doi:10.1111/j.1420-9101.2008.01629.x
Evaluation of offspring size–number invariants in 12 species
of lizard
T. ULLER,*, G. M. WHILE,à E. WAPSTRA,à D. A. WARNER,§,– B. A. GOODMAN,**
L. SCHWARZKOPF,** T. LANGKILDE,§, P. DOUGHTY,àà R. S. RADDER,§
D. H. ROHR,§§ C. M. BULL,–– R. SHINE§ & M. OLSSON *Department of Zoology, Edward Grey Institute, University of Oxford, Oxford, UK
School of Biological Sciences, University of Wollongong, Wollongong, NSW, Australia
àSchool of Zoology, University of Tasmania, Hobart, Tas., Australia
§School of Biological Sciences, University of Sydney, Sydney, NSW, Australia
–Department of Ecology, Evolution, and Organismal Biology, Iowa State University, Ames, IA, USA
**School of Marine and Tropical Biology, James Cook University, Townsville, Qld, Australia
Biology Department, Penn State University, University Park, PA, USA
ààDepartment of Terrestrial Zoology, Museum of Western Australia, Perth, WA, Australia
§§Applied Ecological Research P ⁄ L, Melbourne, Vic., Australia
––School of Biological Sciences, Flinders University, Adelaide, SA, Australia
Keywords:
Abstract
life history;
lizards;
oviparous;
resource allocation;
size–number invariants;
viviparous.
The optimal division of resources into offspring size vs. number is one of the
classic problems in life-history evolution. Importantly, models that take into
account the discrete nature of resource division at low clutch sizes suggest that
the variance in offspring size should decline with increasing clutch size
according to an invariant relationship. We tested this prediction in 12 species
of lizard with small clutch sizes. Contrary to expectations, not all species
showed a negative relationship between variance in offspring size and clutch
size, and the pattern significantly deviated from quantitative predictions in five
of the 12 species. We suggest that the main limitation of current size–number
models for small clutch sizes is that they rely on assumptions of hierarchical
allocation strategies with independence between allocation decisions. Indeed,
selection may favour alternative mechanisms of reproductive allocation that
avoid suboptimal allocation imposed by the indivisible fraction at low clutch
sizes.
Introduction
Life-history theory is one of the cornerstones of evolutionary biology. Its goal is to explain why organisms
differ in the timing of life-history events such as the onset
of breeding, reproductive investment and patterns of
senescence (reviewed in Roff, 1992, 2002; Stearns,
1992). For mathematical convenience, most theoretical
optimality models rely upon a continuous function to
describe the relationships between variables. However,
many interesting biological phenomena do not easily
lend themselves to the assumptions of continuity and
need to be modelled using a discrete approach (DownCorrespondence: T. Uller, Department of Zoology, Edward Grey Institute,
University of Oxford, Oxford OX1 3PS, UK.
Tel.: +44(0) 1865 281194; fax: +44(0) 1865 271168;
e-mail: tobias.uller@zoo.ox.ac.uk
hower & Charnov, 1998). For example, in species with
small clutch sizes, the size–number trade-off only
approximates continuous functions for relatively large
clutch sizes. Thus, a female with a resource availability
that is intermediate to that required to produce two or
three eggs can either produce two eggs of larger than
optimum size or three eggs of smaller than optimum size
(Ebert, 1994; Charnov et al., 1995). This has a number of
potential consequences. For example, it implies that the
‘optimal’ division of resources is not available for most
values of total reproductive effort, and this could have
consequences for the relationship between offspring size
and number (Ebert, 1994; Charnov et al., 1995; Downhower & Charnov, 1998).
Small clutch sizes are very common throughout the
animal kingdom. For example, a substantial proportion
of lizards has a clutch size below 8 (Fitch, 1970). Thus,
most size–number models are of doubtful applicability for
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T. ULLER ET AL.
these species. However, models for small clutch sizes
have been developed. Early discussions of the problem by
Ricklefs (1968) and Nussbaum (1981) were formalized by
Ebert (1994) and Charnov and co-workers (Charnov &
Downhower, 1995; Charnov et al., 1995; Downhower &
Charnov, 1998). Under the assumption that animals
differ in the amount of resources that are available for
reproduction, these models suggest that the variance in
offspring size in a population should decrease with
increasing clutch size. This pattern arises because, as
clutch size increases, the total reproductive effort can be
divided into offspring sizes that are closer to the ‘optimal’
value. Furthermore, Charnov et al.’s (1995) model specifically predicts that variance in offspring size will follow
a simple, invariant, relationship with clutch size (i.e.
independent of model parameters, such as functions
relating offspring size to survival; reviewed in Charnov,
1993). We here reiterate the most important aspects of
their model (Charnov & Downhower, 1995; further
discussions can also be found in Guinnee et al., 2004;
Kasparian et al., 2005; Uller & Olsson, in press).
