A M . ZOOLOGIST, 11:309-325 (1971).
Some Aspects of Natural Selection in Arrhenotokous Populations
DANIEL L. HARTL
Department of Genetics and Cell Biology, University of Minnesota,
Saint Paul, Minnesota 55101
SYNOPSIS. Obligate arrhenotoky is known to have evolved only eight times, once each in
the rotifer order Monogonanta, the arachnid order Acarina, the insect orders
Hymenoptera and Thysanoptera, and in two families of Homoptera (Coccidae and
Aleyrodidae) and Coleoptera (Scolytidae and Micromalthidae).
In this paper it is argued that in .male haploids the population genetic parameter
which is most nearly analogous to the average fitness in ordinary diploids is the
geometric mean of the average fitness in the two sexes. The geometric mean fitness is
maximized by selection in a random mating population, provided one takes into
account the fact that the system of inheritance imposes constraints on the gene
frequencies in males and females. Furthermore, under sufficiently small selection
intensities, the rate of change in geometric mean fitness is approximately equal to the
additive genetic variance in fitness, where the additive variance is defined as the
variance due to the regression on a model of gene action which assumes no dominance
in females and complete dosage compensation in males.
Some problems concerning the sex ratio in male haploids when there is extreme
inbreeding are also discussed.
My assignment in this Symposium is to one extreme, the orders Hymenoptera and
discuss natural selection in arrhenotokous Monogonanta are almost entirely male
populations. An important precursory is- haploid; at the other extreme, only the
sue is why, apart from intellectual curiosi- coccid tribe Iceryini and the coleopteran
ty, there should be concern about natural genera Xyleborus and Xylosandrus include
selection in male haploids. The number of species which are male haploid.
higher taxa in which male haploidy is
Not all species that once were male
known is exceedingly small. It occurs in haploid are still male haploid. Thelytoky
the rotifer order Monogonanta, in the is well represented in the hymenopterans
arachnid order Acarina, and in the four and acarines. Micromalthidae comprises
insect orders Hymenoptera, Thysanoptera, one rare species, Micromalthus debilis,
Homoptera, and Coleoptera. These or- which actually reproduces paedogeneticalders represent eight independent evolu- ly. The cottony cushion scale, Icerya purtions of male haploidy. It has evolved in chasi, is ambiguous. A few haploid males
two homopteran families, Aleyrodidae and do occur but reproduction is also possible
Coccidae, in two coleopteran families, via self-fertilization because, other than
Scolytidae and Micromalthidae, and once
the few males, the adult population conin each of the other four orders (Darlingsists
entirely of hermaphrodites which are
ton, 1932; Whiting, 1945; White, 1954,
diploid
except for haploid cells in the tes1957; Donner, 1966; Takenouchi and Taktis
(Chapman,
1969). (The tribe Iceryini
agi, 1967). The number of male haploid
is
basically
male
haploid, however, and
species in these groups varies widely. At
hermaphroditism has apparently evolved
independently several times. HughesThis work was supported by National Science
Schrader and Monahan, 1966).
Foundation Grant number G15I8786 and by a grant
from the graduate school of the University of
Judged from the number of evolutions of
Minnesota. >fy thanks to Dr. Everett Dempster for
male haploidy, the mode of reproduction
his detailed and perceptive comments on the
does not appear to be particularly attracmanuscript. Thanks also to Drs. S. \V. Brown, J. F.
Crow, R. Crozier, and U. Nur for their suggestions. tive. But judged on the basis of numbers of
309
310
DANIEL L. HARTL
species which are currently reproducing
as male haploids, the system is certainly
successful. Of 26 insect orders listed by
Borror and DeLong (1954), Hymenoptera
is the third largest (only Coleoptera and
Lepidoptera are larger). As of about 1950
some 684,000 species of insects had been
described; 103,000 were hymenopterans. By
the same criterion, the monogonant rotifers are successful. One reason to study the
evolutionary properties of arrhenotoky is
to decide whether these groups are successful because of their male haploidy or in
spite of it.
Male haploids are important as pests, for
example, the sawflies, horntails, spider
mites, and ambrosia beetles; they are important also as predators or as parasites of
pests (particularly the ichneumon and
chalcid hymenopterans); and bees are especially important economically and ecologically because of their production of a
food supplement and their role in pollination.
Altogether, male haploidy can be
thought of as an evolutionary "mutation"
which has recurred under different conditions some half a dozen times. This alone
makes it an interesting phenomenon. In
the same way that a great deal can be
learned about normal development by
studying mutations, so too can we expect
to learn more about evolution by studying
its aberrations.
This paper is divided into three sections. The first and second are some results
of a population genetic analysis of a model of an arrhenotokous population. Evolutionary statics establishes some conditions
under which a genetic polymorphism will
be maintained; evolutionary dynamics examines the rate of change in fitness of
male haploids. These results are not restricted to male haploids, but apply as well
to sex-linked genes in species in which
both sexes are diploid (diplo-diploids).
The third section deals with the sex ratio
in male haploids. I have made no attempt
to re\iew all aspects of natural selection in
male haploids — to have tried would have
been foolhardy. Thus, \ery few experimen-
tal or field studies are cited. Some aspects
of the theory, such as the formal consequences of inbreeding, I do not discuss at
all, but the reader will find quite detailed
discussions in the recent books by Wright
(1969) and Crow and Kimura (1970).
The subject of the evolution of male
haploidy has been discussed extensively by
Whiting (1945) and Hartl and Brown
(1970). The possible relationship between
arrhenotoky and the evolution of complex
social organization has been discussed by
Hamilton (1963, 1964a, 1964&; see also
Crozier, 1970).
EVOLUTIONARY STATICS
The analysis of selection models in male
haploids is algebraically quite messy because there is an asymmetry in gene frequencies between the sexes. Suppose that
at some locus in a random mating arrhenotokous population there are two alleles, At
and A 2, and suppose that the frequencies
of At and A% in eggs are pt and p2, and
that the frequencies of At and A2 in sperms
are qt and qs (pi+pa = qi+qa = 1). Let
the fitnesses (more precisely, viabilities) of
AiAjj AtAz,
A2A2 females be wlt, w12, w22,
respectively, and let the fitnesses of At and
A2 males be vt and v2. These fitnesses could
all be measured on some kind of absolute
scale but it is quite impossible to know
what scale to use. A more convenient measure is to define wn> w12, and w22 as the
relative viabilities in females and vu v2 as
the relative viabilities in males. This convention eliminates all meaningful comparisons between the fitnesses in females
and males. If, for example, the viabilities
in females are 4:7:10 and in males are 3:1,
then as Li (1967a) has pointed out, this in
no way suggests that females are more
viable than males. Indeed, the same model
gives fitnesses of 0.4:0.7:1 in females and
30:10 in males since the fitnesses in either
sex can be multiplied by a constant without changing the selective system.
The parameters in the model are given
in Table 1. This and similar models have
been studied by many authors (Bennett,
311
NATURAL SELECTION IN MALE HAPLOIDS
TABLE 1. Selection model of a random mating arrhenotokous population.
Males
Females
Genotype
Frequency
Relative fitness
A,A,
w
1957, 1958, 1963; Cannings, 1967; Cormack,
1964; Edwards, 1961; Haldane, 1923, 1926a,
1926&, 1927; Haldane and Jayakar, 1964;
Kirkman, 1966; Li, 1967a; Mandel, 1959;
Shaw, 1959; Sprott, 1957; Wright, 1969).
