ALLOMETRIC SCALING OF BODY LENGTH: ELASTIC OR
GEOMETRIC SIMILARITY IN MAMMALIAN DESIGN
MARINA SILVA
Department of Biology, University of Prince Edward Island, 550 University Avenue,
Charlottetown, Prince Edward Island CiA 4P3, Canada
Allometric scaling of body length was examined for 1.733 estimates of body length for a
broad taxonomic and size range of mammals. Mammal species were classified by locomotion, habitat, and taxonomic relatedness. Scaling exponents found in different groups of
mammals were compared to those predicted by geometric (0.333) and elastic (0.250) similitude models. The scaling exponent for the length: mass relationship 0.359 agreed better
with the geometric-similitude model. However, the relationship between body length and
body mass was not linear as previously postulated, and differences in the scaling exponent
were found between marine and terrestrial mammals and between volant and non-volant
forms. Allometric scaling of body length also varied among orders and families of mammals.
Key words:
length, mass, mammals, scaling, morphology
Several studies have examined the relationship between body length and body
mass and suggested that this may be an important tool for the investigation of ecology,
biology and physiology in organisms
(Economos, 1983; Eisenberg, 1981; Niklas.
1994; Wolff and Guthrie, 1985). The allometric exponent of the relationship between
body length and body mass has been used
to explain morphological design of organisms by relating that design to the elastic or
geometric model (Alexander, 1988; Economos, 1983; Pollock and Shadwick, 1994:
)Vainwright et aI., 1982). According to the
elastic model, trunk diameter increases faster than body length, so that body length
scales to the 0.250 power of body mass.
Alternatively, the geometric model assumes
that trunk diameter increases isometrically
with body length, such that the value of the
exponent of the relationship between body
length and body mass is ca. 0.333.
Although some studies have dealt with
the relationship between body length and
body mass in mammals (Calder, 1984; Jerison, 1970: Radinsky, 1978), few of them
(Cristoffer, 1991; Economos, 1983; Niklas,
Journal of Mamrnalogy, 79(1):20-32. 1998
1994) investigated the importance of the
scaling exponent as a tool for the examination of morphology and physiology in
mammals. Furthermore, the few studies that
did address the relationship between body
length and body mass generally were based
on small data sets or were restricted to one
or few taxonomic orders (Economos, 1983;
Jerison, 1970; Radinsky, 1978: Wolff and
Guthrie, 1985). Most studies show that
mammals, except the Bovidae, conform
better with the geometric-similitude model
than with the elastic one (Alexander, 1977,
1988; McMahon, 1973). However, several
studies suggested that terrestrial mammals
should conform better to the elastic model
due to gravity (Alexander, 1977, 1988;
Economos, 1983; McMahon, 1973). If this
is true, aquatic mammals should deviate
less from the geometric model than terrestrial forms because they live in a weightfree environment. Yet, for marine mammals, Economos (1983) showed that the
scaling exponent of the length: mass relationship was closer to the exponent predicted by the geometric similarity model.
Economos' study (1983) was, however,
20
February 1998
21
SILVA-SCALING OF BODY LENGTH
based on a small data set of marine mammals from the order Cetacea. Marine mammals include two other mammalian taxonomic orders, Pinnipedia and Sirenia. Unlike cetaceans, pinnipeds inhabit both terrestrial and aquatic environments. It is
unknown, however, if pinnipeds show a
scaling exponent closer to the one of cetaceans or to the one of terrestrial mammals.
Economos (1983) also suggested that the
relationship between mammalian body
mass and body length may be non-linear.
Econom~s noted that phyletic increase in
body mass of terrestrial mammals was accompanied initially by an increase in linear
dimensions according to the principle of
geometric similarity because gravitational
loading does not present a structural problem. Therefore, at a certain body mass, we
should expect to find a reduction of the
scaling exponent of the length: mass relationship. Although Economos (1983) did
not provide any statistics, he proposed a
threshold for the change in the exponent at
ca. 20 kg. If this is valid, I should expect
to find a significant decrease in the slope of
the length: mass relationship around this
body mass.
