735
q 2006 The Society for the Study of Evolution. All rights reserved.
Evolution, 60(4), 2006, pp. 735?745
SEX RATIO, LIFE-HISTORY INVARIANTS, AND PATTERNS OF SEX CHANGE IN A
FAMILY OF PROTANDROUS GASTROPODS
RACHEL COLLIN
Smithsonian Tropical Research Institute, Apartado Postal 0843-03092, Balboa, Anco?n, Republic of Panama
E-mail:collinr@naos.si.edu
Abstract. Application of optimality theory to the evolution of life histories has been broadly successful in predicting
the conditions favoring sex change, the type of change, and the timing of such changes. The size advantage hypothesis
predicts that the optimal size at which an individual should change sex is a function of its size and the size and sex
of its potential mates. I collected data on the size, sex, and grouping of individuals of 27 populations of 19 species
of the calyptraeids, a family of protandrous marine gastropods that includes Crepidula. These data are used to test
the following predictions about variation in size at sex change: (1) sex ratio is biased toward the first sex; (2) the
ratio of the size at sex change to the maximum size is a life-history invariant; and (3) species that form variable-
sized groups or stacks have more variation in size at sex change than species that show less variation in stack formation.
Across all 19 species, sex ratio was not significantly more often biased toward the first sex than it was toward the
second sex, although sex ratios were significantly male biased more often than they were significantly female biased.
Sex ratios ranged from 0.05 to 0.91, and this variation was related to mode of development, skew in size distribution,
and frequency of stacking, but not with maximum body size. There was little evidence that the ratio of size at sex
change and maximum size is invariant. There is evidence that one of the main underlying assumptions of this life-
history invariant, that male fertility increases with the same function of size in all species, is invalid for calyptraeids
and probably for other animals. Finally, species that form larger stacks or mating groups had more variation in size
at sex change within a population than species that were generally solitary. These results suggest that information
about individual groupings should be included in predictions of life-history theory and that more information about
the relationship between male fitness and size is also needed.
Key words. Bostrycapulus, Calyptraeidae, Crepidula, protandry.
Received April 12, 2005. Accepted January 27, 2006.
Sex change is a widespread phenomenon that has been the
subject of a variety of theoretical and empirical evolutionary
studies (e.g., Warner 1975; Leigh et al. 1976; Hoagland 1978;
Charnov 1979, 1982; Shapiro and Lubbock 1980; Policansky
1981). Predictions of the conditions that favor sex change
and of the size/age at sex change are often cited as successful
examples of the application of optimality theory to the evo-
lution of life histories (Charnov 1982; Hardy 2002). Because
detailed demographic and breeding data are difficult and
time-consuming to obtain, detailed studies of these predic-
tions have usually been limited to single model species (e.g.,
bluehead wrasse, Thalassoma bifasciatum: Warner and Hoff-
man 1980; Warner 1987; Schultz and Warner 1989, 1990;
Warner and Swearer 1991; Warner and Schultz 1992; Peter-
son et al. 2001; peppermint shrimp, Lysmata wurdemanni:
Bauer and Holt 1998; Lin and Zhang 2001; Bauer 2002a,b,
2005; Baldwin and Bauer 2003; Baeza and Bauer 2004).
Recent theoretical developments have used simplifying as-
sumptions to make general predictions that can be applied
across disparate taxa (see below). This allows comparative
studies to use compilations of published data from distantly
related species to test generalizations about patterns of sex
change (e.g., all sex changing animals in Allsop and West
2003a). When strong support for predicted patterns is found,
the simplifying assumptions of these studies appear to be
justified. Because these studies have tended to combine a
haphazard assortment of disparate taxa with few species
drawn from a single genus or family, it is difficult to deter-
mine if the commonly observed high levels of variation and
deviations from expectations are due to a general failure of
the theory, or if there are differences among clades that each
independently conform to theoretical expectations. For ex-
ample, a recently compilation of sex ratio for 40 protandrous
animals (drawn from five phyla) demonstrated large range in
sex ratio (0.11?0.89; Allsop and West 2004). More detailed
sampling within more closely related monophyletic groups
is necessary to determine whether such large variation is due
to the differences in natural history between animals as dif-
ferent as sea cucumbers, marine isopods, and fishes, or if
such variation is also typical of more closely related species.
Here I use data from 27 populations of 19 species of calyp-
traeid gastropods to evaluate three predictions of sex change
theory and compare the patterns observed in this monophy-
letic family to those of a compilation of animals from the
literature (Allsop and West 2003a, 2004).
Background Theory
The size-advantage hypothesis predicts that sex change is
favored when reproductive output increases more quickly
with size or age in one sex than in the other (Warner 1975;
Leigh et al. 1976; Charnov 1979, 1982). The optimal size at
sex change occurs when the potential subsequent lifetime
reproductive output as the second sex exceeds that of re-
maining as the first sex. Because the expected reproductive
success depends on both size and/or age and the pool of each
animal?s potential mates, the optimal size at sex change
should vary in response to the size distribution or age struc-
ture of the population. Modifications of the size at sex change
in response to changes in demographics have been demon-
strated in pandalid shrimp and certain fishes after fisheries
have significantly decreased survival probabilities (e.g.,
Charnov 1981; Hannah and Jones 1991; Buxton 1993; Char-
nov and Hannah 2002).
736 RACHEL COLLIN
Extension of the optimality approach shows that the sex
ratio in sex-changing species should be biased toward the
first sex. At the evolutionary stable size at sex change the
adult sex ratio (r; males/total) is given by the equation (eq.
4 of Charnov and Bull 1989):
r/(1 2 r) 5 [F/f (t)]/[M/m(t)], (1)
where F is average female fertility, M is average male fer-
tility, f(t) is the fertility of a female at the size of sex change,
and m(t) is the fertility of a male at the size of sex change.
If it is assumed that both male and female fitness increase
with size (as is often the case in protandry) then M/m(t) ,
1 , F/f(t) (eq. 5 of Charnov and Bull 1989). As a result r/
(1 2 r) . 1 and therefore sex ratio is expected to be greater
than 0.5. For protogyny sex ratio is predicted to be less than
0.5. Data from protogynous coral reef fishes show more spe-
cies and populations with female-biased than male-biased sex
ratios (data from various sources reviewed in Charnov 1993).
Statistical deviations from 0.5 for each species were not ex-
amined, but the overall interspecific distribution appeared to
be biased toward the first sex. A recent comparative analysis
of published studies of various marine animals demonstrates
that, despite a wide range of sex ratios, protogynous species
are generally female biased and that protandrous species are
generally male biased (Allsop and West 2004).
