J. Genet., Vol. 64, No. I, July 1985, pp. 41-58. (!;) Printed in India.
Evolution of sex ratios in social hymenoptera: kin selection, local mate
competition, polyandry and kin recognition
, N V JOSHI and RAGHA VENDRA GADAGKAR *
Centre for Theoretical Studies and .Centre for Ecological Sciences,
Indian Institute of Science, Bangalore 560012, India.
I- MS received 24 May 1985
Abstract. A model is constructed to study the effects of local mate competition and multiple
mating on the optimum allocation of resources between the male and female reproductive
brood in social hymenopteran colonies from the 'points of view' of the queen (parental
manipulation theory) as well as the workers (kin selection theory). Competition between pairs
of alleles specifying different sex investment ratios is investigated in a game theoretic frame
work. All other things being equal, local mate competition shifts the sex allocation ratio in
favour of females both under queen and worker control. While multiple mating has no effect
on the queen's optimum investment ratio, it leads to a relatively male biased investment ratio
under worker control. Under queen control a true Evolutionarily Stable Strategy (ESS) does
not exist but the 'best' strategy is merely immune from extinction. A true ESS exists under
worker control in colonies with singly mated queens but there is an asymmetry between the
dominant and recessive alleles so that for some values of sex ratio a recessive allele goes to
fixation but a dominant allele with the same properties fails to do so. Under multiple mating,
again, a true ESS does not exist but a frequency dependent. region emerges. The best strategy
here is one that is guaranteed fixation against any competing allele with a lower relative
frequency. Our results emphasize the need to determine levels of local mate competition and
multiple mating before drawing any conclusions regarding the outcome of queen-worker
conflict in social hymenoptera. Multiple mating followed by sperm mixing, both of which are
known to occur in social hymenoptera, lower average genetic relatedness between workers and
their reproductive sisters. This not only shifts the optimum sex ratio from the workers' 'point of
view' in favour of males but also poses problems for the kin selection theory. We show that kin
recognition resulting in the ability to invest in full but not in half sisters reverts the sex ratio
back to that in the case of single mating and thus completely overcomes the hurdles for the
operation of kin selection.
Keywords. Sex ratios; kin selection; local mate competition; polyandry; kin recognition;
social hymenoptera.
1. Introduction
Fisher (1930) showed that in outbreeding populations natural selection would favour
~ equal parental investment in the offspring of each sex. If the population is inbreeding
however, competition for females is predominantly between brothers (local mate
competition) so that natural selection favours a female biased sex ratio (Hamilton
1967). In the extreme case, when there is complete sibmating, a parent 'should' produce.
just enough sons necessary to inseminate all the daughters. These predictions of Fisher
and Hamilton have been repeatedly verified, both theoretically and empirically
(Charnov 1982; Metcalf 1980; Owen 1983; Werren 1980, 1983). Indeed the theory of sex
allocation has in recent years become a cornerstone of evolutionary biology, a status
achieved primarily because of the success with which precise quantitative and
empirically testable predictions have been made (Charnov 1982).
41
42 N V Joshi and Raghavendra Gadagkar
The theory of sex allocation has assumed importance for yet another reason.
Predictions concerning sex allocation appear to be powerful in choosing between
competing theories purporting to explain the evolution of sociality in insects (for a
recent review see Gadagkar 1985a). A hidden assumption in Fisher's argument of equal
allocation between the two sexes is that a parent is equally related to his or her sons and
daughters. In haplodiploid social insects such as wasps, bees and ants, sterile female
workers often feed and care for their siblings instead of producing their own offspring.
Haplodiploidy, a genetic system where males arise from haploid unfertilized eggs and
females from fertilized, diploid eggs, creates asymmetries in genetic relatedness.
