992. Sex ratio and maternal investment in the multivoltine large carpenter bee Xylocopa sulcatipes (Apoidea: Anthophoridae). Ecol Entomol 17:160–166. Behavioral Ecology Vol. 10 No. 2 Taylor PD, 1981. Intra-sex and inter-sex sibling interactions as sex ratio determinants. Nature 291:64–66. Taylor PD, 1990. Allele-frequency change in a class-structured population. Am Nat 135:95–106. Toro MA, 1981. Birth control—sex preference and sex ratio. Heredity 47:417–423. Toro MA, 1982. Altruism and sex ratio. J Theor Biol 95:305–311. Trivers RL, Willard DE, 1973. Natural selection of parental ability to vary the sex ratio of offspring. Science 179:90–92. Trivers RL, Hare H, 1976. Haplodiploidy and the evolution of the social insects. Science 191:249–263. Vollrath F, 1986. Eusociality and extraordinary sex ratios in the spider Anelosimus eximius (Araneae: Theridiidae). Behav Ecol Siociobiol 18:283–287.