A life-history invariant approach (Charnov 1993) has also
been applied to the study of sex-changing animals (Charnov
1993; Charnov and Sku?lado?ttir 2000; Allsop and West
2003a,b). This approach uses comparisons of dimensionless
ratios to search for constant relationships between traits that
might shed light on underlying organizing principles of evo-
lution. Optimality theory has been applied to sex change to
predict the following dimensionless life-history invariants
(Charnov and Sku?lado?ttir 2000): (size at sex change)/(max-
imum size) 5 constant and (age at maturity)/(age at sex
change) 5 constant. This theory appeared to receive strong
empirical support for marine animals on the basis of regres-
sion analyses (Allsop and West 2003a,b). Regression anal-
yses of data where the range of the dependent variable is
bounded by the independent variable (e.g., size at sex change
can never exceed maximum size) have since been shown to
be flawed (Buston et al. 2004; Cipriani and Collin 2005; Nee
et al. 2005) because the pattern predicted by the null hy-
pothesis is the same as predicted by the hypothesis of life-
history invariants (Cipriani and Collin 2005). Gardner et al.
(2005) conducted sensitivity analyses that held constant two
of the three values that are assumed by life-history invariance
theory to be constant (k/M, d, and aM, where k 5 von Ber-
talanffy growth coefficient, M 5 instantaneous mortality rate,
a 5 age at maturity, and d 5 exponential scaling coefficient
of male reproductive success with size; Charnov and Sku?-
lado?ttir 2000) and varied the third parameter. Their subse-
quent sensitivity analyses were also based on the flawed re-
gression analyses (Gardner et al. 2005, p. 563) and therefore
suffer from the same weakness as the previous, less sophis-
ticated analyses.
The incorrect application of statistical analyses to test for
invariants does not, however, invalidate the general idea or
hypotheses put forward by advocates of life-history invariants
or the usefulness of the dimensionless approach. Cipriani and
Collin (2005) suggested that simulation approaches could be
used to directly examine the variation in the observed ratios
with the idea that ratios showing less variation than expected
at random would offer some support of an underlying rela-
tionship between the variables. Such a result could possibly
be interpreted as support for life-history invariants. Appli-
cation of this test to previously published datasets showed
that several of them indeed had less variation in such di-
mensionless ratios than is expected at random (Cirpriani and
Collin 2005).
Predictions of sex ratio and life-history invariants are de-
rived using a number of basic simplifying assumptions. For
example, they assume a stable age distribution (see Charnov
1982) and equivalent mortality rates for males and females
(Charnov 1982, 1986). Life-history invariant theory makes
other more complicated assumptions about demographics
(e.g., k/M, d, and a M are invariant, Charnov and Sku?lado?ttir
2000). For most groups of sex-changing animals, data are
not available to test these assumptions.
Studies seeking to test these predictions often make another
assumption for which data may more often be available. The
most common assumption is that the individuals sampled
represent a single population and that there is a single optimal
size at sex change across that population. However sex-
changing species often have populations with structured mat-
ing groups. For example, labrid fishes often form harems,
anemone fish live in breeding pairs, and calyptraeid gastro-
pods often form clusters or stacks. Therefore, a large sample
of individuals, as would be collected to assess population
sex ratio, probably represents a pool of several groups. If the
individual composition of such groups varies, the optimal
size of sex change should also vary among the groups. In
species with labile sex change, where animals change sex in
response to cues from conspecifics, a population sample that
includes several groups would be expected to show consid-
erable within-population variation in size at sex change. Such
variation in size at sex change is common among calyptraeid
gastropods (e.g., Collin 1995, 2000).
Here I use comparative data on size-sex distributions of
calyptraeid gastropods to test the following predictions of
sex allocation theory: (1) sex ratios are biased toward the
first sex; (2) size at sex change/maximum size is less variable
than expected at random; and (3) intraspecific variation in
size at sex change is associated with social environment, such
that species with more variation in the social environments
to which different individuals are exposed show more vari-
ation in size at sex change than species in which all indi-
viduals are exposed to similar social environments.
MATERIALS AND METHODS
Natural History of Calyptraeids
Calyptraeid gastropods, or slipper limpets, are protandrous,
filter-feeding limpets that occur commonly in the intertidal
and shallow subtidal. The snails reproduce via copulation
and females can store sperm. The females brood embryos
that hatch as crawling juveniles in some species and plank-
tonic larvae in other species (Collin 2003a). Several species
form stacks of two or more individuals, where a small male
usually attaches to the shell of a larger female. Large stacks
737CALYPTRAEID SEX CHANGE
TABLE 1. Results of CONTINUOUS likelihood-ratio tests for significant differences between the maximum likelihood (ML) value of
l and the null hypothesis.
Variable
Null hypothesis
k (branch
length scaling)
l 5 0 (no
phylogenetic effect) l 5 1
ML estimate
of l
Sex ratio 0 P 5 1 P 5 0.01 0
1 P 5 1 P 5 1.5 3 1025 0
L50 and Lmax 0 P 5 0.47 P 5 0.002 0.14
1 P 5 0.83 P 5 3.3 3 1027 0.18
L50/Lmax 0 P 5 0.89 P 5 0.011 0.43
1 P 5 0.52 P 5 0.006 0.04
Proportion stacked 0 P 5 1 P 5 0.021 0
1 P 5 1 P 5 0.0035 0
Relative size overlap 0 P 5 1 P 5 0.03 0
1 P 5 1 P 5 0.0032 0
Proportion stacked and relative overlap 0 P 5 1 P 5 0.005 0
1 P 5 1 P 5 3.12 3 1025 0
of up to 10 or 12 individuals are known from several species,
most notably Crepidula fornicata. These groups are often
attached to limited substrates (e.g., shells or small cobbles)
in muddy habitats and to other gastropods in rocky habitats.
There are sometimes several stacks on each of these small
substrates. Other species more commonly occur in mono-
specific layers or individually on any hard substrate.
Calyptraeids are generally sedentary. Small individuals are
relatively mobile, but larger ones grow to fit the substrate to
which they are attached and seldom move. This suggests that
large males can only copulate with females in their immediate
vicinity (i.e., adjacent individuals on the same substrate or
stack-mates). Smaller males can move over hard substrates
in the laboratory, but do not tend to move unless disturbed.
The differences in potential mobility have been used to sup-
port the idea that small males may have an advantage over
larger males by moving between females and may monop-
olize matings at the expense of the large males. However,
there is no direct evidence that this occurs in the field. In-
dividuals of most species live for several years, but the av-
erage life span or growth rate in the field are not know for
any species.