Furthermore, study of taxonomic deviations in the allometric exponent of the
length: mass relationship from the global
trend may be a powerful tool in the investigation of mammalian adaptations. However, little is known about the relationship
between body length and body mass across
and within taxonomic groups in mammals.
For example, Eisenberg (1981) noted that
volant mammals are smaller than non-volant forms. This adaptation may allow bats
to minimize wing loading, which is basically body mass (length cubed) divided by
area (length squared). Animals with low
wing loading may be better able to carry
more mass, and thus, a scaling exponent
significantly smaller than the one expected
from geometric-similitude model might be
selectively advantageous for volant mammals considering aerodynamic consequences of flight (Cristoffer, 1991; Hayssen
and Kunz, 1996). In addition, if selection
for a reduction in body mass in the Chiroptera increases progressively with increasing
body length (Eisenberg, 1981), then the
scaling exponent of the length: mass relationship in bats would not only be steeper
than the one of non-volant forms, but it also
should differ significantly between small
and large bats. No analyses across species
were conducted to examine these predictions, although Cristoffer (1991) examined
rodents and bats from both Nigeria and
Thailand and found that large bats are less
massive than rodents of comparable body
length.
Other differences with respect to the scaling exponent across mammalian orders and
within them also can be predicted when one
considers ecology or behavior. For example, Wainwright et al. (1982) suggested that
mammals like cats (Carnivora: Felidae) and
gazelles (Artiodactyla: Bovidae) reduce
their mass to outrun other mammals of similar body length. This predicts scaling exponents significantly steeper than the one
expected from the geometric-similitude
model in these families.
My purpose was to examine scaling of
body length in mammals and provide equations that can be used for predicting body
length. In the context of these scaling exponents, I compared mammals that use different means of locomotion and belong to
different taxonomic groups.
METHODS
Data on body mass and body length were obtained from the literature (Anderson et al., 1993;
Ceballos and Miranda, 1986; Eisenberg, 1989;
Emmons. 1990; Hufnagl, 1972; Husson. 1978;
Hutterer, 1986; Mares et al., 1989; Marshall,
1967; Nowak, 1991; Straham, 1983). Trunk diameter was not used because those values were
rarely reported in the literature. Body mass (g)
and body length (rom) were mean values for
adults; when the mean values were unavailable,
I used the midpoint of ranges. Body length was
length of the head and body (Le.. excluding
length of tail); for cetaceans, total length of the
body was used. Only a small number of sources
22
Vol. 79. No.1
JOURNAL OF MAMMALOGY
reported the method used to measure body
length. Therefore, bias associated with the use
of different methods to measure body length was
difficult to evaluate accurately, although I suspect it was negligible due to strong correlations
found in this study.
Mammals were grouped by taxon, habitat, and
means of locomotion. Comparisons were performed by classifying mammals into the following categories: marine (Cetacea, Pinnipedia, and
Sirenia); wholly-marine (live strictly in the sea:
Cetacea and Sirenia); and terrestrial. Terrestrial
mammals were sorted into volant (Chiroptera)
and nonvolant fonns. Taxonomic analyses were
performed for both order and family following
the classification of Corbet and Hill (1991).
All variables were transformed logarithmically to stabilize variance, linearize responses, and
normalize residuals. When detransformation
from the logarithmic to the original scale were
necessary, I corrected values following Zar
(1996). Model II regressions were thought to be
most appropriate for determining scaling exponents (La Barbera, 1989; Martin and Barbour,
1989); thus, I employed major axis regression
(Jolicoeur, 1990) in this analysis. Although there
are other model II regression methods, in this
case major-axis regression was the most appropriate because both body length and body mass
were log-transformed (Jolicoeur, 1990; Mazer
and Wheelwright, 1993; Mesple et a1., 1996).