Provided that a number of assumptions hold (Table 1),
the models predict that the mean offspring size should be
independent of clutch size. However, the range in
offspring size should decline with clutch size. More
specifically, the ratio of the ranges of investment per
offspring at clutch size n + 1 and n should be inversely
proportional to the ratio of clutch sizes, i.e.
Imax nþ1 Imin nþ1
Cn
¼
:
Imax n Imin n
Cnþ1
ð1Þ
Thus, the maximum and minimum per-offspring
investment at each clutch size should converge upon
the optimal level of investment per offspring (Fig. 1).
Empirical tests of this prediction are restricted to
natural or laboratory populations of some species of
parasitic wasps (West et al., 2001; Guinnee et al., 2005),
one mammal (Kasparian et al., 2005), one fish (Charnov
et al., 1995), one lizard (Uller & Olsson, in press), and an
experimental approach on one clone of Daphnia (Guinnee et al., 2004). Whereas all studies found a decrease in
Table 1 Key assumptions for the Charnov et al. (1995) small clutch
size model.
Maternal control over resource allocation
Resources available for reproduction are fixed for a given female
Resources should represent an individual’s total reproductive investment
at each given time of reproduction
Conversion of resources into per-offspring investment is linear
Selection favors a single optimal offspring size (e.g. independently of
clutch size)
Females do (or can) not vary the per-offspring investment within clutches
No lower or upper limit on per-offspring investment
References: Charnov & Downhower, 1995; Charnov et al., 1995;
Downhower & Charnov, 1998; Guinnee et al., 2004; Kasparian
et al., 2005; Uller & Olsson, in press.
Offspring size
144
1
2
3
4
5
Clutch size
Fig. 1 Predicted relationship between clutch size and the range in
average per offspring investment (e.g. offspring mass). Figure
modified from Charnov et al. (1995).
variance in offspring size with increased clutch size, the
fit to eqn 1 was relatively poor in most cases. However,
many of these species are likely, or have been shown,
to violate one or more assumptions (e.g. mammals;
Kasparian et al., 2005). Furthermore, there is only one
published study that is set in a comparative framework (Guinnee et al., 2005), despite that this could
substantially further our understanding of the models’
accuracy in predicting size–number patterns. For example, if species differ in at least one trait that is likely to
violate assumptions to a lesser or greater degree, a
comparative approach can constitute a direct test of the
role of specific assumptions for poor fit to theory or, at
the least, suggest some reasons for any observed
among-species variation. Here, we test the fit to the
invariant relationship described in eqn 1 using data from
12 species of Australian lizards with clutch sizes in the
range of 1–9.
Materials and methods
Brief overview of lizard reproduction
Most lizards are oviparous. Eggs are soft-shelled and are
either buried in soil or sand or attached to the substrate.
However, approximately 20% of the world’s >4000 lizard
species are viviparous with viviparity having evolved
independently around 100 times among squamates
(reviewed in Blackburn, 1982; Shine, 1985; Pough et al.,
2001). The degree of placental structure differs substantially among species (Weekes, 1935). Those with no or
very limited post-ovulatory exchange between mother
and offspring are frequently referred to as ovoviviparous.
However, most species are not sufficiently studied to
allow any detailed classification regarding the degree of
transfer across the placenta and we therefore limit
ourselves to the classification oviparous vs. viviparous.
These two categories could differ in a number of ways
that could influence the degree to which they fit the
assumptions of the models, in particular the extent to
which offspring size is affected by maternal effects
post-ovulation (see Discussion). Thus, we may expect
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Tests of size–number invariants in lizards
Table 2 Data summary and references to primary literature on the Materials and methods used to catch, house and measure reproductive
traits for each of the 12 lizard species.