Females are related to their sisters by 3/4 and to their brothers by 1/4. This asymmetry is
in fact an important factor in favour of Hamilton's (1964a, b) theory of the evolution of
social behaviour because it genetically predisposes a hymenopteran worker towards the
evolution of altruistic behaviour. Trivers and Hare (1976) argued, however, that a
worker gains nothing in fitness if she invested equally in brothers and sisters because
her average relatedness to her siblings is 1/2, the same as her average relatedness to her
own offspring. Sterility in workers would be selected if they can capitalise on the
asymmetries in genetic relatedness by investing in their sisters and brothers in the ratio
3: 1. In other words, when workers invest in siblings who are related to them in the ratio
3: 1, natural selection would favour a ratio of allocation paralleling the ratio of
relatedness. The queen who is fertile and who produces the sons and daughters would
on the contrary favour an equal investment in brood of the two sexes in her colony. In
this context there would be a conflict of interests between the queen and the workers in
the optimum ratio of allocation of resources between brood of the two sexes.
The theory of kin selection argues that workers are sterile and act altruistically
towards their sibs because this is the strategy that maximises their inclusive fitness.
Inclusive fitness of an individual may be defined as its total contribution to the gene
pool of the next generation obtained both by the production of offspring and by aiding
genetic relatives. The theory of parental manipulation (Alexander 1974) on the other
hand suggests that workers are sterile because they are manipulated into this state by
their parents. The two theories make mutually opposing predictions regarding the
expected allocation between the two sexes in the reproductive brood in social
hymenopteran colonies. Ifkin selection is responsible for worker sterility and altruism,
then the workers should win in the conflict over the investment ratio and the resultant
sex-investment ratio observed should be 3: 1. If worker sterility and altruism are a
consequence of parental manipulation instead, queens should be successful in
manipulating the workers into investing equally in reproductives of the two sexes and a
1: 1 ratio of investment is expected. Trivers and Hare (1976) weighed male and female
reproductives in a number of monogynous ant colonies and showed that the
observed ratios of investment were significantly closer to 3: 1 than 1: 1 and concluded
that their data are uniquely explained by kin selection theory.
The predictions and conclusions of Trivers and Hare (1976) depend on the
assumption that the social insects under consideration are outbreeding and that the
queens mate only once. Under inbreeding or local mate competition (LMC), the queens
too would prefer a female biased investment ratio (Hatnilton 1967) and the 3: 1 ratio
seen by Trivers and Hare may have nothing to do with workers realising their optimum
investment ratio as opposed to the queens' optimum value (Alexander and Sherman
1977). Similarly, if the queens mate W,ith more than one male then the relatedness
between the workers and the reproductive sisters they rear will no longer be ~/4 but
distributed anywhere between 3/4 (full sisters) and 1/4 (half sisters), Under these
7"cyc,,!;/~~. ~,t.,-:::::::Cc
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;
Evolution of sex ratios in social hymenoptera 43
conditions the predictions used by Trivers and Hare (1976) are no longer valid
(Alexander and Sherman 1977). B(jth LMC and multiple mating by queens are known to
occur in social hymenoptera although their intensities might vary widely (Alexander
and Sherman 1977; Page and Metcalf 1982; Crozier 1980). While multiple mating has
long been recognised to be common in hymenoptera (see Wilson 1971) it has often been
assumed that sperms from different males clump in the spermatheca of the femaies
leading to use of sperm from a single male for extended periods of time (see Orlove
1975, for example; see Crozier and Bruckner 1981, and Starr 1984, for detailed
discussions of other available data). However, in the only case where careful
investigation has gone into this question it is clear that sperms do not clump (Page and
Metcalf 1982).
When multiple mating results in a lowered average genetic relatedness between
workers and reproductives in a colony this not only alters the expected sex-investment
ratio from the workers' 'point of view' but may also be considered as a factor against kin
selection (Hamilton 1964b; Wilson 1971). It may be argued however that 'workers can
circumvent the problem of multiple insemination' by kin recognition leading to
investment in full sisters but not in half-sisters (Page and Metcalf 1982).