With the exception of Canning's article
all discussions have concentrated on the
conditions for the existence of a nontrivial
equilibrium and on the conditions for the
stability of the equilibrium. Cannings
(1967) has gone on to show that the gene
frequencies converge to any stable nontrivial equilibrium.
The equilibria are well known and can
be obtained by a simple procedure. Long
ago, Wright (1942) gave an equation for
the change in gene frequency in a randommating diplo-diploid population in terms
of the change in the average fitness of that
population. From this equation it follows
immediately that the average fitness of the
population is maximized at points of stable
gene frequency equilibria and minimized
at points of unstable equilibria (Li, 1955) .
An analogous theory can be developed for
male haploids. Assume that the selection
coefficients are sufficiently small that the
gene frequencies in the sexes will be approximately equal. Then pi=qi and p2=*
q2 and therefore Apt = (pip^/w,) \fit (wlt
—Wit) +Pa (WiS—w2S)]= (pipt/2) (dwf/dpj)
(l/u>f) = (pipn/2) (d log Wf/dpj where w,
i2- In the same
_
^ ) (l/wm) =p1ps (d
log wm/dpj) where wm = ptV^p^v^. Then
if p is the average gene frequency of At
(p = (2/3)p1+ (l/3)qt) we have Ap^Ap
= (2/3) Afc+ (1/3)A^ = (l/3)/^ 2 [(d log
«>r/dpi)+ (d log wm/dpt)} =
log wmwf/dp,]= (l/i) ptp^d log
where w—iumwf. This is a special case of a
general differential formula used by
Wright, Aq = q (l—q)[d log Hi/dq]/2k, in
which k = 1 is appropriate in zygogenetic
U>lt
A,A,
w,,
A,
A.
P-
P.
v,
species, k — 3/2 in arrhenotokous ones.
This suggests that if the average fitness
of a male haploid population is defined as
the geometric mean of the fitnesses in the
two sexes, then this will be maximized at
stable equilibria even when the gene frequencies differ in the sexes.
This turns out to be the case, provided
that one takes into account the fact that
the genetic system itself imposes a functional relationship between the gene frequencies in the sexes. Note first in Table 1
that pi and qt are not independent. At
equilibrium qt = pjVj/v where v = piVt-\p2v2. This does not depend on any particular definition of average fitness but is one
of the inevitable consequences of this system of inheritance.
We therefore maximize yjwv (w = p1q1
Wi1+(fi1q^-\-p2qi)wlz->rplqiwii) subject to
the constraint qt = p{Oi/v using the Lagrangian method of undetermined multipliers. To do this we find the extremum of
/
pjVi
M = V"^^— ^ I qt
\
) treating pt
and q± as independent and A as a constant.
Then
dM
(2)
where <S>=qt (wu—wlt)-\-qt
Pi (w11—w12)+p2 (ro12—wn),
—v2.
{wllt—w2i),
and * =
x =r
v,
Setting (2) equal to 0 we find \=
2yjwv. Substituting this into (1) and recalling q^ptVj/v at equilibrium leads to
dM
dp,
1 f r
— ~^=A Pi wnvt
y/wv I
L
wu (w,+w,) "I
2
J
312
DANIEL L. HARTL
~^
n>««f]j.
by going through it in detail. The solutions can be found in Cannings (1968), and
the appropriate methods in Kimura (1956)
and Wright (1969).
When there are two alleles a stable equilibrium exists if and only if w12 (vt-\-v2)/2
The extremum is thus at
TV,
^>iu11v1 and w12 (v1-\-v2)/2^>w2Sv2.
w
l
This is precisely the equilibrium gene
frequency (Bennett, 1957; Mandel, 1959;
Cannings, 1967).
Determination of whether the equilibrium is a maximum or a minimum can be
made by substituting qt=zp,Vj/v directly
into the equation for w. One then obtains
-\-p22w22v2] V*
dy/wv
£
dp1
y/wv
d2y/wv
wv (d£
With
morphological traits it often happens that
the phenotypes of the hemizygous males
are the same as those of the corresponding
homozygous females. (Some of the mechanisms of dosage compensation in male haploids are discussed in Hartl and Brown,
1970.) Assume that the w's and v's have
been measured on some common scale so
that comparisons can be made between the
sexes. If dosage compensation occurs at the
level of fitness then we may put zu11=v1
and w22—v2. A stable equilibrium exists if
and only if wn(w11-\-wS!2)/2>wi11
and
(4)
where t = Pi V"i{"i—w12 ivt + r; 2 )/2] + p2
[w12 (Vt+vd/i—w^Vi]
and
d^/dp1—w11v1
-\-w22v2—wi2 {vi-\-vz)- At equilibrium | = 0
and d2^/iuv/dp21<^ 0 or > 0 according to
whether iv12 (yl-\-v2)—wllv1—w2iv2 > 0 or
< 0. The first case represents a maximum
of \fwv, the second case a minimum. Li
(1967rt) has performed calculations similar
to those in (4) but the approach via Lagrange multipliers is preferable because it
shows directly that an equilibrium point
\/wv is maximized if the equilibrium is
stable and minimized if the equilibrium is
unstable. (For a rigorous proof of the conditions for stability and convergence, see
Cannings, 1967.) This does not imply that
•\/wv is unconditionally maximized at
equilibrium nor that \/wv always increases.
It merely says that of all the sets of gene
frequencies that satisfy the equation 17,=
piVt/v, -\/J0v is maximized at the one set
that represents the stable equilibrium of
gene frequency.
This procedure can readih be extended
to multiple alleles but little is to be gained
Assume now that w iJ >w 22 (no generality is lost by this assumption). Then since
wn (wu+wu)/2 > w12 (wu-\-w22)/2 > iu-u
we have IL>12^>WJ, and also w12>iu22. Thus
overdominance in females is a necessary
condition for a polymorphism to be maintained. But it is not sufficient, for if iutl=
1.0, wy2=1.5, w22=0.2, then w12 (iun-\-wS2)/
2=0.9<w 2 i ! . Therefore one should expect
to find fewer polymorphisms maintained
by overdominance in a male haploid population than in a comparable diplodiploid
population.t
EVOLUTIONARY DYNAMICS
Since V"w is a satisfactory definition of
average fitness from the standpoint of its
t In an article to be published in Genetica, Dr.
Ross Crozier has reached just the opposite conclusion, namely, that the fraction of loci maintained
by overdominance should be the same in male
baploids as in ordinary diploids. Croner's argument
is based on the assumption that there is no
correlation betueen the sexes in the effects of genes
which act on characters strongly related to
fitness. I, on the contrary, have assumed dosage
compensation of these genes, which is just another
u j \ of saying that then elfects in males and females are completely correlated. Which is in fact
the better assumption is a matter that can be
decided only by experiment.
313
NATURAL SFXF.CTION IN M A L E HAPLOIDS
TABLE 2. Regression
model
for slow
selection.