Species
Reproductive
mode
No.
clutches
Clutch size
range
Eulamprus tympanum
Viviparous
356
1–5
Eulamprus heatwolei
Eulamprus quoyii
Tiliqua rugosa
Egernia whitii
Niveoscincus microlepidotus
Niveoscincus ocellatus
Lampropholis mirabilis
Bassiana duperreyi
Amphibolurus muricatus
Ctenophorus pictus
Ctenophorus fordi
Viviparous
Viviparous
Viviparous
Viviparous
Viviparous
Viviparous
Oviparous
Oviparous
Oviparous
Oviparous
Oviparous
53
35
41
106
148
558
34
62
122
143
270
1–4
4–8
1–3
1–3
1–5
2–6
2–4
5–9
3–8
3–6
2–5
oviparous lizards to more closely follow the predictions of
the Charnov et al. (1995) models.
Data collection
We obtained data on 12 Australian lizards (Table 2), five
oviparous and seven viviparous. In most studies, animals
from natural populations were brought into the laboratory to oviposit or give birth just before oviposition or
parturition. The holding facilities differ between species
and studies and we direct the reader to key papers in the
primary literature for further information for each
species (Table 2). We exclusively used offspring mass at
hatching or parturition as our dependent variable and
consistently pooled data within species whenever possible to increase sample sizes per species. The likelihood
that this should cause error is small considering that the
prediction from eqn 1 is independent of variation in
assumptions regarding trade-off functions and offspring
survival (that could fluctuate across time and space) and
because measures of mass should show limited variation
across settings (under the assumptions that balances are
accurately calibrated).
The prediction that mean offspring mass should not be
correlated with clutch size was tested by carrying out a
Spearman’s rank correlation on mean offspring mass (per
clutch) and clutch size. We tested for a convergence of
maximum and minimum mean offspring mass by applying the ordered heterogeneity (OH) test (Rice & Gaines,
1994) following a Bartlett’s test of homogeneity of
variance (following Guinnee et al., 2004). This allowed
us to evaluate hypotheses regarding the increase (or
decrease) in the variation in mean offspring mass with
increasing clutch size in the context of a nondirectional
test of the difference in the variation in mean offspring
mass between clutch sizes (Rice & Gaines, 1994). Finally,
the main prediction of an invariant relationship between
the range in offspring mass and clutch size was tested by
Mean offspring
mass (g)
0.81 ± 0.01
0.98
0.94
101.53
1.36
0.36
0.59
0.16
0.34
1.16
0.96
0.59
±
±
±
±
±
±
±
±
±
±
±
0.01
0.02
2.75
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
References
Schwarzkopf (1992), Rohr (1997),
Doughty & Shine (1997)
Langkilde et al. (2005)
Schwarzkopf (2005)
Bull et al. (1993)
While et al. (2007)
Olsson & Shine (1998, 1999), Shine & Olsson (2003)
Wapstra et al. (1999), Wapstra & Swain (2001)
B. Goodman, unpublished
Radder & Shine (2007a,b)
Warner & Shine (2006, 2007)
Uller et al. (2006)
Uller & Olsson (2006, in press)
calculating the left-hand side of eqn 1, comparing it with
the theoretical predictions for clutch sizes of {n + 1,n} for
each of the species, and testing if predictions were within
the 95% confidence interval. Confidence intervals of the
observed values were estimated using bootstrap with
10 000 random sample events with replacement
(Poptools add-ins for Microsoft Excel, Hood, 2002). We
used the smallest of the sample sizes of the two clutch
sizes to avoid bias due to differences in sample size (see
e.g. West et al., 2001).
Results
Mean offspring mass differed significantly among clutch
sizes for most species (Table 3). Data for two of the five
oviparous and for four of the seven viviparous species
Table 3 Results from a Spearman’s rank order correlation, testing
the prediction that mean offspring mass is independent of clutch
size, and a fixed factor analysis of variance, testing for significant
differences in offspring mass (dependent variable) across clutch sizes
(independent variable), for each of the 12 lizard species.