Experimental studies of sex change in calyptraeids have
demonstrated that the size at sex change is labile and is in-
fluenced by other animals in the immediate vicinity (Gould
1917, 1919, 1952; Coe 1938, 1948, 1953; Collin 1995, 2000;
Warner et al. 1996; Collin et al. 2005). Small males grown
alone or housed with another male changed sex more quickly
and at smaller sizes than those housed with females (Collin
et al. 2005). A modeling study concluded that models based
on simple assumptions about the relationship between male
size and reproductive output do not accurately predict the
size at sex change in C. fornicata (Collin 1995). The size at
sex change was consistently underestimated, suggesting that
there is either an advantage to being male, such as higher
growth rate or lower mortality rate, or that, because sex
change is irreversible, the snails were bet hedging by chang-
ing later than appeared to be optimal (Collin 1995). There is
some evidence that male C. fornicata grow faster than females
of comparable size in caged field experiments; however, lab-
oratory experiments did not find a similar relationship for
three other species (Collin et al. 2005).
Data Collection
Samples of 19 species of abundant calyptraeids were col-
lected from areas of less than 15 m2 in the intertidal or sub-
tidal from North and South America, New Zealand, and South
Africa (see Appendix, available online only at http://
dx.doi.org/10.1554/05-210.1.s1). For six of the species (Cre-
pidula atrasolea, Crepipatella dilatata, C. fornicata, Crepi-
dula norrisiarum, Crepidula ustulatulina, and Bostrycapulus
calyptraeformis) two or three spatially distinct populations
were sampled. In addition, B. calyptraeformis were sampled
from two separate quadrats in a single location (Playa Chum-
ical, Panama). All animals that were encountered from each
sampled area were collected. Species that live attached to the
shells of other gastropods were collected with their hosts,
those on cobbles were collected attached to their substrates,
and those that attach to large rocks were removed from the
substrate in the field.
Shell-length of all animals collected from each site was
measured to the nearest 0.1 mm with vernier calipers. Sex
was determined by viewing each animal under a dissecting
microscope and recording the presence or absences of a penis
and/or a female genital papilla. Species of Crepipatella and
Bostrycapulus, genera in which animals do not form a female
genital papilla, were removed from the shell and the presence
or absence of the female capsule gland and albumin gland
were recorded. Small juveniles lacking both penis and female
reproductive structures and transitional animals with both or
neither structure were observed in most species. Since they
do not contribute to the current breeding population, both
juveniles and transitionals were excluded from subsequent
analyses. For most species, transitional individuals accounted
for less than 5% of the population (Table 1); however, tran-
sitional animals reached up to 22% of one population of C.
dilatata. Populations of all the species except for C. fornicata
(which were collected in the winter) included brooding fe-
males, indicating that the size and sex distributions represent
snap-shots of a breeding population.
Analyses were conducted including all sampled popula-
tions because different populations within a species often
differed significantly from each other and because most of
the predictions should hold for all populations of all sex-
738 RACHEL COLLIN
changing species. All analyses were repeated with only a
single sample (that with the largest sample size) representing
each species (Table 1). The results did not vary.
ANALYSIS
Sex Ratios
Sex ratios were calculated as the number of males divided
by the total number of males and females. Significant de-
viations from 0.5 were tested for in each population using a
x
2
-test. The overall bias toward the first sex was tested for
in the datasets of 27 populations or 19 species by comparison
to a binomial distribution. Logistic regression was used to
test for associations between sex ratio and other population
parameters (Wilson and Hardy 2002).
Size at Sex Change/Maximum Size
The size at which 50% of the animals were female was
considered to be the size at sex change (referred to here as
L50). This was estimated using probit analysis (which gives
identical results to the logistic regression approach used by
Allsop and West 2003a,b). Maximum size was the size of
the largest individual in the samples collected from that lo-
cation.
Previous studies seeking to find the invariant relationship
between size at sex change and maximum size (L50/Lmax) have
relied on a regression approach. If the slope of the regression
of ln(L50) on ln(Lmax) is equal to one, the ratio has been
considered invariant because the relationship between the two
variables is linear (Charnov 1993; Allsop and West 2003a,b).
However, the same relationship is expected from random data
for data bounded by Y , X, and therefore the recovery of a
slope of one is not informative (Cipriani and Collin 2005).
Cipriani and Collin (2005) suggested that variables with an
underlying linear relationship would be expected to produce
a regression with a higher r2-value than random data and that
the r2 rather than the slope of the regression could be used
to determine if there is an underlying relationship. This ap-
proach may be sensitive to the choice of the null model (Bus-
ton et al. 2004), and the regression model varies significantly
when the range of the Y-values is smaller than the range in
X or when the ranges only partially overlap (Cipriani and
Collin 2005). These problems can be circumvented by com-
paring the variance in the observed L50/Lmax to the variance
of bootstap datasets generated from the observed values of
L50 and Lmax (Cipriani and Collin 2005).
Here I use three approaches to generate null expectations
of the variation in L50/Lmax. I generated null expectations by
producing 1000 datasets of 19 and 27 pairs of randomly
generated L50 and Lmax drawn from a uniform distribution,
with Lmax constrained to vary within the observed range of
Lmax and L50 free to range between Lmax and zero. Size at
maturity could increase with maximum size and therefore
further increase the strength of association between randomly
derived expectations (Buston et al. 2004). Therefore, these
analyses were repeated with the additional constraint that L50
. 0.18Lmax (the average size at maturity for the calyptraeid
species used here is 18% of Lmax). Finally, I generated 1000
datasets of 19 and 27 L50/Lmax subject to Y , X by drawing
each L50-value from the list of observed L50-values and each
Lmax from the observed Lmax-values. For each randomly gen-
erated dataset of 19 or 27 values, I calculated the mean and
standard deviation of the L50/Lmax. The grand means and two-
tailed 95% confidence intervals for the mean L50/Lmax and
standard deviations of L50/Lmax were calculated from the 1000
samples from each simulation and compared to the observed
mean and standard deviations of L50/Lmax.