Females
Genotype
Relative frequency (weighted)
Relative fitness (u)
Additive model (a)
//,///
2pV3
x,,
2
AtA,
4p,pJ3
x,,
1
Males
AtAt
2paf/3
x t,
0
A,
p,/S
y,
2
A,
p,/l
y,
0
being maximized at stable equilibria, the same relation to one another as did the
question of its rate of change becomes w's; similarly for the y's and v's. But the
worthwhile. The rate of change of average transformed fitnesses have the useful propfitness in diplo-diploids has been studied erty that ~=y {x—p^.x
extensively under the general title "fundamental theorem of natural selection" (FishTo obtain the variance if fitness is due
er, 1930, 1941; Crow and Kimura, 1956,
to the additive action of genes we must
1970; Kimura, 1958; Moran, 1962; Li,
specify an additive model of gene action.
19676, 1967c, 1969; Mandel, 1968; Ewens, The easiest way to do this is to specify an
1969; Turner, 1969; Wright, 1969; Hard, additive regression model and then obtain
1970). In its simplest form, the funda- the variance in fitness due to the regression.
mental theorem states that the change in The model is shown in Table 2 where it is
average fitness of a population at any time assumed that each A gene adds one unit
t
is equal to the variance in fitness which of fitness in females. Note
that dosage comcan be attributed to the additive action of pensation is also assumed. That is, one At
genes at that time. An analogous theorem gene in a male adds the same amount of
exists for arrhenotokous species (Haiti, fitness as two in a female.
unpublished). The general theory which
In performing the regression the females
permits any amount of inbreeding is rather
are
given twice as great a weight as the
complicated, but if mating is random there
males
(cf. Table 2). This is because the
is a simpler approach which leads to the
weight
attached to each sex is the relative
same result.
The model to be analyzed is the one in contribution of that sex to remote generTable 1 with the additional assumption ations. Since the males contribute i/s of
that the selection differentials are suffi- the genes to future generations they reciently small so that p1^*q1 and p2s*q2 ceive 1/3 of the weight. (This is correct
(This assumption is a reasonable one over whatever the sex ratio as long as selection
a rather wide range of selection intensities. is slow and of about the same intensity and
Hartl, 19706). To obtain an expression for direction in both sexes.)
From Table 2, Z=2pt, T'—ipj (2+p,)/3,
the rate of change in fitness, it is necessary
to have thefitnessesin each sex measured and Var (a)=8p1pa/3. Also u— (2/3) (p^x,,
on the same scale. Yet the w's and v's are
i+P*y») and
measured on different scales. One way to and ua = (2/3) (2p"1xu+2plptxlt)+
(1/3)
rectify the situation is to multiply the fit- (2p1y1). Therefore Cov (u^)=2p1p2 (2/3+
ness of each genotype of female by the y )/3 where
p=p1(x11—x12)-\-p2{xn—x22)
average fitness of the males, and to multi- and y=yi—y2. The regression coefficient $
ply the fitness of each genotype of male by of u on a is 0=Cov (w,a)/Var (<x)= (2/3+
the average fitness of females. Algebraically y)/4 and the variance in u attributable to
we p u t x11=w1^v, x12=w12vJ
x22—wi2Uj the linear regression (additive variance) is
y1=v1w, y2=v2w where 'v—p1v1-\-p2v2 and o2A=fl2Var (u)=p1p, (2/3+7)76.
w—p21iu11-\-2p1p2w12-\-p22w22.
Switching to
The change in gene frequency in females
the new fitnesses doesn't change the selecAp1**=(l/x)(p»1x,1+ptptxlt)
tive system because the x's stand in the is given by
—pt=pip;fi/x~; in males we have Apt*=
314
DANIEL L. HARTL
(l/y)(Piyi)—Pi=Pip2y/yOverall Ap,=
( 2 / 3 ) 4 ^ **+_(l/3)A^* (Wright, 1969),
and since x=y we have &pi=pipt (2/?+y)/
To obtain an expression for
A y 3c assume that selection is sufficiently
slow so that (Apt)3 and higher powers are
very close to 0. Then Ay/x can be expanded
into a Taylor series
x \ (Apj)2
— )
J
\-o\ (Ap!
2 ^
assumption that the gene frequencies are
equal in the sexes, which they seldom are
exactly. Also the rate of change in fitness
has a component due to dominance and
deviations from dosage compensation. Both
of these effects become smaller as selection
becomes slower because (Apt)2 approaches
0.
A random mating diplo-diploid population with fitnesses Wu^n, and w2i in AJAJ,
AtAt, and A2Ae genotypes has in the absence of dominance a change in fitness
given by
when the initial average fitness is set equal
(wJS—wst)=:
to 1 and p=p, (wn—Wjg^pt
(wltiu^)/2.
When there is no dominance and complete dosage compensation in the male haploids then y=2/} and from equation (5) we
have
dx \ Apt
( dp, I 2yF
dp
3
Substituting dxjdp1=v (dw / dpt)-\-w (civ/
dp1)=2(3+y and dhc/dp21=2
(xh-\-2py)/x
where 8=x11—2xlg-\-xM leads to the equation
PiP* (2£+y) 2
Gxyjlc
?y)-(2/H-y)2\
8xy¥~
'
Sx
I
which is correct to the order of (A/?,)3.
In any single generation one can always
choose a set of w's and v's such that x=l.
Furthermore, if there is no dominance and
complete dosage compensation then 8=0
and 8 ^ 7 = (2y3-f-y)2. Making these substitutions one obtains
A\/J?i'=:oJiA.
(5)
This is the analogue of the fundamental theorem. It is not exact because of the
— — AwDipioia-
3
(6)
If the evolutionary process leading to
the substitution of amino acids in polypeptides is primarily selective and if the number of substitutions that occur is limited by
the speed at which they can be substituted
(instead of by the unliklihood of their
occurrence), then equation (6) implies
that the number of amino acid substitutions which occur in a polypeptide during
the evolution of a male haploid line (such
as a hymenopteran) should be 1/3
greater than the number of substitutions in
the homologous polypeptide of a comparable diplo-diploid line. On the other
hand, if the amino acid substitutions come
about primarily by genetic drift, then the
number of amino acid substitutions will be
roughly the same in haplo-diploids and
diplo-diploids. This is because in a finite
population the probability that a single
gene selected at any moment will eventually supplant by drift all the other genes at
that locus is I/A" where N is the total
number of genes at the locus in the population. At the same time the expected
NATURAL SELECTION IN MALE HAPLOIDS
315
number of new neutral mutations at the
locus is ATp where /j. is the neutral mutation rate. The rate of substitution of neutral mutations is therefore (l/iV)A'ju = p
(Crow and Kimura, 1970) which is independent of the population size and whether the population is haplo-diploid or diplo-diploid.