Correlation
ANOVA
Species
rs
P-value
F
Eulamprus tympanum
Eulamprus heatwolei
Eulamprus quoyii
Tiliqua rugosa
Egernia whitii
Niveoscincus microlepidotus
Niveoscincus ocellatus
Lampropholis mirabilis
Bassiana duperreyi
Amphibolurus muricatus
Ctenophorus pictus
Ctenophorus fordi
)0.16
)0.32
)0.28
)0.47
)0.40
0.27
)0.19
)0.35
)0.09
)0.28
)0.10
)0.06
< 0.01
0.02
0.08
< 0.01
< 0.01
< 0.01
< 0.01
0.04
0.48
< 0.01
0.22
0.30
F4,351 = 2.18
F3,49 = 3.84
F5,33 = 1.52
F2,38 = 6.04
F2,103 = 10.61
F4,143 = 9.03
F4,553 = 7.47
F2,31 = 2.59
F4,57 = 3.89
F5,116 = 2.47
F3,139 = 0.98
F3,265 = 0.95
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P-value
<
<
<
<
<
0.07
0.01
0.21
0.01
0.01
0.01
0.01
0.09
0.01
0.04
0.40
0.41
146
T. ULLER ET AL.
Table 4 Results of the ordered heterogeneity test of a decrease in
variation in average offspring mass with clutch size for each of the 12
lizard species.
Species
N
Pc
rs
rsPc
P-value
Eulamprus tympanum
Eulamprus heatwolei
Eulamprus quoyii
Tiliqua rugosa
Egernia whitii
Niveoscincus microlepidotus
Niveoscincus ocellatus
Lampropholis mirabilis
Bassiana duperreyi
Amphibolurus muricatus
Ctenophorus pictus
Ctenophorus fordi
5
4
6
3
3
5
5
3
5
6
4
4
0.8375
0.7023
0.9237
0.0926
0.9967
0.9999
0.9870
0.8312
0.1297
0.9291
0.8958
0.9997
)0.70
)0.40
)0.94
)0.50
)1.00
0.60
)1.00
0.50
)1.00
)0.95
)0.80
)1.00
)0.5863
)0.2809
)0.8683
)0.0463
)0.9967
0.5999
)0.9870
0.4156
)0.1297
)0.8826
)0.7166
)0.9997
NS
NS
**
NS
**
NS
***
NS
NS
**
*
***
NS, P > 0.05, *P < 0.05, **P < 0.01, ***P < 0.001. N is the number of
clutch sizes for each species. Pc is the complement of the P-value
from the Bartlett’s test of homogeneity of variance (i.e. Pc =
1 ) PBartlett). rs is the Spearman’s rank correlation coefficient.
failed to show the convergence of the maximum and
minimum mean offspring mass (Table 4). The confidence
intervals were very large for some species for which we
had limited data, in particular at low and high clutch
sizes. However, it is clear that the fit to quantitative
predictions was poor, with the predicted value falling
outside of the confidence interval for at least one test of
eqn 1 for five of the 12 species (Appendix 1). This is
clearly seen from plots of the means and 10th and 90th
percentiles for each clutch size for each species (Fig. 2),
showing that several species failed to show convergence
and, contrary to the most fundamental prediction, some
species actually showed an increase in the variation in
offspring mass with increasing clutch size.
Discussion
Our evaluation of invariant offspring size–number relationships in 12 species of lizards showed an overall weak
fit to theoretical expectations. In particular, almost half of
the species had at least one test where the confidence
interval did not include the ratio predicted by eqn 1.
Thus, our results provide an important extension of
previous single-species tests and strongly suggest that the
Charnov et al. (1995) model does not capture the
complexities of size–number strategies in lizards with
small clutch sizes. We provide an overview of the most
likely reasons for the lack of fit and their implications for
the evolution of reproductive investment.
When models fail to capture reality
When optimality models fail to predict empirical data,
one or several assumptions are likely to be incorrect or
too simplistic. The critical assumptions of the small clutch
size models are listed in Table 1 in the Introduction. We
will discuss what we consider to be the most problematic
when applying these to lizards, but our discussion has
applicability also to other taxa.
Virtually all size–number models, including the ones
for small clutch sizes, assume that reproductive investment is a step-wise optimization problem with independence of decisions at each step. In other words, females
first decide on how much to invest into reproduction,
and secondly, how to divide these resources into
offspring size vs. number (additional steps may also
occur, including whether to produce sons or daughters,
Hardy, 2002; Wapstra et al., 2007). Thus, because the
available resource pool is fixed when size–number
allocation occurs, employing a certain clutch size strategy
necessarily generates a mean offspring size and vice versa
(Smith & Fretwell, 1974; reviewed in Roff, 1992). There
are several problems with the logic of this approach. First,
total allocation to reproduction may not be independent
of the division of resources to offspring size or number.