Regression analyses were performed for taxonomic groups (orders or families) represented by
at least six species. Thus, for several orders the
sum of their respective familial samples for
which regression was perfonned was not equal
to the sample size when all species of the order
were pooled. Non-linearity of the relationship
between body length and body mass was determined using locally weighted sequential
smoothing (LOWESS; Cleveland, 1979), a model-free method for detennining the unbiased
fonn of the relationship between two variables,
while guarding against the influence of deviant
points (Cleveland and McGill, 1985). LOWESS
was developed by Tokey (Cleveland and McGill,
1985) and is designed to be robust to variations
in distribution and provide an unbiased indication of trends in large refractory data clouds. I
used polynomial regression analysis to test for
statistical significance of nonlinearities indicated
by LOWESS (sensu Silva and Downing, 1995).
4.5
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,
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4.0
3.5
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c
3.0
~
2.5
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<D
~
2.0
C>
2
1.5
1.0
-4
-3
-2
-1
0
1
2
3
4
5
log10 Body Mass (kg)
-
4.5
~
3.5
.c
b
C> 4.0
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~
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0
<D
3.0
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•>0
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0
2.0
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o
::r
1.5
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P < 0.0001
1.0 ':--:-'::--:":-----::':---:':--:::--:":-.....-:-'
1.0 1.5 2.0 2.5 3.0 3.5 •. 0 •. 5
log10 Estimated Body length (mm)
FIG. 1.-(a) Relationship between logarithm
of body length (nun) and logarithm of body
mass (kg) of all mammals and (b) between predicted body length and observed body length using the all-mammals scaling fonnula (Table I)
based on 1,733 estimates of body length and
body mass.
RESULTS
I collected 1,733 published estimates of
body length and body mass. Body length
varied from 20 mm in the southern pygmy
mouse (Baiomys musculus) to 15,200 mm
in the spenn whale (Physeter catodon).
Body mass varied from 0.002 kg in Kitti's
hog-nosed bat (Craseonycteris thonglogyai)
to 42,500 kg in the spenn whale (Fig. la).
February 1998
SILVA-SCALING OF BODY LENGTH
23
TABLE l.-Major axis regression analyses of the relationships between the logarithm of body
length (L, mm) and the logarithm of body mass (M, kg) in mammals.
Group
Slope (el)
0.359
0.340
0.316
0.357
0.317
0.334
All mammals
Marine mammals
Wholly-marine mammals
Terrestrial mammals
Volant mammals
Non-volant mammals
(0.357-0.362)
(0.311--0.369)
(0.275--0.358)
(0.354--0.360)
(0.304-0.330)
(0.331-0.338)
Intercept
nO
2.501
2.587
2.717
2.498
2.369
2518
1,733
85
34
1,648
479
1,169
• Number of observations included in the analysis.
Pearson's coefficient of correlation; all correlations are significant at
b
The regression slope of the log-log relationship between body length and body
mass for all mammals pooled was 0.359
(Table 1). The confidence interval of this
value did not include expected values of
0.250 and 0.333, corresponding to the elastic- and the geometric-similitude models,
respectively. Correlation between body
length and body mass was extremely strong
suggesting that body mass was a good pre-
dictor of variation in body length (Table I).
The slope of the relationship between predicted body length and observed body
length was not significantly different from
I (slope = 0.998; Fig. Ib) showing that the
all-mammals equation (Table 1) may be
used for predicting body length.
The LOWESS analysis revealed that the
'.5
~
E
E
-
4.0
3.5
""'"
3.0
"-
2.5
C
~
"0
0
<D
2.0
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~
1.5
-
-,
1.0
-3 -2 -1
0
Lowess
Malor Qxls
2
3
4
5
10910 Body Mass (kg)
FIG. 2.-The log-log linear fit (solid line) to
the 1,733 estimates bf body length and body
mass of mammalian species shown in Fig, 1; the
dotted line represents LOWESS.