This suggests that comparisons of the
rate of amino acid substitution in homologous proteins in haplo-diploid and diplodiploid populations could supply important information on whether the substitutions are primarily selective or primarily
neutral. The logic behind the comparisons
assumes that the number of substitutions
of selectively favored mutations is limited
by the speed at which they can be substituted rather than by the rarity of their
occurrence. We ordinarily think in terms
of the latter, but some recent competition
experiments in bacteria involving a mutator strain (Gibson, et al., 1970) suggest
that favorable mutations may not be as
rare as often assumed. On the other hand,
if it is true that the rate of substitution of
favored alleles is limited by the number
that occur, then the number of substitutions in sex-linked polypeptides in diplodiploids should be 1/4 smaller than the
number of substitutions in autosomal polypeptides because a diplo-diploid population has 1/4 fewer sex-chromosomes than
autosomes. Under the hypothesis that the
substitutions are primarily brought about
by the random fixation of neutral alleles,
the rates of substitution in haplo-diploids,
in the sex-linked polypeptides of diplodiploids, and in the autosomal polypeptides of diplo-diploids should all be about
the same. (For some recent discussions of
whether the substitutions are primarily selective or primarily neutral, see Kimura,
1968, 1969; King and Jukes, 1969; Smith,
1970).
assumes special importance in male haploids is the selection of an optimum sex
ratio. This is an intricate but thoroughly
engrossing problem which has been solved
only in certain special cases. In a random
mating population the equilibrium sex ratio turns out to be 1/2 (Fisher, 1930;
Hamilton, 1967; Hartl and Brown, 1970).
On the other hand, Hamilton (1967) has
observed that "among small arthropods,
whenever reproduction is quite regularly
by brother-sister mating there seems to be
extreme economy in the production of
males. . . ." In all cases which have
been investigated cytologically, moreover,
this biofacies is accompanied by arrhenotokous reproduction. Intuition tells us that
males should be rare in such species because the species are parasitic and the
reproductive potential of a female depends
on the number of fertilized daughters she
produces; a female's best strategy is to produce only enough males to fertilize all of
her daughters. When some outcrossing occurs, however, it is to a female's advantage
to produce a few more sons so that they
might fertilize somebody else's daughters.
Thus there will be conflicting tendencies
and an optimum sex ratio will exist somewhere between 0% and 50% males.
THE SEX RATIO IN MALE HAPLOIDS
One drawback of Hamilton's models is
that they implicitly assume that each infection gives rise to an infinite number of
progeny. This can be seen in the equation
above; as n -»1 (that is, one female
The results of the preceding two sections
apply as well to the sex-linked genes in
diplo-diploids as they do to haplo-diploids.
One population genetic problem which
The general solution is not known.
Hamilton (1967), in a provocative paper
on the subject of sex ratio, containing
ingenious models and an extensive review,
has produced an interesting approach for
the case in which control of the sex ratio is
through the males. He also provides an
approximate solution appropriate when
the sex ratio is controlled by autosomal
genes. His results are all plausible but it
would be desirable in addition to have
them derived by different methods. In the
case of autosomal control the optimum
proportion of males is given as (n—l)/2n
where n is the number of females which
parasitize an uninfected host.
316
DANIEL L. HARTL
parasitizes an uninfected host), the optimum proportion of males among the progeny goes to zero, which makes sense only
if the number of progeny from each infected host is exceedingly large. It is interesting to ask how the optimum sex ratio is
affected by the fact that the brood sizes are
finite.
The model to be considered is the following. Assume a number of hosts which is
very large compared to the population
size of the parasite. Assume that uninfected
hosts become infected by at most one female. The females are assumed to have
mated prior to the infestation and are assumed to have stored in their spermathecae, sperm capsules, and associated organs
sufficient numbers of sperms to fertilize all
of their eggs. The eggs are laid and develop concurrently. Upon eclosion mating
occurs. We assume that the males can mate
repeatedly and can therefore inseminate
all of their sisters. After mating, the fertilized females emigrate and infect new hosts.
The above idealization abstracts some
of the major aspects of what Hamilton
(1967) has called "a biofacies of extreme
inbreeding and arrhenotoky." It is not
strictly necessary that hosts become infected by single females; all that is required is
that the body size of the host be large
relative to the body size of the parasites so
that when superinfection occurs there will
still be only a negligible amount of outcrossing (e.g. species of Xyleborus and the
trees they parasitize). In jDiactice these
postulates partition the parasite population into a large number of reproductively
isolated sibmating families. The absence of
gene flow between the families will tend
to accentuate differences and maximize
competition between them. One may assume that each family is highly homozygous. To the extent that there is genetic
variation in the sex ratio, natural selection
will favor those females which produce
the optimum one.
In the simplest case assume that every
egg laid survives and that all host;, are
equally productive. In this situation the
reproductive potential ol a female is pro-
portional to the number of fertilized
daughters she produces. This assumes that
broods consisting entirely of females make
no contribution to future generations.
This is not entirely correct, as the females
of many arrhenotokous species are sufficiently long-lived that they can mate with
their parthenogenetic sons. Hamilton
(1967) notes that "this ability is widespread." As examples one can cite Melillobia acasta (Browne, 1922), Cephalonomia
quadridentata (Van Emden, 1931), Podapolipus diander (Volkonsky, 1940),
Xyleborus compactus (Entwistle, 1964),
and Xyleborus ferruginens (Norris and
Baker, 1968). On the other hand, although
uninseminated females are not completely
doomed, they must suffer a substantial loss
in reproductive value. This is because of
the time lag that occurs while the virgin
females are waiting for their sons to mature. In fact, this time lag is likely to be
approximately equivalent to a doubling of
the developmental time. That is to say,
the reproductive value of an uninseminated female is of the same order of magnitude as that of an inseminated female
whose progeny take twice as long to develop as normal. Lewontin (1965) has elegantly shown that an effective doubling of
developmental time in a colonizing species
may be practically equivalent to sterility.
Thus, although all-female broods do contribute to future generations, they may do
so only to a limited extent. (It is interesting to note in this context that in Podapolipus diander the son with which an uninseminated female mates is a paedogenetic
parasite of its mother, precocious in development, and morphologically distinct
from the normal males in the population.
Volkonsky, 1940). The quantitative question of how infertile uninseminated females are is a difficult one. It depends on
the population sizes and intrinsic rates of
increase of the host and parasite, on the
prevailing sex ratio of the parasites, and
on other factors as well. Without much
difficult), one can imagine conditions in
which uninseminated females have an advantage over inseminated ones (See
317
NATURAL SELECTION IN MALE HAPLOIDS
TABLE 3. Examples of the ideal extreme biofneies (from Hamilton, 1967, copyright by the American Association for the Advancement of Science, with permission).
Typical brood
Species
Camphractus cinctus
Anaphoidea nitens
Cephalonomia quadridenlata
Dusmetia sangwani
Thysanus elongatus
Anaphoidea calendrae
Telenomus fariai
Prestwichia aquatica
Xlyeborus compactus
Perisierola emigrata
Elasmus hispidarum
Monodontomerns species
Acarophenax tribolii
Plenrotropis parvuhis
Nasonia vitripennis
Limothrips denticornis
Sderodermus imrnigrans
Caraphractns cinctus
Melittobia acasta
Asolcus species
Melittobia chalybii
Tarsonemoides species
Trichogramma semblidis
Pyernotes ventricostis
Siteroptes graminum
Blasophaga psenes
• broods on two different hosts.
Males
1
1
1
1
1
1
1
1
]
2
4
1
1
4
9
i
4
5
1
8
2
5
10
4
7
22
Browne, 1922, for example).
In the simplest case, the maximum
reproductive potential of a brood of x offspring occurs when the brood consists of
exactly 1 male and x—1 females. This is
undoubtedly why Hamilton (1967) places
such emphasis on the tendency in the extreme biofacies toward the inclusion of one
male per brood.
Table 3 is abstracted from Hamilton's
(1967) article. The references appropriate
to each species can be found in his paper.