For example, if allocation occurs continuously during
embryonic development, as in mammals, allocation to
individual offspring (and hence total allocation) can
continue once clutch size has been fixed. More generally,
resource allocation to offspring in direct response to
clutch size, or vice versa, could reduce the variation in
offspring size at low clutch sizes. Indeed, we argue that
this particular allocation strategy should be favoured
when clutch sizes are small, as it would allow better
precision and avoid suboptimal allocation (i.e. avoid
producing unnecessarily small or large offspring) by
allowing flexible adjustment of total allocation directly in
response to clutch size. Thus, this allocation mechanism
would to some extent resolve the dilemma imposed by
the integer change-over problem (sensu Downhower &
Charnov, 1998) that a fixed, hierarchical, allocation
strategy would incur. Indeed, a model based on continuous functions that linked total reproductive effort and
per-offspring investment found that changes in parameter values had consequences for both the total reproductive effort and the per-offspring investment (Winkler
& Wallin, 1987; Caley et al., 2001). Thus, it would be
useful to explore similar models that explore the importance of flexibility in allocation decisions for discrete
cases.
The above reasoning suggests that species with continuous allocation throughout offspring development,
such as mammals, should show a poorer fit than would
species that more closely follow the standard allocation
assumption (fixed total reproductive effort and subsequent size–number decision). Many reptiles, including
lizards, rely on stored resources for allocation (i.e. they
are capital rather than income breeders; Schwarzkopf,
1994; Bonnet et al., 1998), suggesting that they do
conform to the assumption of a fixed pool of resources.
Thus, it would be interesting to compare data from
mammals with a similar range of litter sizes to our data
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Tests of size–number invariants in lizards
1.1
1.2
Eulamprus tympanum
1.0
Eulamprus heatwolei
1.1
0.9
1.0
0.8
0.9
0.7
0.6
0.8
1
1.4
2
3
4
5
Eulamprus quoyii
1.2
110
1.0
90
0.8
70
0.6
50
4
5
6
1.8
7
2
3
4
Tiliqua rugosa
1
8
Egernia whitii
1.6
Average offspring mass (g)
1
130
0.8
2
3
Niveoscincus microlepidotus
0.6
1.4
0.4
1.2
1.0
0.2
1
2
1
3
0.7
0.3
0.6
0.2
0.5
2
3
4
5
Lampropholis mirabilis
0.1
Niveoscincus ocellatus
0.4
0.0
2
0.5
3
4
5
6
2
1.6
Bassiana duperreyi
3
4
Amphibolurus muricatus
1.4
0.4
1.2
0.3
0.2
1.0
5
6
7
1.2
Fig. 2 Means and 10th and 90th percentiles
of average offspring mass for each clutch
size for each of the 12 species of lizard. See
Appendix 1 for sample sizes. Scales on y-axes
differ due to variation in offspring mass
between species.
8
0.8
9
Ctenophorus pictus
1.1
0.7
1.0
0.6
0.9
0.5
0.8
3
4
5
Clutch size
on reptiles. Unfortunately, only one mammal has been
investigated in this respect (Cavia aperea; data deviated
significantly from predictions; Kasparian et al., 2005).
Alternatively, as oviparous and viviparous reptiles differ in their potential for continuous maternal allocation throughout offspring development, comparisons
between the two reproductive modes may also give us
valuable insights into the importance of a fixed pool of
resources. Our data showed that many viviparous
6
3
4
0.8
0.4
5
6
7
8
Ctenophorus fordi
2
3
4
5
Clutch size
(43%) and oviparous (40%) species had confidence
intervals that did not include the predicted value from
eqn 1. The majority of viviparous lizards also failed to
show a general convergence of maximum and minimum offspring mass with increasing clutch size,
whereas the majority of oviparous species did show
significant convergence. However, this difference could
be driven also by alternative factors (see below) and
data for more species would be required to accurately
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test for a quantitative difference between reproductive
modes, preferably controlling for phylogeny. Furthermore, the majority of viviparous lizards only have
limited resource allocation to offspring post-ovulation
(Thompson et al., 2001; Thompson & Speake, 2006),
which make them less suitable for evaluating the role of
fixed vs. nonfixed per-offspring allocation at the timing
of clutch size decisions. However, post-ovulatory offspring provisioning is not necessary to cause nonindependence between allocation steps. For example, studies
of the lizard Uta stansburiana suggest that egg size and
number are physiologically, and potentially genetically,
linked (Sinervo & Licht, 1991), which would cause a
similar link between total reproductive effort and the
size–number strategies (see also Caley et al., 2001;
Fischer et al., 2006). Deviations from model predictions
could also arise if either egg size or clutch size is
constrained within females (Christians, 2002) or if
clutch size directly affects offspring size, for example,
via differential uptake of water during incubation
(reviewed in Brown & Shine, in press). Nevertheless,
we argue that post-ovulatory maternal effects on
offspring size could explain deviations from model
predictions for at least some of the species in this study.