In recent years a number of theoretical investigations relating to the evolution of sex
ratios in social hymenoptera have been undertaken. Stimulated in part by the work of
Trivers and Hare (1976)who predicted a 3: 1 investment ratio under worker control and
a 1: 1 ratio under queen control, several authors have confirmed these predictions by
rigorous methods (Oster et al 1977; MacNair 1978; Craig 1980a; Uyenoyama and
Bengtsson 1981; Charnov 1978a). Other factors such as LMC and worker-queen conflict
have also occasionally been considered (Taylor and Bulmer 1980; Oster et al 1977;
Bulmer 1981; Benford 1978). Our intention here is to simultaneously consider the
effects of LMC, polyandry and kin recognition and generate predictions regarding
optimum sex allocation ratios in the frame work of the kin selection and the parental
manipulation theories. In this paper we are only working within the context of the
Trivers and Hare (1976) and Alexander and Sherman (1977) arguments in our efforts to
recalcul;lte what in fact should have been the theoretical predictions of the former
authors, had they taken the points made by the latter into consideration. It should be
mentioned that the importance of female biased sex ratios in the evolution of
eusociality is not universally agreed upon. Craig's (1979, 1980b) models for instance
suggest that female biased sex ratios are not likely to have been very useful for the
origin of eusoci~l behaviour. In the maintenance of already existing eusocial behaviour,
Charnov (1978b) points out that selecti°.n for egg laying by workers is very strong even
if they are rearing a "seemingly advantageous" combination of brothers and sisters.
2. The model
We consider an infinite population of social insect colonies. Each colony is initiated by a
single inseminated female (queen), whose all-female first brood consists entirely of
workers. The subsequent brood consisting of both male and female reproductives is fed
and cared for by the workers. The queen dies after the emergence of reproductives.
From each nest a fraction d of both male and female reproductives disperses to join a
mating aggregate where random mating takes place. On the other hand, sibmating
takes place within the fraction (1 -d) remaining at each nest. After mating the males die
and each inseminated female initiates a new nest. Thus d parametrises local mate
44 N V Joshi and Raghavendra Gadagkar
competition, with d = 1 corresponding to complete outbreeding, and d = 0 complete
inbreeding. We have of course modelled LMC in this fashion for convenience. LMC could
occur even in the absence of sibmating if, for instance, the males from one nest all
attempt to mate with the females of a neighbouring, though unrelated nest.
The sex ratio trait is modelled by a one-locus-two allele system. The allele A being
dominant, individuals with genotype AA and AB produce a fraction r A of males among
their reproductive progeny, while those of genotype BB produce a fraction r B of males.
For simplicity we have assumed that investment ratio is directly translated into sex
ratio.
2.1 Queen control of the investment ratio
Even if the optimum investment ratio is different for the queen and the workers, the
queen could in principle manipulate the workers into feeding her reproductive
offspring in the ratio optimum for her. This is modelled by adjusting the sex ratio of the
reproductive offspring according to the genotype of the queen.
If queens mate only once, there will be six types of inseminated females, AA.A, AA.B,
AB.A, AB.B, BB.A and BB.B (where the first two letters refer to the queen's genotype
and the third to the genotype of the male she has mated with i.e., of the sperm stored in
her spermatheca). Each inseminated female contributes genes to the next generation by
three pathways: (1) through the sons which join the mating aggregate, (2) the daughters
which join the mating aggregate, (3) the sibmated daughters. The quantitative details
about the contributions from each of the six types of inseminated females are described
in table Al (appendix).
Thus knowing the frequencies P AA.A(n), P AA.B(n). ..., PBB.B(n) of each of the
classes in the nth generation, one can obtain the frequencies P AA.A
(n + 1), ...etc. in the next generation. Since all the frequencies add up to unity, the
dynamics is described by a system of five coupled nonlinear difference equations (see
appendix).