P =
'"
0.99
0.93
0.94
0.99
0.92
0.99
Average
log L
Average
log M
2.365
3.400
3.567
2.312
1.803
2.520
-0.379
2.427
2.723
-0.523
-1.818
0.007
0.001.
log-log relationship between body length
and body mass was distinctly nonlinear
(Fig. 2). Polynomial regression analysis
showed that the nonlinearity indicated by
LOWESS was significant (P < 0.0001).
The allometric exponent behaved differently over ranges of body mass from 0.002 to
0.1 kg, 0.1-100 kg, and 100-42,500 kg.
Separate scaling exponents were 0.410,
0.346 and 0.344, respectively. Confidence
intervals of the regression slopes suggested
that small manunals differ significantly
from intermediate and large mammals (P <
0.05), but the difference between intermediate and large mammals is not significant
(P > 0.05). Confidence intervals also indicated that scaling values for small and intermediate mammals differ significantly
from both the geometric- and the elasticsimilitude models, while the values for
large manunals were consistent only with
the geometric-similitude model.
ANOVA showed that marine mammals
were larger than terrestrial mammals (P <
0.05) and mean body mass for the respective groups differ significantly (Table I).
However, this may not indicate that the marine environment requires that mammals be
larger, because marine mammals do not occur randomly across all taxa but are restricted to three orders. Thus, difference in size
could reflect taxon and means of locomotion, instead of habitat. Differences in the
scaling exponent also existed between terrestrial and marine mammals (Table 1).
When marine mammals (Cetacea, Pinnipe-
24
Vol. 79, No.1
JOURNAL OF MAMMALOGY
TABLE 2.~Major axis regression analyses of the relationships between the logarithm of body
length (L, mm) and the logarithm of body mass (M, kg) in several mammalian taxonomic orders and
families.
log L
Average
logM
2.34
-0.58
Average
Taxon
Marsupialia
Dasyuridae
DideJphidae
Macropodidae
Xenarthra
Dasypodidae
Mynnecophagidae
Insectivora
Erinaceidae
Soricidac
Talpidae
Tenrecidae
Scandentia
Chiroptera
Emballonuridae
Hipposideridae
Mo1ossidae
Mormoopidae
Noctilionidae
Phyllostomidae
Pteropodidae
VespertiIionidae
Primates
Callitrichidae
Cebidae
Ccrcopithecidac
Lorisidae
Carnivora
Canidae
Felidae
Herpestidae
Mustelidae
Procyonidae
Ursidae
Viverridae
Pinnipedia
Otariidae
Phocidae
Cetacea
Perissodactyla
Artiodactyla
Bovidae
Camelidae
Cervidae
Suidae
Tayassuidae
Rodentia
Caviidae
Ctenomyidae
Dasyproctidae
Echimyidae
Erethizontidae
Slope (el)
0.311
0.355
0.301
0.313
0.350
0.306
0.375
0.307
0.242
0.375
0.272
0.299
0.297
0.317
0.332
0.309
0.355
0.288
0.400
0.303
0.293
(0.292-0.330)
(O.31l-DAOJ)
(0.251-0.352)
(0.240-0.390)
(O.30fKl.394)
(0.278-0.334)