Importantly, maternal effects on offspring size driven by
female basking during pregnancy are common in
viviparous lizards and documented in several of the
species (or closely related species) included in this data
set (e.g. Shine & Harlow, 1993; Swain & Jones, 2000;
Wapstra, 2000; Caley & Schwarzkopf, 2004). Unfortunately, at present too few lizards with low clutch sizes
seem to have been studied in sufficient detail to
generate the large data sets required to test invariant
predictions, which precludes a detailed comparative
analysis. Experimental manipulations could potentially
also directly address the importance of specific assumptions in generating deviations from model predictions,
although this approach will only be feasible for a limited
number of species.
The small clutch size model tested here also assumes
that selection on offspring size does not differ within
clutches and across clutch sizes. The first assumption is
doubtful in species with selection for differential sex
allocation (Wapstra et al., 2007), but evidence for sexspecific offspring size has not been clearly documented in
the present species (but see Langkilde & Shine, 2005).
Finally, physical or physiological constraints may limit
the lower and upper size of offspring. For example, the
size of the pelvic girdle may constrain maximum
offspring size, as has been described in turtles (Congdon
& Gibbons, 1987) and been suggested for lizards (Sinervo
& Licht, 1991; Ji et al., 2006). Alternatively, there may be
a minimum size below which offspring are unviable
(Charnov et al., 1995). These patterns would be evident
as asymmetries in the convergence of the 10th and 90th
percentiles in Fig. 1. Although results are qualitative
rather than quantitative, there is little evidence for wide-
spread asymmetric convergence. In fact, convergence
was poor overall as supported by both the OH test and
the test of eqn 1 (summarized in Table 4 and Appendix 1).
Thus, we suggest that the assumption of hierarchical, and
independent, allocation steps is a more serious problem
than the existence of upper and lower limits on offspring
mass for the deviations from invariant predictions in
lizards.
Conclusions
Our data represent the most comprehensive evaluation
of invariant predictions from size–number models for
species with small clutch sizes. There is, however, a
general agreement among this and previous tests that
the discrete models fail to capture the complexities of
offspring size–number strategies. We have argued that
the most likely reason is that there exists greater
flexibility and links between different steps of resource
allocation than is allowed by this, and most other,
optimality models. This would violate assumptions of a
hierarchical allocation sequence and the independence
among allocation decisions. We agree with several
other workers (e.g. Christians, 2002; Glazier, 2002;
Williams, 2005) that a better integration of mechanisms
of allocation with life-history theory could provide
insights into the evolution of reproductive allocation.
Despite that it is over three decades since the publication of one of the most influential papers in this field
(Smith & Fretwell, 1974), our understanding of the
evolution of the division of resources into offspring size
and number remains surprisingly incomplete. We argue
that the evidence is strong that current offspring size–
number models for small clutch sizes fail to capture
empirical patterns and therefore must be revised.
Invariants may be less common in nature than is
frequently believed (Nee et al., 2005; Savage et al.,
2006).
Acknowledgments
This paper is testimony to the positive attitude towards
data sharing and collaborations that is the hallmark of
herpetological research in Australia and elsewhere. We
dedicate this paper to one of its authors, Raju Radder,
who died while the work was in progress. Raju was an
inspiration to us all, and his generosity of spirit, enthusiasm and insights will be sorely missed by his colleagues.
We emphasize that important research is conducted on
many more Australian lizard species than the ones that
met the requirements for inclusion in this paper and
apologize for any unintended omissions. Two anonymous reviewers provided valuable comments on the
manuscript. Financial support was provided by the
Wenner-Gren Foundations (TU), a James King of Irrawang Traveling Scholarship (TL) and the Australian
Research Council (TU, EW, CMB, RS, MO).