The striking point about Table 3 is the
large number of cases in which there is
exactly one male in the typical brood. This
is especially true when the brood size is
small. Table 3 includes 17 cases in which
the typical brood is less than 25 progeny;
of these, 11 have exactly one male per
brood.
It is difficult to imagine that such a virtual triumph of theory should pose a problem, but it does. Although a female may be
able to lay only one unfertilized egg, she
cannot ensure with certainty the survival
Females
2
3
3
5
5
6
6
8
9
8
8
12
14
13
19
20
20
25
46
40
50
65
60
86
140
235
Total
3
4
4
6
6
7
7
9
10
10
12
13
15
17
21
23
24
30
47
48
52
70
70
90
147
257
Proportion
of males
.33*
.25
.25
.17
.17
.14
.14
.11
.10
.20
.33
.08
.07
.24
.10
.13
.17
.17*
.02
.17
.04
.07
.14
.04
.05
.09
of that egg'. Cannibalism is widespread in
the extreme biofacies (For examples, see
Browne, 1922; Volkonsky, 1940; Entwistle,
1964). This fact alone implies that not all
eggs can possibly survive. (I must make an
exception of certain parasitic hymenopterans which lay only a few eggs all of
which survive, e.g. Thysanus elongatus;
Clausen, 1924. In these cases the simple
model above is appropriate. The subsequent discussion applies primarily to mites
and ambrosia beetles, and to those hymenopterans which do have some attrition
in the hosts.) In addition, there will always be statistical variation among hosts in
the sizes of broods they will support.
These two elements of randomness, random survival of the individuals within a
brood and the randomness of the brood
sizes themselves, should be taken into account in understanding the adjustment of
the sex ratio. The question to be posed is
whether the data in Table 3 are consistent
with what would be expected on. the basis
of a more realistic model.
318
DANIEL L. HARTL
Let me first replace the assumption that
every egg survives with the possibly more
realistic assumption of binomial survival.
Assume that the proportion of unfertilized
eggs laid by a female is r and that the
proportion of fertilized eggs laid by a female is 1—r. Assume also that every host
produces an equal number, x, of progeny.
Then if survival is binomial, the probability that a brood consists at maturity of
exactly a females and b males (a-\-b = x) is
given by P{a,b) — (x\
(1—r)«r6. The rea
productive potential of a brood, W, can
be set equal to the number of inseminated
females in the brood. Thus, unisexual
broods have a reproductive potential of 0,
and if a single male is capable of fertilizing
x—1 females then a brood of a females and
b males (0<&<x) has a reproductive potential of a.
The average reproductive potential of
females which produce a fraction, r, of
unfertilized eggs is therefore
x-l
(1—r) a r b .
W(r) =
IB
The sum Va/xWl—r) a r b =x (1—r) is
a— I
the mean of the binomial distribution.
Equation (7) is equal to this except for the
last term, x (1—r)x, which is lacking in (7).
Thus we may write
W(r)=x(l—
r)—x(l —r)*.
(8)
To find the optimum sex ratio maximize
(8) with respect to r:
dW(r)
r
fi
dr
i—1]=0.
This has the solution
f—1— e [(ln*)/(l-z)].
The solution r is a maximum because
d'HV (r)/dr* < 0. Note also that dW (r)/
dr = x (.v— 1) > 0 when r . 0 and dW (>•)/
dr = —x < 0 when r = 1. Thus f is always
between 0 and 1.
The curve labelled A in Figure 1 gives
MEAN NJM3ER OF PROGENY PER BROOD
FIG. 1. Optimum sex ratios when the number of
progeny per brood is finite. Curve B assumes (1) no
attrition within broods, and (2) all hosts equally
productive; curve A assumes (1) binomial survival
within broods, and (2) all hosts equally productive;
curve C assumes (1) binomial survival within
broods, and (2) a Poisson distribution of progeny
among hosts. The dotted curve gives the probability
of all-female broods (the ordinate) under the assumptions of curve A. The points are from Table
3; a number in parentheses near a point is the
number of species corresponding to that point.
the values of f corresponding to various
values of x. This is to be compared with
curve B, which is the function r = l/x
and corresponds to the simple model of no
attrition. The points in the Figure are
taken from Table 3.
It is obvious from comparing curves A
and B that the observed sex ratios are
considerably smaller than would be expected on the basis of binomial survival.
Yet it is also clear that the optimum sex
ratio should be greater when survival is
binomial than it is when no attrition occurs. In the binomial model a fraction
(1—r)x of all broods consists entirely of females. This is a substantial fraction. When
r = l / x , (1—r)* ranges from 25% to
around 36%. The probability of an allfemale brood when r = l/x is shown as
the dotted curve in Figure 1 (the ordinate
of the dotted curve should be interpreted
as a probability, not a sex ratio).
This is where the problem arises. The
sex ratios observed are substantially lower
than the optima predicted by the theory.
One way of rationalizing the discrepancy is
to argue that all-female broods are not
319
NATURAL SELECTION IN MALE HAPLOIDS
effectively sterile, that the reasoning based
on the analogy with developmental time is
inappropriate; another possibility is that
the presence of host-to-host variability in
the number of parasites supported might
tend to lower the optimum sex ratio.
If we assume that the distribution of
brood sizes is Poisson, then when the average brood size is A the probability that a
brood consists of x individuals is given by
e~x (\x/x!). The reproductive potential of
a brood consisting of x individuals with a
sex ratio of r is still x (1—r)—x (1—r)x, so
the average reproductive potential of a female which produces a fraction r of unfertilized eggs is
W(r)=y
e->^[x(l—r)—x (1— r)*].
(9)
Poisson distribution the variance is equal
to the mean. In actual progeny distributions, the variance in progeny number is
ordinarily larger than the mean (See, for
example, Crow and Morton, 1955). Thus,
the distribution of progeny must have at
least two parameters. An additional property that would be desirable is that, like
the Poisson, the form of the distribution
should not change throughout the period
of random survival. One distribution possessing these properties is the negative
binomial or Pascal distribution. This distribution is preferable to the Poisson distribution when the broods are likely to
live or die as a group, which is probably
close to what actually occurs.
When progeny are distributed according
to the negative binomial the probability
of a brood of size x is
Differentiating (9) with respect to r and
ts-\-x—\\qxp'
00
using
00
the
identities V e- m (m'/i/)=:l,
^
where p = 1—q and where the mean L and
variance V of the distribution are given by
00
t=0
P
t=0
m(l-J-m) leads to
(10)
When equation (10) is equal to 0, r satisfies the equation
e~^[A(l—r)+l] = l.
(11)
Note when r = 0, dW (r)/dr — A2>0
and when r = 1, dW(r)/dr = _ A [ 1 — e ~ x ]
<0. Therefore equation (11) always has a
solution f such that 0 < f < l .
The curve labelled C in Figure 1 gives
the optimum sex ratios for the various values of A, the mean number of progeny per
brood. These values were obtained from
equation (11) by means of an interval
bisection procedure. As can be seen, a
Poisson distribution of progeny does not
reduce the optimum sex ratio by very
much and only when the average brood
size is quite small.