There is an error associated with the estimates of both L50
and Lmax that is not usually taken into account in studies of
life-history invariants. To determine if the observed L50/Lmax
differed significantly between species, the error of the esti-
mate of L50/Lmax was calculated for each species. Such dif-
ferences would provide evidence that the differences between
ratios are not statistical errors, but real differences between
species. Because there is no adequate method for calculating
the error on an extreme value (i.e., maximum size) from a
single sample, errors on this ratio were calculated in two
ways. One method included only an error on the size at sex
change estimated by the probit analysis. This is an under-
estimate of the actual error because it does not include the
error on the maximum size. Second, a program in R (written
by R. Cipriani, Universidad Simone Bolivar) was used to
calculate the size at sex change and L50/Lmax from 5000 boot-
strap analyses of the data for each species. This has the ad-
vantage of calculating a compound error including both the
error on the size at sex change and the error on the estimate
of maximum size. The value of the maximum size is nec-
essarily slightly underestimated in the bootstrap samples, and
therefore the calculated values and errors will be biased up-
ward. However, the uncertainty in the estimate due to dif-
ferences in sample size will be captured. In practice there
was little difference between the two estimates or the size of
their errors.
Characterization of Sociality and Size Overlap
I used a correlation approach to determine if intraspecific
variation in size at sex change is associated with intraspecific
variation in social environment. The standardized size over-
lap between males and females was used as a proxy for var-
iation in size at sex change. This was calculated for each
species as the difference in size between the largest male and
smallest female divided by the average size. Variation in
social environment was assessed by four different measures
because it is not known how calyptreaids assess their social
situation. The population that was part of a stack would reflect
the animals? perception of their social situation if snails as-
sess their situation as a simple distinction between being
stacked or unstacked. If the stack size is important, then the
average number of snails in a stack, the range in stack size,
and the standard deviation in stack size, would all reflect
different aspects of the social environment (see Table 3).
Because solitary individuals were collected in all species, the
maximum number of individuals in a stack is equal to the
range of stack size. Multiple regression analysis with step-
wise removal of independent variables and a P-to-remove of
0.05 (Sokal and Rohlf 1995) was used to determine which
of the four variables is most closely associated with variation
in size at sex change.
739CALYPTRAEID SEX CHANGE
Logistic regression was used to determine if animals in
stacks changed sex at a different size than animals that lived
alone. The significance of effects of body length and stacking
status on sex were tested.
Phylogenetic Effects
Because shared evolutionary history may cause close rel-
atives to be more similar than expected, it has become a
common practice to correct for such phylogenetic effects by
using methods such as independent contrasts. However, these
procedures should not be employed if there are no phylo-
genetic effects in the data. Here I looked for phylogenetic
effects by conducting a phylogenetic generalized least
squares analysis with CONTINUOUS (Pagel 1997, 1999) and
using a likelihood ratio test to test for a significant phylo-
genetic correlation (l) and by examining the data to deter-
mine how the within-species variation compares to variation
among species. CONTINUOUS analysis was based on a
neighbor-joining tree of the 19 species that showed the same
topological relationships as a pruned version of a larger anal-
ysis of calyptraeids (Collin 2003b). For traits evolving along
branches following a Brownian motion model of evolution,
the phylogentic correction (l) of the trait values is expected
to be one, while traits that are independent of the phylogeny
show a phylogentic correlation (l) of zero. The maximum
likelihood value of l was estimated using CONTINUOUS
and likelihood ratio tests were used to test for significant
differences from both l 5 0 and l 5 1, as recommended by
Freckleton et al. (2002). Freckleton et al. (2002) demonstrat-
ed that for phylogenies of 20 or more taxa, the power of l
to detect phylogenetic dependence is greater than 90%.
Therefore this test should have sufficient power to detect
phylogenetic correlation for the datasets examined here.
RESULTS
Phylogenetic Effects
There were no statistically significant phylogenetic cor-
relations for sex ratio, Lmax, L50, L50/Lmax, or stacking be-
havior (Table 1). In all cases the inclusion/exclusion of
branch length information did not change the results. There
was always enough power to reject the hypothesis that the
phylogenetic correlation was as expected from a Brownian
motion model of evolution along the tree topology (i.e., l 5
1). In most cases the maximum likelihood estimate of l was
zero (Table 1), showing that there is no evidence for a phy-
logenetic effect. In addition, these traits often varied among
populations of the same species (Table 2), as would be ex-
pected for a labile trait. Therefore, there is little evidence to
justify the use of phylogenetically corrected data, and all
subsequent analyses treated each species as an independent
datapoint.
Sex Ratios
Thirteen of the 27 populations had sex ratios that could
not be distinguished from 0.5 on the basis of a x2-test (Table
2). Eight populations (seven species including two popula-
tions of C. fornicata) had a statistically significant male (first
sex) bias, and six populations (four species including three
populations of C. dilatata) had significant female bias. Ex-
amination of all populations or a single population per species
shows that sex ratios across calyptraeids are not significantly
more biased toward the first sex than expected at random (P
. 0.05 for binomial expectation for 13 vs. 12 for 25 popu-
lations or 10 vs. eight for 18 species). Samples with a 0.5
sex ratio were excluded from the analysis. The range and
extremes of observed sex ratios (0.05?0.91; Fig. 1, Table 2)
was greater for these calyptraeids than for the 40 species of
various protandrous fish and invertebrates included in a re-
cent review of sex-changing animals (0.11?0.89; Allsop and
West 2004).
Different populations from seven of the eight species for
which there were multiple samples showed the same bias.
The single exception was for B. calyptraeformis, which had
one significantly male-biased population and two populations
in which there was no significant bias. Populations of C.
ustulatulina and C. fornicata had sex ratios that differed sig-
nificantly from each other, while showing the same direction
of bias. In addition, one of the pairwise comparisons among
three C. dilatata populations and one among three B. calyp-
traeformis populations showed significantly different sex ra-
tios (P , 0.05).
Logistic regression across all populations showed that sex
ratio was significantly associated with proportion stacked (P
, 0.0001), skew in the size distribution (P , 0.0001), mode
of development (P , 0.0001), L50/Lmax (P , 0.0001), but
not with maximum body size (P . 0.1). Skew in the size
distribution and L50/Lmax had the largest effect (Fig. 2), but
all of the effects remained significant according to likelihood
ratio tests with stepwise addition or subtraction from the
multiple regression model. When only a single population
for each species was included, sex ratio was significantly
associated with proportion stacked, skew in the size distri-
bution, mode of development, L50/Lmax, and maximum body
size (P , 0.001).
Size at Sex Change/Maximum Size
The range of L50/Lmax values for calyptraeids (0.33?0.72;
Table 2; Fig. 3) is lower and slightly less variable than that
reported for 77 sex-changing animals (0.47?0.96) and for 52
fishes (0.52?0.95; data from Allsop and West 2003a,b).