ª 2008 THE AUTHORS. J. EVOL. BIOL. 22 (2009) 143–151
JOURNAL COMPILATION ª 2008 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY
Tests of size–number invariants in lizards
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Received 17 June 2008; revised 3 September 2008; accepted 4 September
2008
Appendix 1 Summary of the tests of eqn 1 for 12 species of Australian lizards.
Species
Reproductive
mode
Eulamprus tympanum
Viviparous
Eulamprus heatwolei
Viviparous
Eulamprus quoyii
Viviparous
Tiliqua rugosa
Viviparous
Egernia whitii
Viviparous
Niveoscincus microlepidotus
Viviparous
Niveoscincus ocellatus
Viviparous
Clutch
sizes
(2,1)
(3,2)
(4,3)
(5,4)
(2,1)
(3,2)
(4,3)
(5,4)
(6,5)
(7,6)
(8,7)
(2,1)
(3,2)
(2,1)
(3,2)
(2,1)
(3,2)
(4,3)
(5,4)
(3,2)
(4,3)
(5,4)
(6,5)
Sample
sizes
Expected
Cn ⁄ Cn + 1
Observed
Imax nþ1 Imax nþ1 =Imax n Imin n
Lower
CI (2.5%)
Upper
CI (97.5%)
(106,33)
(135,106)
(65,135)
(17,65)
(13,5)
(25,13)
(10,25)
(10,5)
(6,10)
(11,6)
(3,11)
(20,8)
(13,20)
(52,26)
(28,52)
(71,14)
(42,71)
(16,42)
(5,16)
(188,85)
(177,188)
(86,177)
(22,86)
0.50
0.67
0.75
0.80
0.50
0.67
0.75
0.80
0.83
0.86
0.87
0.50
0.67
0.50
0.67
0.50
0.67
0.75
0.80
0.67
0.75
0.80
0.83
0.91*
0.93*
1.16*
0.76
0.83
1.00
1.53
0.71
1.29
1.10
0.86
1.34
0.82
0.71
0.86
1.03
1.14*
1.60*
0.57
0.90*
1.11*
0.95
0.45
0.54
0.79
0.85
0.46
0.20
0.48
0.51
0.15
0.42
0.39
0.00
0.46
0.39
0.38
0.48
0.34
0.69
1.05
0.03
0.74
0.87
0.55
0.39
1.38
1.17
1.61
1.36
2.04
1.76
3.30
2.26
3.15
2.60
4.00
3.68
1.52
1.11
1.51
2.65
1.90
2.40
1.77
1.11
1.29
1.37
1.07
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Tests of size–number invariants in lizards
151
Appendix 1 (Continued)
Species
Reproductive
mode
Lampropholis mirabilis
Oviparous
Bassiana duperreyi
Oviparous
Amphibolurus muricatus
Oviparous
Ctenophorus pictus
Oviparous
Ctenophorus fordi
Oviparous
Clutch
sizes
Sample
sizes
Expected
Cn ⁄ Cn + 1
Observed
Imax nþ1 Imax nþ1 =Imax n Imin n
Lower
CI (2.5%)
Upper
CI (97.5%)
(3,2)
(4,3)
(6,5)
(7,6)
(8,7)
(9,8)
(4,3)
(5,4)
(6,5)
(7,6)
(8,7)
(4,3)
(5,4)
(6,5)
(3,2)
(4,3)
(5,4)
(23,7)
(4,23)
(11,9)
(19,11)
(17,19)
(6,17)
(14,6)
(34,14)
(37,34)
(16,37)
(15,16)
(57,14)
(54,57)
(18,54)
(134,69)
(58,134)
(9,58)
0.67
0.75
0.83
0.86
0.87
0.88
0.75
0.80
0.83
0.86
0.87
0.75
0.80
0.83
0.67
0.75
0.80
2.63*
0.84
0.95
1.16
1.14
0.81
0.92
1.37
0.70
0.90
0.78
1.25
0.78
1.03
0.96*
0.73
0.62
1.16
0.06
0.45
0.38
0.35
0.36
0.31
0.64
0.43
0.53
0.51
0.56
0.34
0.54
0.76
0.48
0.20
4.84
2.50
2.27
2.27
3.16
2.04
3.55
2.83
1.21
1.37
1.20
3.63
1.45
1.66
1.22
1.00
1.24
*Significant deviation from the predicted value.
ª 2008 THE AUTHORS. J. EVOL. BIOL. 22 (2009) 143–151
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