On the other hand, a Poisson distribution of progeny is not very realistic. In the
Assuming binomal survival the reproductive potential of a female that produces a
fraction r of unfertilized eggs is
W(r) =
[x (1—r)—x (1—r)*]
ts+x—\\qxp:
(12)
Differentiating (12) with respect to r
and noting the identities
- l \ q x = q s ( l — q ) - ' - 1 , and
3=0
x2/s-\-x—l\q*=s2q2
X—0
-\-qs(l—q)~s~2
leads to
(l—q) —
8-1
320
DANIEL L. HARTL
MEAN NUMBER OF PROGENY PER BROOD
FIG. 2. Optimum sex ratios when survival within
broods is binomial and when the distribution ot
progeny is negative binomial. A-G correspond to
distributions with increasingly large variance. When
the mean number of progeny per brood is L, the
curves A-G correspond to variances of It. (A), 5L
(B), I0L (C), 20L (D), 50L (E), 70/- (F), and
100Z. (G). Points as in Figure 1.
dW{r)
—sq
dr
p
(13)
Notice that dW(r)/dr at r = 0 is s (l+s)q2/
[)- > 0 and dW(r)/dr at r = l is sq (p^1
— 1)/P < 0- Therefore (13) has at least one
root in (0,1).
Equation (13) has been solved numerically by the interval bisection technique.
The sex ratios which produce the maximum reproductive potential when the
progeny distribution is negative binomial
arc given in Figure 2. Also in Figure 2 are
the observed sex ratios from Table 3. The
curves labelled A through G are the "unbeatable sex ratios" when the variance in
progeny number is 2, 5, 10, 20, 50, 70, and
100 times the mean, respectively. Except
for the two points which are too high, the
observed sex ratios fall well within this
family of distributions.
The result is somewhat disturbing, however, because the variances required to account for the observed sex ratios seem unreasonably large. Whether they are in fact
too large is a matter ior determination
from field studies. Crow and Morton
(1955) have published data on the mean
and variance of the adult progeny distribution of Drosophila melanogaster strains
raised under laboratory conditions. Crow
and Morton point out that the mean number of progeny per mating pair per generation cannot be very far from 2 "without
the population increasing explosively or
quickly becoming extinct." They have used
as a parameter for interpopulational comparisons the ratio of the variance in progeny number to the mean progeny number
when the mean is adjusted to a value of
2. They call this the "index of variability,"
and when survival of individuals is random they show the index of variability to
be equal to
2
(cf. their equation 14).
In their Drosophila laboratory populations the adjusted index of variability
ranged from 1.2 to 6.1. Using as a rough
estimate of the variance in progeny number the variance of the curve closest to
each point in Figure 2, and excluding the
two points which are obviously too high, I
find the adjusted index of variability of the
points in Figure 2 to be approximately 8.
This is substantially larger than in Drosophila. On the other hand, families of
parasites are likely to survive or to die as a
group, and this will tend to raise the index of variability. Moreover, one would
expect, even in the case of random survival
of individuals, that the index of variability of populations in nature will be larger
than that observed in the laboratory.
The assumption that the number of progeny has a negative binomial distribution
is probably not unreasonable, but cases are
rare indeed in which data are reported in
such a way as to allow a test of this hypothesis. Entwistle's (1964) data on the
progeny distribution of Xyleborus compactus look suspiciously negative binomial but
because of the way the classes are pooled
it is impossible to be certain.
In the negative binomial distribution
the effects of increasing the variance while
321
NATURAL SELECTION IN MALE HAPLOIDS
TABLE 4. The effects of increasing the variance in a negative binominal distribution with mean
10.
Mean
10
10
10
10
10
Variance
500
50
100
250
500
Proportion of broods
containing 0-3 progeny
.05
.16
.30
.49
.63
Brood size for which
cumulative
probability = 09%
30
40
50
80
110
holding the mean constant at some fairly sterile as assumed. Although unfertilized
small value are twofold. First, the classes females are generally easy to produce in
with very few or no progeny become more the laboratory, some real doubt exists as to
common and, second, the right-hand tail how often they occur in nature. This
of distribution is stretched out. This is probably depends on the species. Entwistle
shown in Table 4 where the distributions (1964) notes that in Xyleborus compacwhich have a mean of 10 and increasingly tus, "of the total number of females, nearly
large variances are summarized. Although 6 per cent did not have access to males in
the right-hand tail is greatly extended, the the parent gallery." Browne (1922), on the
probability of each individual class is contrary, speaking of Melittobia acasta,
about the same (not exactly the same, but says, "I have never, under natural condireasonably close). Therefore if progeny tions, come across an unfertilized female;
are distributed this way in nature, with a that is, every female which has been allarge variance, then one would expect to lowed to emerge from the pupal stage in
find: (1) a great many infestations which the cell where it grew up mates before
lead either to no progeny at all or to some leaving that cell." So it is important to
exceedingly small number, and (2) that distinguish between unfertilized females
when an infestation is successful it is al- having the ability to mate with their armost as likely to have a great many pro- rhenotokous sons and subsequently progeny as it is to have some number near the ducing daughters, and whether any signifimean. This is a proposition that may be cant fraction of females in nature are
difficult to test in the field; most studies called upon to use this ability.
have not been designed to detect infections
Perhaps a more important reason why
which are unsuccessful.
the observed sex ratios are lower than exAlthough I consider it likely that the pected is that survival may not be binomivariance in progeny number in nature will al. That is, perhaps an unfertilized egg is
be very large, it would be surprising if it more likely to survive than a fertilized egg.
turned out to be large enough to account Hamilton (1967), in his empirical "outline of an ideal extreme biofacies" makes a
for the low observed sex ratios.
The possibility of some outcrossing point of noting that adult males eclose
makes matters even worse. As Hamilton first. This may be to ensure that the
(1967) points out, outcrossing tends to in- daughters will be mated before they emicrease the proportion of males in the pop- grate, but it could also provide an advanulation. Inasmuch as the theoretically pre- tage in viability to the males. Browne
dicted sex ratios are already too high, one (1922) notes that in Melittobia acasta the
is reluctant to make the discrepancy higher males usually emerge first in a brood and
still by including the possibility of signifi- concludes that either the first eggs laid by
a female are unfertilized, a situation which
cant outcrossing.
There remain two ways to account for would be exactly the reverse of that in
the observed sex ratios. Perhaps uninsemi- Megacliile rotundata (Cerber and Klosternated females are not as reproductively meycr, 1970), or that males develop more
322
DANIEL L. HARTL
rapidly than females. In either case the have served a useful purpose by calling
developing males would not be subject to attention to those aspects of the life cycle
as much intrabrood competition as the fe- that are especially worth examining.
males.