Comparisons of the observed data with randomly generated
data and the bootstrap datasets made from randomizations of
the observed L50- and Lmax-values shows that the mean of
L50/Lmax for calyptraeids does not differ from the random
expectations, but the standard deviation of these values is
significantly smaller than expected (Table 3). Repeating the
analysis for all 104 animals for which data are currently
available (calyptraeids plus the 77 other animals from Allsop
and West 2003) shows that mean L50/Lmax is significantly
larger than random and the standard deviation is significantly
smaller than random (Table 3). The significantly small stan-
dard deviation suggests that L50 and Lmax may have an un-
derlying relationship; however, the standard deviation is only
slightly smaller than expected, suggesting that the effect is
minimal. If L50 was further constrained to be less than 85%
or 90% of the maximum recorded size (a realistic assump-
tion), the observed ratios would be difficult to distinguish
740 RACHEL COLLIN
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.
741CALYPTRAEID SEX CHANGE
FIG. 1. Histogram showing the distribution of sex ratios of (A)
all 27 populations examined and (B) the 19 species. Data are from
Table 1. Sex ratios .0.5 are male bias and those ,0.5 are female
biased.
FIG. 2. Scatter plot of skew of the size distribution for each pop-
ulation versus sex ratio for that population. See text for details of
logistic regression analysis. The black points are the values for
single populations chosen to represent each species.
FIG. 3. Scatter plot of size at sex change versus maximum size.
OLS r2 5 0.82, P 5 0.005, n 5 27. The black points are the values
for the single populations chosen to represent each species.
from random. Further evidence that statistical invariance is
minimal comes from the combined dataset of calyptraeids
and other animals. There is a significant difference in L50/
Lmax between protandrous and protogynous species, with pro-
tandrous species changing sex at a smaller size relative to
their maximum size (protandrous L50/Lmax 5 0.61, proto-
gynous L50/Lmax 5 0.78, t-test, P 5 0.0001). This difference
was predicted by Gardner et al. (2005).
Confidence intervals estimated on the basis of probit anal-
ysis alone showed that many of the L50/Lmax values are sig-
nificantly different from each other (Fig. 4). Bootstrap anal-
ysis of the same data gave slightly higher values for each
ratio (as expected from the bootstrap replicates slightly un-
derestimating the maximum size) and the confidence intervals
were slightly larger. In a few cases where several taxa had
similar values of L50/Lmax, the rank order of the taxa changed,
but the overall pattern remained the same (Fig. 4). There are
numerous values of L50/Lmax for which the confidence inter-
vals do not overlap and in the cases of C. atrasolea, C. di-
latata, and C. fornicata the values for different populations
are significantly different. Again, this provides support for
the idea that L50/Lmax varies within and between calyptraeid
species.
Social Structure
Variation in size at sex change as measured by size overlap
between males and females was correlated with social struc-
ture. All four variables reflecting intraspecific variation in
social experience showed a significant association with this
measure of variation in size at sex change when examined
independently across all populations (proportion stacked: r2
5 0.23, P , 0.05; range in stack size: r2 5 0.43, P , 0.0005;
average number in a stack: r2 5 0.21, P , 0.05; standard
deviation of stack size: r2 5 0.42, P , 0.001). Multiple
regression with stepwise removal of nonsignificant factors
showed that only range in stack size had a significant effect
in combination with the other factors (Fig. 5). When only a
single population was included for each species, range in
stack size and standard deviation in stack size were the only
factors that were significant alone, and range in stack size
was the only significant factor in the multiple regression
(range in stack size: r2 5 0.36, P , 0.01; standard deviation
of stack size: r2 5 0.42, P , 0.05).
742 RACHEL COLLIN
TABLE 3. L50/Lmax for 27 populations or 19 species of calyptraeids compared to random expectations.
Dataset Sample size L50/Lmax mean (CI) L50/Lmax SD (CI)
Calyptraeid data
Bootstrapped observed data 27 0.47 (0.398, 0.552) 0.20 (0.151, 0.247)
19 0.47 (0.389, 0.567) 0.20 (0.143, 0.253)
104 0.31 (0.256, 0.331) 0.27 (0.235, 0.293)
Random simulated data 27 0.51 (0.405, 0.615) 0.28 (0.233, 0.330)
19 0.51 (0.384, 0.635) 0.28 (0.217, 0.342)
104 0.51 (0.459, 0.563) 0.28 (0.258, 0.306)
Random simulated data with size at maturity 27 0.59 (0.499, 0.678) 0.23 (0.192, 0.278)
19 0.59 (0.481, 0.699) 0.23 (0.184, 0.284)
Populations of calyptraeids 27 0.51 0.10*
Species of calyptraeids 19 0.51 0.10*
All animals 104 0.68* 0.16*
* Significantly different (P , 0.05) from random expectations.
FIG. 4. L50/Lmax for all 27 populations of calyptraeids. Error bars indicate 95% confidence intervals. Points in black show the error of
the results using probit analysis, and points in gray show the estimate and compound error based on resampling.
Analyses treating stacked and unstacked individuals sep-
arately showed that there is often (nine of 20 populations) a
significant effect of stacking on sex change (Table 4). Of the
20 populations for which appropriate data were available,
five showed a significant difference in sex ratio between
stacked and unstacked animals. There was no consistent di-
rection of change with C. adunca and C. naticarum having
higher sex ratios in the unstacked animals and C. incurva,
C. norissiarum, and C. ustulatulina having higher sex ratios
in stacked animals (Table 4). Logistic regression detected an
effect of stacking on size at sex change (L50) in six of 20
populations. In this case, all populations with significant dif-
ferences showed a larger size at sex change in stacked than
in solitary individuals animals. The power to detect a dif-
ference was low for several of the datasets, where no effect
was detected because there were few animals in one of the
categories (Table 4).
DISCUSSION
Variation in Sex Ratio and L50/Lmax
Compilations of data from the literature on sex-changing
fishes, molluscs, crustaceans and echinoderms show large
variation in sex ratio and L50/Lmax (Allsop and West 2003a,b,
2004). It could be argued that this variation is due largely to
combining protandrous with protogynous, sedentary with
mobile, and social with solitary organisms and that restricting
the data to animals from a single clade would improve the
fit to theory. This idea receives some support from the sig-
nificant difference in L50/Lmax for protandrous and protogyn-
ous species recorded here and from the difference in sex ratio
between the two types of sex change (Allsop and West 2004).
Even when protandrous and protogynous species are treat-
ed separately, sex ratio and L50/Lmax are still highly variable.
When the analysis is limited to calyptraeids, a monophyletic
743CALYPTRAEID SEX CHANGE
FIG. 5. Scatter plot of size overlap between males and females as
a proxy for variation in size at sex change versus the range in stack
size; OLS regression r2 5 0.40, P 5 0.005, n 5 26. The black
points are the values for the single populations chosen to represent
each species.