(0.351-0.399)
(0.283-0.332)
(0.134-0.356)
(0.297-0.458)
(0.195-0.351)
(0.041-0.599)
(0.193-0.408)
(0.304-0.330)
(0.294-0.371)
(0.190-0.437)
(0.286-D.426)
(0.202-0.379)
(0.292-0.517)
(0.285-0.322)
(0.210-0.380)
0.300 (0.271-0.3311
0.298
0.254
0.250
0.263
0.373
0.306
0.393
0.304
0.346
0.286
0.235
0.317
0.285
0.291
0.271
0.331
0.315
0.249
0.290
0.279
0.274
0.295
0.200
0.168
0.332
0.323
0.258
0.351
0.212
0.283
(0.283-0.313)
(0.155-0.358)
(0.208-0.292)
(0.193-0.336)
(0.294-0.457)
(0.294-0.318)
(0.345-0.442)
(0.275-0.333)
(0.235-0.466)
(0.259-0.313)
(0.167-0.305)
(0.170-0.476)
(0.239-0.333)
(0.264-0.318)
(0.247-0.295)
(0.276-0.388)
(0.273-0.359)
(0.192-0.308)
(0.270-0.310)
(0.252-0.307)
(0.160-0.395)
(0.260-0.332)
(-0.040-0.465)
(0.023-0.320)
(0.324-0.340)
(0.205-0.451)
(0.007-0.543)
(0.264-0.442)
(0.141-0.286)
(0.164-0.409)
Intercept
n'
2.520
2.532
2.543
2.506
86
12
37
14
37
23
9
45
8
19
7
6
9
479
2.460
2.429
2.515
2.477
2.447
2.595
2.425
2.475
2.494
2.369
2.425
2.299
2.442
2.331
2.472
2.333
2.374
2.329
2.491
2.457
2.522
2.522
2.511
2.571
2.516
2.577
2.560
2.572
2.568
2.554
2.608
2.645
2.688
2.554
2.721
2.740
2.623
2.633
2.704
2.642
2.718
2.749
2.504
2.512
2.433
2.547
2.480
2.538
31
7
65
15
IO
214
23
99
123
20
49
24
II
227
40
47
16
71
19
8
21
51
26
23
31
16
138
59
6
48
7
II
455
10
6
12
16
II
"
P
0.96
0.99
0.90
0.94
0.94
0.98
0.99
0.97
0.91
0.92
0.97
0.85
0.93
0.92
0.96
0.95
0.79
0.89
0.95
0.91
0.84
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0016
0.0001
0.0003
0.0334
0.0003
0.0001
0.0001
0.0013
0.0001
0.0001
0.0001
0.0001
0.0001
0.90
0.0001
0.96
0.78
0.87
0.85
0.96
0.96
0.94
0.95
0.87
0.93
0.87
0.90
0.95
0.95
0.98
0.94
0.94
0.93
0.93
0.94
0.96
0.93
0.69
0.66
0.97
0.91
0.82
0.94
0.86
0.87
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0021
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0028
0.0001
0.0857
0.0278
0.0001
0.0003
0.0464
0.0001
0.0001
0.0005
2.11
-1.l9
2.28
2.64
2.67
2.58
2.82
2.07
2.31
1.88
2.09
2.32
2.22
1.80
1.74
1.67
1.85
1.78
1.94
1.82
2.01
1.72
2.59
2.35
-0.90
2.64
2.76
2.34
2.78
2.84
2.94
2.60
2.64
2.67
3.21
2.78
3.29
3.24
3.32
3.57
3.40
3.14
3.15
3.28
3.14
3.12
2.98
2.22
2.39
2.30
2.69
2.38
2.63
0.43
0.60
0,49
0.80
-1.31
-0.59
-1.95
-1.24
-0.52
-0.95
-1.82
-2.08
-2.04
-1.77
-1.91
-1.35
-1.73
-1.27
-2.07
0.32
-0.43
0.47
0.92
-0.47
0.68
0.84
1.19
0.1I
0.25
0.44
2.10
0.59
2.23
2.05
2.36
2.74
2.66
1.81
1.87
2.12
1.72
2.08
1.41
-0.86
-0.39
-0.53
0.42
-0.49
0.34
TABLE
Taxon
Heteromyidae
Muridae
Sciuridae
Lagomorpha
25
SILVA-SCALING OF BODY LENGTH
February 1998
Slope (Cl)
0.383
0.333
0.341
0.337
(0.312--0.458)
(0.318--0.347)
(0.315-0.368)
(0.286--0.390)
2.-Continued.
Intercept
2.569
2.501
2.514
2.542
n"
6
289
60
18
,.