So far I have discussed only those species
One other point that should be made is that have fairly small brood sizes (less
that the reported sex ratios in Table 3 may than 25). When the broods are very large,
be biased downward. This is certainly then there will be a tendency for the inclutrue of Melittobia acasta in which Browne sion of more males than before in order to
(1922) has found that in addition to the ensure that all of the females will be fertilmale eclosing first in the cells, "two ized. This case can be handled as the premales, on meeting, usually engage in mor- vious ones. Assume that a single male is
tal combat." He does concede, however, able to fertilize a number -q of females.
that when the cell is full of emerging fe- Then if all hosts give rise to exactly x
males the males are usually too busy to pay progeny and if survival within a brood is
much attention to one another. In any binomial, the average number of fertilized
event, the great pugnacity of the males of females per brood is
some species (See Hamilton, 1967) will
x—\
bias the observed sex ratios. The most diffix
cult problem is to decide how much confi- W(r) = ~y min[ (x—i)rj, i] (x\ (1—r)h -\
dence should be placed in data such as
«=i
'
(14)
those in Table 3. Clausen (1939) has arEquation (14) has been maximized for
gued that "the idea of the sex ratio of any various values of x and rj. Some of the
given [male haploid] species being even results are presented in Table 5, along
approximately a fixed figure is entirely un- with the optimum sex ratios predicted if a
tenable. It will vary: (1) with the sex ratio single male could fertilize all the females
of the host; (2) with successive gener- in a brood. It can be seen that if the males
ations upon the same or a different host are able to fertilize only a small number
generation; (3) with different hosts; (4) of females relative to the total brood size,
upon the same host and in the same sea- then the optimum sex ratio is substantially
son, but in different geographical regions; greater than when there is no limit to the
and (5) in successive years when the host number of fertilizations by each male.
population is increasing or declining rapidly.
SUMMARY
In summary, the observed sex ratios can
be accounted for if either: (1) the distribution of progeny numbers among hosts
has a very large variance, or if (2) unfertilized eggs are more likely to survive than
fertilized eggs, or if (3) newly emergent
males tend to kill each other. Carefully
collected data from the field would be
most welcome in assessing the relative importance of these possibilities. It could
turn out that some important aspect of the
extreme biofacies has not yet been taken
into account. Nature is sometimes so complex that, as Salt (1936) has cautioned, it
"should give pause to those who still think
to solve the problems of parasite and host
relations on paper by mathematical
speculation." In such a rase the theory will
Male haploidy is only known to have
evolved eight times, once each in the rotifer order Monogonanta, the arachnid order
Acarina, the insect orders Hymenoptera
and Thysanoptera, and in two families of
Homoptera (Coccidae and Aleyrodidae)
and Coleoptera (Scolytidae and Micromalthidae). Judged on this standpoint,
male haploidy is not a very attractive evolutionary innovation, but judged on the
basis of numbers of male haploid species
the mode of reproduction is a success.
About 1/7 of all insect species are hymenopterans, an order which is exclusively
male haploid except for some thelytokous
species. Thus it is of some real interest to
deteiminc whether the leproducthe s\stem
itself has evolutionary advantages which
323
NATURAL SELECTION IN MALE HAPLOIDS
TABLE 5. Optimum sex ratios when the number of fertilizations per male is limited.
Brood
size
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
Maximum number of
fertilizations per male
Optimum
sex ratio
5
5
Optimum sex ratio
when no limit to
fertilizations per male
.33
!28
.26
3.5
.24
.33
.23
.18
.15
.13
10
10
10
10
10
.33
.23
.21
.19
.18
.33
.23
.18
.15
.13
15
15
15
15
15
.33
.23
.18
.17
.15
.33
.23
.18
.15
.13
5
5
5
have contributed to the hymenopterans'
success.
This paper is divided into three sections. In the first, the conditions for the
existence of a stable equilibrium at a diallelic locus under random mating are determined by means of the maximization of
geometric mean fitness, subject to the constraints imposed upon the gene frequencies
by the system of inheritance. Overdominance is a necessary but not a sufficient
condition for the maintenance of a stable
polymorphism when there is dosage compensation in fitness. From this it is argued
that the number of polymorphisms maintained by overdominance should be less in
a male haploid population than in a diplodiploid population.
In the second section it is shown that the
increment in geometric mean fitness of a
male haploid population is approximately
equal to the additive genetic variance in
fitness, where the additive variance is
defined as the variance due to the regression on a model of gene action which assumes no dominance in females and complete dosage compensation in males. Under
these assumptions, and random mating, a
male haploid population evolves 1/3 faster
than the corresponding diplo-diploid
population. The implications of this result
are discussed in relation to experiments to
determine whether the amino acid substitutions that occur during the evolution of
a polypeptide are primarily selective or
primarily neutral.
The third section of the paper deals
with an extension of Hamilton's (1967)
theory of the sex ratio in the ideal extreme
biofacies of inbreeding and arrhenotoky.
The extreme biofacies comprises parasitic
species which reproduce primarily by
brother-sister mating. If all eggs laid survive and if all hosts are equally productive, then the optimum sex ratio is produced when a brood contains exactly one
male. This theory has been extended to
include random survival of individuals
within broods (assumed to be binomial)
and random numbers of progeny among
hosts (assumed to be either Poisson or negative binomial). In the latter case the
optimum proportion of males produced by
a female, that is, the proportion of males
which maximizes her reproductive potential, is given by the solution r of the equation
-sq
sq[\+sq {\-r)]p«
qs
where p= 1—q and — is the mean of the
P
distribution of progeny among hosts and
qs (
q \
— I 1 + — ) is the variance.
p\
p)
324
DANIEL L. HARTL
gy. J. & A. Churchill, London.
The sex ratios predicted by the theory
Donner,
J. 1966. The Rotifers. Frederick Warne and
are compared with values observed in the
Company, London.
field, and the most important finding is Edwards, A. W. F. 1961. The population genetics of
that the observed sex ratios are too low to
"sex-ratio"
in
Drosophila
pseudoobscura.
Heredity 16:291-304.
be accounted for by the theory unless either: (1) the variance of the progeny dis- Entwistle, P. F. 1964. Inbreeding and arrhenotoky
in the ambrosia beetle Xyleborus compactus
tribution is very large, or (2) unfertilized
(Eichh.) (Coleoptera: Scolytidae). Proc. Roy.
eggs are more likely to survive than fertilEntomol. Soc. (London), Ser. A. 39:83-88.
ized eggs, or (3) newly emergent males Ewens, W. J. 1969. A generalized fundamental theorem of natural selection. Genetics 63:531-537.
tend to kill each other and thus bias the
Fisher, R. A. 1930. The genetical theory of natural
observed ratios downward.
selection. Clarendon Press, Oxford.
REFERENCES
Bennett, J. H. 1957. Selectively balanced polymorphism at a sex-linked locus. Nature 180:13631364.
Bennett, J. H. 1958. The existence and stability of
selectively balanced polymorphism at a sex-linked
locus. Australian J. Biol. Sci.l:598-602.
Bennett, J. H. 1963. Random mating and sex linkage. J. Theoret. Biol. 4:28-36.
Borror, D. J., and D. M. DeLong. 1954. An introduction to the study of insects. Rinehart and
Co., New York.
Browne, F. B. 1922. On the life-history of Melittobia acasta, Walker; a chalcid parasite of bees and
wasps. Parasitology 14:349-371.
Cannings, C. 1967. Equilibrium, convergence and
stability at a sex-linked locus under natural selection. Genetics 56:613-617.
Cannings, C. 1968. Equilibrium under selection at a
multi-allelic
sex-linked
locus.
Biometrics
24:187-189.
Chapman, R. F. 1969. The insects. American Elsevier Publ. Co., New York.
Clausen, C. P. 1924. The parasites of Pseudococcus
maritimus (Ehrhorn) in California (Hymenoptera, Chalcidoidea). Part II. Biological studies
and life histories. Univ. Calif. (Berkeley) Publ.