TABLE 4. Propensity of calyptraeids to form stacks and aggregations.1
Species
Sex ratio of stacked vs.
unstacked animals (P of x2-test)
L50 unstacked vs. L50 stacked
(P of logistic regression)
Crepidula adunca 0.53 vs. 0.80 (P , 0.0001) ns
Crepipatella dilatata 3 ns 16.7 vs. 19.8 (P 5 0.04)
Crepipatella fecunda ns 29.1 vs. 37.5 (P 5 0.002)
Crepidula fornicata 1 ns 23.9 vs. 30.8 (P 5 0.0035)
Crepidula incurva 0.53 vs. 0.23 (P , 0.001) 7.1 vs. 8.5 (P , 0.0001)
Crepipatella lingulata ns P 5 0.0482
Crepidula naticarum 0.81 vs. 0.99 (P , 0.001) ns
Crepidula norrisiarum 1 0.56 vs. 0.37 (P 5 0.04) ns
Crepidula ustulatulina 2 0.51 vs. 0.33 (P 5 0.05) 5.2 vs. 5.6 (P 5 0.07)
1 Species with appropriate data but no significant results: Bostrycapulus aculeatus, Crepidula atrasolea 1, Crepidula atrasolea 2, Crepidula complanata,
Crepidula depressa, Crepipatella dilatata 2, Crepidula fornicata 2, Crepidula norrisiarum 2, Crepidula onyx, Crepidula ustulatulina 1, Maoricrypta monoxyla.
2 Sample of stacked too small to estimate L50 separately.
family in which all species are sedentary and protandrous,
both quantities remain highly variable. In fact, the range of
sex ratios is larger for the 27 observed populations of ca-
lyptraeids than it is for 121 sex-changing animals (Allsop
and West 2004). Because the sex ratios reported by Allsop
and West were averages of available populations, the range
is likely to be slightly underestimated (the original raw data
are not given). Data from different calyptraeid populations
were not averaged here, since one of the basic assumptions
of sex allocation theory for environmentally mediated sex
change is that sex can change at different sizes or ages in
response to environmental conditions. Therefore, there is not
a single sex ratio or L50/Lmax characteristic of each species.
The idea that these values can vary intraspecifically is sup-
ported by the calyptraeid data that show significant differ-
ences among populations for sex ratios and L50/Lmax.
Assumptions of the Theory
The considerable, unpredicted variation in sex ratio and
L50/Lmax in samples from a single taxon suggests that calyp-
traeids violate the basic assumptions of sex allocation theory.
The prediction that sex ratio should be biased toward the first
sex is based on minimal assumptions and is not altered by
size-specific mortality or growth rates (Charnov and Bull
1989). The main underlying assumption is that both male and
female fertility increases with size (Charnov and Bull 1989).
Data on egg production for several species of calyptreaids
make it clear that female fertility increases with size (e.g.,
Collin 1995, 2000; Chaparro et al. 1999; Chaparro and Flores
2002). Unfortunately no data are available for male fertility.
A male-biased sex ratio, consistent with the idea that both
sexes have increasing fertility with size, is observed for sev-
eral calyptraeids. The female-biased sex ratios observed for
other species imply that male fertility at the size of sex change
is less than average male fitness. Therefore, it is likely that
male fertility decreases with size in these species. Such a
decrease could be explained by the difference in mobility
between large and small males that has been suggested for
some species. Whatever the mechanism, it is clear that the
shape of the relationship between male fertility and size prob-
ably differs between species, increasing in some and decreas-
ing in others. The predictions of first-sex-biased sex ratios
could also be altered by high mortality during the transition
phase. This is unlikely to explain the lack of fit for calyp-
traeids, as laboratory studies have not found different mor-
tality rates in transitional animals (Collin et al. 2005).
The theoretical derivation of L50/Lmax as an invariant relies
on the assumptions that k/M, aM, and d are constant across
the species examined. The large variation in L50/Lmax among
calyptraeids suggests that at least one of these three values
is not constant across species. Unfortunately, there are no
data currently available on natural mortality rate (M), von
Bertalanffy growth coefficients (k), or age at maturity (a) for
calyptraeids. There are also no data pertaining directly to d
values for calyptraeids. However, the sex ratio data reported
here indicates that there are differences in the shape of the
male fertility curves, providing indirect evidence that one of
the basic assumptions of the original theoretical derivation
of L50/Lmax as a life-history invariant is violated. The sim-
ulations of Gardner et al. (2005) suggested that the derivation
of L50/Lmax as a life-history invariant is relatively robust to
variation in d. However, a number of their simulations used
the flawed regression approach to the analysis of bounded
variables. In addition, their sensitivity analyses follow their
formal model and varied each presumed constant (k/M, aM,
and d) one at a time. Because the formal derivation that
744 RACHEL COLLIN
produced the prediction of an invariant L50/Lmax also pre-
dicted an invariant breeding sex ratio (Gardner et al. 2005,
p. 564), which is a demonstrably highly variable in both
calyptraeids (this paper) and across sex-changing animals
(Allsop and West 2004), something about the model clearly
does not accurately capture what is going on in nature. Stud-
ies designed to examine the relationship between male size
and fertility directly, as well as real demographic data and
not derived averages from a few distantly related species, are
necessary before the interspecific variation in k/M, aM, and
d can be assessed and included in simulations of size at sex
change.
Life-History Invariants in Sex Changing Animals
Life-history invariants are based on the idea that two fea-
tures of life histories may have a relationship that is constant
under a variety of transformations. This is similar to familiar
physical or physiological allometries such as the scaling be-
tween body size and metabolic rate. Body size explains a lot
of the variation in metabolic rate despite variation in, for
example, color, family, or brood size. The invariants specif-
ically examined by Charnov (1993) are dimensionless ratios
(implying that the relationship between the two is linear) such
as yearly clutch size/instantaneous mortality. Just as physical
scaling relationships may explain a large or small portion of
the variance in the data, the proposed life-history invariants
may be tight or noisy. In cases where the variables are mu-
tually unconstrained the r2-value reflects the strength of the
relationship, and high r2-values reflect a good fit to the theory
if the relationship is a linear.