To study the competition between two alternative sex ratio strategies r A and rB, we
consider a population purely of type AA.A, into which a small proportion of
inseminated females containing the B gene is introduced. For this situation, the above
system of five nonlinear coupled equations can be approximated by a system of five
linear difference equations. As shown in the appendix, the elements of the' relevant
transformation matrix G' are functions of r A' r Band d alone.
If the dominant eigenvalue of the matrix G' is greater than unity the proportion of B
increases with time and we say that A is invadable by B. Conversely, if the eigenvalue is
less than unity, A is uninvadable by B. Similarly, we can investigate whether a pure
population of B is invadable by A.
Ifpure A is un invadable by B while B is invadable by A then A would be selected for
and would go to fixation. If both pure A and pure Bare un invadable, frequency
dependent selection is implied (whichever establishes itself first wins). Finally if both
pure A and pure B are invadable, co-existence of the two alleles is indicated.
For different values of r A (the proportion of males specified by the dominant allele),
rB (proportion of males specified by the recessive allele) and d, the parameter
characterizing local mate competition, we have investigated the dynamics of the system
to determine which of the above conditions prevail viz. one of the two alleles going to
fixation, the two alleles coexisting, or frequency dependent selection.
Evolution of sex ratios in social hymenoptera 45
2.2 Worker control of investment ratios
Trivers and Hare (1976) assumed that "the offspring is capable of acting counter to its
parents' best interests" and thus workers should be able to feed the reproductive brood
in the ratio that optimises their inclusive fitness. Once again making the simplifying
assumption that investment ratios are directly translated into sex ratios, this is
modelled simply by adjusting the sex ratio of the reproductive brood in accordance
with the genotype of the workers. For instance queens of the type AB.B will produce
workers of the type AB and BB in equal proportions. Although some investigators have
considered the possibility of workers of one genotype behaviourally dominating over
workers of other genotypes (Charnov 1978; Craig 1980; Pamilo 1982; Bulmer 1983) we
agree with Bulmer (1983) that additivity seems biologically more likely. Thus in a
colony with AB and BB workers in equal numbers, the proportion of males in the
reproductive brood is taken to be (r A + r ~/2. Thus (r A + r ~/4 males each of type A and
Band t[l -(r A + r ~/2] females each of type AB and BB are produced in a colony
initiated by an inseminated female of type AB.B.
2.3 Polyandry
When queens mate with more than one male, they are assumed to mate with males of
different genotypes in the proportion that males of these genotypes are represented in
the population (at the nest site in the case of sibmating or at the mating aggregate in the
case of outbreeding). Equal numbers of sperms of each male are assumed to be stored in
the spermatheca which are then used randomly. Thus in a system of 2 alleles, if every
female mates twice there will be 9 types of inseminated females, AA.A.A, AA.A.B,
AA.B.B, AB.A.A, AB.A.B, AB.B.B, BB.A.A, BB.A.B, and BB.B.B, where the first two
letters refer to the genotype of the female and the last two letters refer to the genotypes
of the 2 males she has mated with. Similarly, one can write down the genotypes of
inseminated females for any specified number of matings. For different numbers of
matings we have investigated the outcome of competition between alternative sex ratio
alleles both under queen and worker control.
2.4 Evolution of kin recognition
In the previous section the workers were assumed to invest in all brothers and sisters
irrespective of their relatedness (i.e., full sisters and half sisters were not distinguished).
Here we investigate the case where workers can distinguish genetic relatedness and will
invest in their full sisters, but not their half sisters. Multiple mating by the queen does
not make any difference in the case of brothers as there can be no half brothers in a
haplodiploid system.
We also model kin recognition by a one-locus-two allele system. Now allele A in
addition to coding for r A proportion of males also confers ability to recognise kin. Bon
the other hand codes for r A proportion of males but does not confer ability to recognise
kin. Instead of assuming that if the ability to recognise kin is dominant, the sex ratio
specified by that allele is necessarily dominant, we have considered all possible
combinations of dominance and recessiveness of kin recognition ability and sex ratio.