Entomol. 3:253-293.
Clausen, C. P. 1939. The effect of host size upon
the sex ratio of hymenopterous parasites and its
relation to methods of rearing and colonization.
J. N. Y. Entomol. Soc. 47:1-9.
Cormack, R. M. 1964. A boundary problem arising
in population genetics. Biometrics 20:785-793.
Crow, J. F., and N. E. Morton. 1955. Measurement
of gene frequency drift in small populations.
Evolution 9:202-214.
Crow, J. F., and M. Kimura. 1956. Some genetic
problems in natural populations. Proc. Symp.
Math. Stat. Prob., 3rd Berkeley 1-22.
Crow, J. F., and M. Kimura, 1970. An introduction
to population genetics theory. Harper and Row,
New York.
Cro/ier, R. H. 1970. Coefficients of relationship and
the identity of genes by descent in the Hymenoptera. Am. Naturalist 101:216-217.
Darlington, C. D. 1932. Recent advances in cytolo-
Fisher, R. A. 1941. Average excess and average
effect of a gene substitution. Ann. Eugen.
11:53-63.
Gerber, H. S., E. C. Klostermeyer. 1970. Sex control
by bees: A voluntary act of egg fertilization
during oviposition. Science 167:82-84.
Gibson, T. C, M. L. Scheppe, and E. C. Cox. 1970.
Fitness of an Escherichia coli mutator gene.
Science 169:686-688.
Haldane, J. B. S. 1923. A mathematical theory of
natural and artificial selection. Part I. Proc. Cambridge Phil. Soc. 23:19-41.
Haldane, J. B. S. 1926a. A mathematical theory of
natural and artificial selection. Part III. Proc.
Cambridge Phil. Soc. 23:363-372.
Haldane, J. B. S. 19266. A mathematical theory of
natural and artificial selection. Part IV. Proc.
Cambridge Phil. Soc. 23:607-616.
Haldane, J. B. S. 1927. A mathematical theory of
natural and artificial selection. Part V. Proc.
Cambridge Phil. Soc. 23:838-844.
Haldane, J. B. S., and S. D. Jayakar. 1964. Equilibria under natural selection at a sex-linked locus. J. Genet. 59:29-36.
Hamilton, W. D. 1963. The evolution of altruistic
behavior. Am. Naturalist 97:354-356.
Hamilton, W. D. 1964a. The genetical evolution of
social behavior. T. J. Theoret. Biol. 7:1-16.
Hamilton, W. D. 1964b. The genetical evolution of
social behavior. II. J. Theoret. Biol. 7:17-52.
Hamilton, W. D. 1967. Extraordinary sex ratios.
Science 156:477-488.
Hartl, D. L. 1970. Population consequences of
non-mendelian segregation among multiple alleles. Evolution 24:415-423.
Hartl, D. L., and S. W. Brown. 1970. The origin of
male haploid genetic systems and their expected
sex ratio. Theoret. Population Biol. 1:165-190.
Hughes-Schrader, S., and D. F. Monahan. 1966.
Hermaphroditism in Icerya zeteki Cockerell, and
the mechanism of gonial reduction in iceryine
coccids (Coccoidea: Margarodidae Morrison).
Chromosoma 20:15-31.
Kimura, M. 1956. Rules for testing stability of a
selective polymorphism. Proc. Natl. Arad. Sci. U.
S. 12:336-310.
Kimura, M. 1958. On the change of population
fitness b\ natural seli-ilion. HeudiH 12:1 4.V167.
Kimura, M. 1968. Evolutionary rate at the molecu-
NATURAL SELECTION IN MALE HAPLOIDS
lar level. Nature 217:624-626.
Kimura, M. 1969. The rate of molecular evolution
considered from the standpoint of population
genetics. Proc. Natl. Acad. Sci. U. S. 63:1181-1188.
King, J. L. and T. H. Jukes. 1969. Nondarwinian evolution. Science 164:788-798.
Kirkman, H. N. 1966. Properties of X-linked alleles
during selection. Am. J. Human Genet. 18:424432.
Lewontin, R. C. 1965. Selection for colonizing ability, p. 77-91. In H. G. Baker and G. L. Stebbins,
[ed.], The genetics of colonizing species. Academic Press, New York.
Li, C. C. 1955. The stability of an equilibrium and
the average fitness of a population. Am. Naturalist 89:281-295.
Li, C. C. 1967a. The maximization of average
fitness by natural selection for a sex-linked locus.
Proc. Natl. Acad. Sci. V. S. 57:1260-1261.
Li, C. C. 19676. Fundamental theorem of natural
selection. Nature 214:505-506.
Li, C. C. 1967c. Genetic equilibrium under selection. Biometrics 23:397-484.
Li, C. C. 1969. Increment of average fitness for
multiple alleles. Proc. Natl. Acad. Sci. U. S.
62:395-398.
Mandel, S. P. H. 1959. Stable equilibrium at a
sex-linked locus. Nature 183:1347-1348.
Mandel, S. P. H. 1968. Fundamental theorem of
natural selection. Nature 220.1251-1252.
Moran, P. A. P. 1962. The statistical processes of
evolutionary theory. Clarendon Press, Oxford.
Norris, D. M., and J. M. Baker. 1968. A minimal
nutritional substrate required by Fusarium solani to fulfill its mutualistic relationship with
Xyleborus ferrugineus. Ann. Entomol. Soc. Am.
61:1473-1475.
325
Salt, G. 1936. Experimental studies in insect parasitism. IV. The effect of superparasitism on populations of Trichogramma evanescens. J. Exptl.
Biol. 13:363-375.
Shaw, R. F. 1959. Equilibrium for the sex-ratio
factor in Drosophila pseudoobscura. Am. Naturalist 93:385-386.
Smith, J. M. 1970. Population size, polymorphism,
and the rate of non-Darwinian evolution. Am.
Naturalist 104:231-327.
Sprott, D. A. 1957. The stability of a sex-linked
allelic system. Ann. Human Genet. 22:1-6.
Takenouchi, Y., and K. Takagi. 1967. A chromosome study of two parthenogenetic scolytid beetles. AnnotaCiones Zool. Japon. 40:105-110.
Turner, J. R. G. 1969. The basic theorems o£
natural selection: A naive approach. Heredity
24:75-84.
Van Emden, F. 1931. Zur Kenntris der Morphologic
und Okologie des Brotkafer-Parasiten Cephalonomia quadridentata Duchaussoy. Z. Morphol.
Oekol. Tiere 23:425-574.
Volkonsky, M. 1940. Podapolipus diander N. sp.
acarien heterostygmate parasite du criquet migrateur (Locusta migratoria L.). Arch. Inst. Pasteur Algerie 18:321-340.
White, M. J. D. 1954. Animal cytology and evolution. 2nd ed. Cambridge University Press, Cambridge.
White, M. J. D. 1957. Cytogenetics and systematic
entomology. Ann. Rev. Entomol. 2:71-90.
Whiting, P. W. 1945. The evolution oE male
haploidy. Quart. Rev. of Biol. 20:231-260.
Wright, S. 1942. Statistical genetics and evolution.
Bull. Am. Math. Soc. 48:223-246.
Wright, S. 1969. Evolution and the genetics of
populations. Vol. 2. University of Chicago Press,
Chicago.