Allsop and West (2003a) interpreted a log-log slope of one
and an r2 of 0.98 as evidence of universal invariance of L50/
Lmax. They comment how amazing it is that this rule holds
over all animals and conclude that k/M, aM, and d must
therefore also be invariant across all the taxa examined. As
described above, linear regression statistics of bounded var-
iables cannot be interpreted literally, but it is true that L50/
Lmax is less variable than expected at random (Cipriani and
Collin 2005; and above), suggesting some underlying rela-
tionship between the two variables. However, there are sev-
eral lines of evidence that question the interpretation of this
ratio as an invariant, not only for calyptraeids, but for the
dataset examined by Allsop and West (2003a,b). First, if the
same invariant relationship holds for both protandrous and
protogynous species there should not be a significant differ-
ence in L50/Lmax between species with each strategy. How-
ever, both Allsop and West?s (2003a,b) original data and their
data combined with the calyptraeid data show significant dif-
ferences in L50/Lmax between protandry and protogyny (orig-
inal data: 0.78 vs. 0.67; P 5 0.0001; n 5 77). Secondly, the
same theory from which the invariance of L50/Lmax is derived
also predicts an invariant sex ratio. Allsop and West (2004)
demonstrated that there is significant variation in sex ratio
across the same sex-changing animals for which they believe
L50/Lmax is invariant and sex ratio also varies significantly in
calyptraeids. It will be difficult to identify the causes of these
inconsistencies, until explicit measures of k, a, M, and d are
available, and it will indeed be surprising if they do not vary
among all sex-changing animals. The significant differences
in L50/Lmax between different populations of the same species
of calyptraeids suggests that these values may not even be
constant across different populations of the same species.
With so much variation in L50/Lmax, the faint signal of an
underlying relationship might more reasonable be considered
a curiosity rather than a biological law.
Sociality and Variation in Size at Sex Change
Experimental evidence shows that associations with con-
specifics affects the size at sex change in several calyptraeid
species (Gould 1917, 1919, 1952; Coe 1938, 1948, 1953;
Collin 1995, 2000; Warner et al. 1996; Collin et al. 2005).
Males experimentally placed with females change sex at larg-
er sizes than those placed with other males or those alone.
The field-collected data presented here is consistent with this,
showing that stacked individuals change sex at larger sizes
than solitary animals of the same species, and that sex ratios
often differ for stacked and unstacked animals. There is also
a correlation between the amount of stacking and variation
in size at sex change, with up to 40% of the interspecific
variation in variation in the size at sex change being explained
by range in stack size. All of these lines of evidence support
the idea that animals in the field experience difference social
environments and alter the size at sex change in response to
their interactions with conspecifics. Therefore, it seems likely
that populationwide analyses may obscure interesting pat-
terns and that analyses on the level of aggregations or stacks
may be a fruitful direction for future research on sex change
in calyptraeids.
ACKNOWLEDGMENTS
I thank O. Chaparro (Universidad Austral, Valdivia, Chile)
for obtaining samples from Yaldad, R. Cipriani for generating
bootstrap confidence intervals, and J. Norenburg and J. Chris-
ty for commenting on the manuscript. Fieldwork was sup-
ported by National Science Foundation dissertation improve-
ment grant DEB 9972555 and the Smithsonian Tropical Re-
search Institute.
LITERATURE CITED
Allsop, D. J., and S. A. West. 2003a. Changing sex at the same
relative body size. Nature 425:783?784.
???. 2003b. Constant relative age and size at sex change for
sequentially hermaphroditic fish. J. Evol. Biol. 16(5):921?929.
???. 2004. Sex ratio evolution in sex changing animals. Evo-
lution 58:1019?1027.
Baeza, J. A., and R. T. Bauer. 2004. Experimental test of socially
mediated sex change in a protandric simultaneous hermaphro-
dite, the marine shrimp Lysmata wurdemanni (Caridea: Hippol-
ytidae). Behav. Ecol. Socbiol. 55(6):544?550.
Baldwin, A. P., and R. T. Bauer. 2003. Growth, survivorship, life-
span, and sex change in the hermaphroditic shrimp Lysmata wur-
demanni (Decapoda: Caridea: Hippolytidae). Mar. Biol. 143(1):
157?166.
Bauer, R. T. 2002a. Tests of hypotheses on the adaptive value of
an extended male phase in the hermaphroditic shrimp Lysmata
wurdemanni (Caridea: Hippolytidae). Biol. Bull. 203(3):
347?357.
???. 2002b. Reproductive ecology of a protandric simultaneous
hermaphrodite, the shrimp Lysmata wurdemanni (Decapoda:
Caridea: Hippolytidae). J. Crust. Biol. 22(4):742?749.
???. 2005. Cost of maleness on brood production in the shrimp
745CALYPTRAEID SEX CHANGE
Lysmata wurdemanni (Decapoda: Caridea: Hippolytidae), a pro-
tandric simultaneous hermaphrodite. J. Mar. Biol. Ass. U.K. 85:
101?106.
Bauer, R. T., and G. J. Holt. 1998. Simultaneous hermaphroditism
in the marine shrimp Lysmata wurdemanni (Caridea: Hippoly-
tidae): an undescribed sexual system in the decapod Crustacea.
Mar. Biol. 132(2):223?235.
Buston, P. M., P. L. Munday, and R. R. Warner. 2004. Evolutionary
biology: sex change and relative body size in animals. Nature
428, 8 April 2004. Available via doi: 10.1038/nature 02512a.
Buxton, C. D. 1993. Life-history changes in exploited reef fishes
on the east coast of South Africa. Environ. Biol. Fishes. 36:
47?63.
Chaparro, O. R., and M. L. Flores. 2002. Reproductive output of
Crepidula fecunda females: distribution of energy in the pro-
duction of gametes and capsular walls. New Zealand J. Mar.
Freshwater Res. 36:661?673.
Chaparro, O. R., R. F. Oyarzun, A. M. Vergara, and R. J. Thompson.
1999. Energy investment in nurse eggs and egg capsules in Cre-
pidula dilatata Lamarck (Gastropoda, Calyptraeidae) and its in-
fluence on the hatching size of the juvenile. J. Exp. Biol. Ecol.
232:261?274.
Charnov, E. L. 1979. Natural selection and sex change in pandalid
shrimp: test of a life history theory. Am. Nat. 113:715?734.
???. 1981. Sex reversal in Pandalus borealis effect of a shrimp
fishery. Mar. Biol. Letters. 2:53?58.
???. 1982. The theory of sex allocation. Princeton Univ. Press,
Princeton, NJ.
???. 1986. Size advantage may not always favor sex change. J.
Theor. Biol. 119:283?285.
???. 1993. Life history invariants: some explorations of sym-
metry in evolutionary ecology. Oxford Univ. Press, Oxford, U.K.
Charnov, E. L., and J. J. Bull. 1989. Non-Fisherian sex ratios with
sex change and environmental sex determination. Nature. 338:
148?150.
