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Behavioral Ecology Vol. 10 No. 2: 141-148
© 1999 International Society for Behavioral Ecology
Cooperative breeding, offspring packaging, and biased sex ratios in allodapine bees
Arbeitsgruppe Michiels, Max-Planck-Institute for Behavioral Physiology (Seewiesen), PO Box 1564, D-82305 Starnberg, Germany, and Department of Zoology and Entomology, University of Pretoria, Pretoria 0002, Republic of South Africa
Address correspondence to J. M. Greef, Department of Zoological Entomology, University of Pretoria, Pretoria 0002, South Africa. E-mail: greeff{at}mpi-seeviesen.mpg.de
Received 12 February 1998; accepted 10 May 1998.
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ABSTRACT |
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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.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONA CLASS-STRUCTURED MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Lloyd (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 female-biased 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 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 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
observed 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. 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 abdomen 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).
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A CLASS-STRUCTURED MODEL |
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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
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.
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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 = 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 =
M(vm) and cf =
F(
vfiui).
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
 | (1) |
vfiui is simply
the average reproductive value of a female, and r =
M/(M + 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 = 0.5 and Rm = 1 in haplodiploids
(Crozier, 1970), we can write:
 | (2) |
 | (3) |
 | (4) |
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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 depicted 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.
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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.
Females help their brothers
A clutch of two eggs can only have a sex ratio of 0, 0.5, and 1. All male
clutches (r = 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 = 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.
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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. 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.
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 
will have a lower fitness than a colony at peak ß, but, because a valley
separates the two, the population will not evolve toward peak ß. 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.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONA CLASS-STRUCTURED MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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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 stably 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) 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.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONA CLASS-STRUCTURED MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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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.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONA CLASS-STRUCTURED MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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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.[Web of Science]
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