Charnov, E. L., and R. W. Hannah. 2002. Shrimp adjust their sex
ratio to fluctuating age distributions. Evol. Ecol. Res. 4:239?246.
Charnov, E. L., and U. Sku?lado?ttir. 2000. Dimensionless invariants
for the optimal size (age) of sex change. Evol. Ecol. Res. 2:
1067?1071.
Cipriani, R., and R. Collin. 2005. Life-history invariants with
bounded variables: Artifacts of an incorrect null hypothesis? J.
Evol. Biol. 18:1613?1618.
Coe, W. R. 1938. Influence of association on the sexual phases of
gastropods having protandric consecutive sexuality. Biol. Bull.
75:274?285.
???. 1948. Nutrition and sexuality in protandric gastropods of
the genus Crepidula. J. Exp. Zool. 108:155?169.
???. 1953. Influences of association, isolation and nutrition on
the sexuality of snails in the genus Crepidula. J. Exp. Zool. 122:
5?19.
Collin, R. 1995. Sex, size, and position: a test of models predicting
the size at sex change in the protandrous gastropod Crepidula
fornicata. Am. Nat. 146(6):815?831.
???. 2000. Sex change, reproduction and development of Cre-
pidula adunca and C. lingulata (Gastropoda: Calyptraeidae). Ve-
liger. 43:24?33.
???. 2003a. World-wide patterns of development in calyptraeid
gastropods. Mar. Ecol. Prog. Ser. 247:103?122.
???. 2003b. Phylogenetic relationships among calyptraeid gas-
tropods and their implications for the biogeography of specia-
tion. Syst. Biol. 52(5):618?640.
Collin, R., M. McLellan, K. Gruber, and C. Bailey-Jourdain. 2005.
Effects of conspecific associations on size at sex change in three
species of calyptraeid gastropods. Marine Ecology Progress Se-
ries. 293:89?97.
Freckleton, R. P., P. H. Harvey, and M. Pagel. 2002. Phylogenetic
analysis and comparative data: a test and review of evidence.
Am. Nat. 160(6):712?726.
Gardner, A., D. J. Allsop, E. L. Charnov, and S. A. West. 2005. A
dimensionless invariant for relative size at sex change in ani-
mals: explanation and implications. Am. Nat. 165(5):551?566.
Gould, H. N. 1917. Studies on sex in the hermaphrodite mollusk
Crepidula plana. 2. Influence of the environment on sex. J. Exp.
Zool. 23:225?250.
???. 1919. Studies on sex in the hermaphrodite mollusk Cre-
pidula plana. 3. Transference of the male producing stimulus
through sea water. J. Exp. Zool. 29:113?120.
???. 1952. Studies on sex in the hermaphrodite mollusk Cre-
pidula plana. IV. Internal and external factors influencing growth
and sex development. J. Exp. Zool. 119:93?163.
Hannah, R. W., and S. A. Jones. 1991. Fisheries-induced changes
in the population structure of pink shrimp Pandalus jordani. Fish.
Bull. 89:41?52.
Hardy, I. C. W. 2002. Sex ratios: concepts and research methods.
Cambridge Univ. Press, Cambridge, U.K.
Hoagland, K. E. 1978. Protandry and the evolution of environ-
mentally mediated sex change: a study of the Mollusca. Mala-
cologia 17:365?391.
Leigh, E. G., E. L. Charnov, and R. R. Warner. 1976. Sex ratio,
sex change and natural selection. Proc. Nat. Acad. Sci. 73:
3655?3660.
Lin, J., and D. Zhang. 2001. Reproduction in a simultaneous her-
maphroditic shrimp, Lysmata wurdemanni: Any two will do?
Mar. Bio. 139(6):1155?1158.
Nee, S., N. Colegrave, S. A. West, and A. Grafen. 2005. The illusion
of invariant quantities in life histories. Science 309:1236?1239.
Pagel, M. 1997. Inferring evolutionary processes from phylogenies.
Zool. Scripta 26:331?348.
???. 1999. Inferring the historical patterns of biological evo-
lution. Nature 401:877?884.
Peterson, C. W., R. R. Warner, D. Y. Shapiro, and A. Marconato.
2001. Components of fertilization success in the bluehead
wrasse, Thalassoma bifasciatum. Behav. Ecol. 12:237?245.
Policansky, D. 1981. Sex choice and the size advantage model in
jack-in-the-pulpit (Arisaema triphyllum). Proc. Nat. Acad. Sci.
78:1306?1308.
Schultz, E. T., and R. R. Warner. 1989. Phenotypic plasticity in
life-history traits of female Thalassoma bifasciatum (Pisces:La-
bridae). 1. Manipulations of social structure in tests for adaptive
shifts of life-history allocations. Evolution. 43:1497?1506.
???. 1990. Phenotypic plasticity in life-history traits of female
Thalassoma bifasciatum (Pisces: Labridae). II. Correlation of
fecundity and growth rate in comparative studies. Environ. Biol.
Fish. 30:333?344.
Shapiro, D. Y., and R. Lubbock. 1980. Group sex ratio and sex
reversal. J. Theor. Biol. 82:411?426.
Sokal, R. R., and F. J. Rohlf. 1995. Biometry. 3rd ed. W. H. Freeman
and Co., New York.
Warner, R. R. 1975. The adaptive significance of sequential her-
maphroditism in animals. Am. Nat. 109:61?82.
???. 1987. Female choice of sites versus mates in a coral reef
fish Thalassoma bifasciatum. Animal Behaviour 35:1470?1478.
Warner, R. R., and S. G. Hoffman. 1980. Population density and
the economics of territorial defense in a coral reef fish Thal-
assoma bifasciatum. Ecology 61:772?780.
Warner, R. R., and E. T. Schultz. 1992. Sexual selection and male
characteristics in the bluehead wrasse, Thalassoma bifasciatum:
mating site acquisition, mating site defense and female choice.
Evolution 46:1421?1442.
Warner, R. R., and S. E. Swearer. 1991. Social control of sex change
in the bluehead wrasse Thalassoma bifasciatum (Pisces: Labri-
dae). Biol. Bull. 181:199?204.
Warner, R. R., D. L. Fitch, and J. D. Standish. 1996. Social control
of sex change in the shelf limpet, Crepidula norrisiarum: size-
specific responses to local group composition. J. Exp. Mar. Biol.
Ecol. 204:155?167.
Wilson, K., and I. C. W. Hardy. 2002. Statistical analysis of sex
ratios: an introduction. Pp. 48?92 in I. C. W. Hardy, ed. Sex
ratios: concepts and research methods. Cambridge Univ. Press,
Cambridge, U.K.
Corresponding Editor: D. Promislow