P
Average
log L
Average
logM
0.99
0.94
0.96
0.96
0.0001
0.0001
0.0001
0.0001
2.02
2.12
2.31
2.59
-1.43
-1.15
-0.59
0.16
, Number of observations included in the analysis.
to Pearson's coefficient of correlation.
dia, and Sirenia) were analyzed alone, the
regression slope of the log-log relationship
between body length and body mass was
0.340, and when wholly-marine marrunals
(Cetacea and Sirenia) were analyzed, the
value was 0.316. Cetacea, however, contributed unequally to this analysis (Cetacea, n
= 31; Sirenia, n = 3). A similar scaling
analysis of terrestrial marrunals resulted in
an allometric exponent of 0.357. Confidence intervals around regression slopes for
these groups indicated that there were no
significant differences between them (P >
0.05). The scaling exponent of marine
mammals was not significantly different
from that predicted by geometric similarity,
while that of terrestrial mammals was significantly different. When terrestrial mammals were sorted into volant and non-volant
forms, analyses indicated that non-volant
mammals complied better with geometric
similarity than elastic similarity; however,
elastic and geometric models must be rejected for volant marrunals (Table 1).
Scaling of body length differed among
taxonomic orders (Table 2). Values of confidence intervals of regression slopes indicated that of the 13 taxonomic orders analyzed for which adequate (n :z:: 6) data sets
were available, nine slopes differed significantly from the one found when all mammals were pooled. For example, confidence
intervals indicated that the scaling exponent
of Rodentia was significantly different from
that of Artiodactyla, Carnivora, Perissodactyla, Pinnipedia, and Primates. On the other
hand, analyses indicated that Lagomorpha
and Rodentia, and Cetacea and Pinnipedia,
did not differ significantly. These results
must be interpreted with caution, because
differences in the allometric exponents
might reflect different ranges of body mass
covered by these orders. Non-linearity of
the log-log relationship between body
length and body mass was less obvious than
when mammals were pooled, but small departures from linearity occurred for the
smallest or the largest organisms in several
orders (Figs. 3 and 4).
Range of body mass for the order Insectivora is similar to that for the Chiroptera,
but body length of Chiroptera increased 5%
faster with body mass. Although the scaling
value found for Pinnipedia did not differ
significantly from that of Cetacea, only Cetacea showed a scaling exponent not significantly different from that predicted by
geometric similarity. With the exception of
Perissodactyla, all taxonomic orders
showed only small deviations from the theoretical value of 0.333 expected on the basis of the geometric similitude. For Perissodactyla, the exponent of the length: mass
relationship was not significantly different
from the theoreti"cal value of 0.250, suggesting that predicted and observed exponents were not significantly different.
Within families, body mass was correlated significantly with body length (Table 2).
Confidence intervals of regression slopes of
the log-log relationship between body
length and body mass suggested that most
of the differences were not significant.
Within the Carnivora, scaling exponents of
Canidae and Felidae differ significantly;
scaling value for Canidae differed signifi-
26
JOURNAL OF MAMMALOGY
'.0 ~---------,
3.5
Vol. 79, No.1
~----------,
,<
Carnivora
Artiodactyla
3.5
2.5
3.0
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.. _-.-
'.'
"'"
E
E
2.5
0
'-'"
•••
..c
+m
'.0
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0
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.,
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/~~o
3.5
, .5
-
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en
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0
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0
•
3
Cetacea
--.J
>-
2
Taxon
•.. _- All mammals
3.0
Q)
"0
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Low •••
._. AU mammal.
3.0
,
-
Taxon
LowI.I
_. All mammals
•
3
2
5
Taxon
--_ ... Low •••
"'-' All mammals
1..
-3
-,
-2
0
3 ••
3.'
Xenarthra
Insectivore
2.5
3.0
2.0
2.'
1.5
-
,
-Taxon
Taxon
......
..... - Low,,,
Low •••
.,,_. All mommlll,
... _. All mammal.
2.0
-2
-,
0
2
10910
3
Body
1.0
-3
-2
Mass
-,
0
(kg)
FIG. 3.-Relationships between the logarithm of body length (rum) and the logarithm of body mass
(kg) of Artiodactyla, Carnivora, Cetacea, Chiroptera, Xenarthra, and Insectivora. The solid lines are
simple log-log linear fits to data of each taxonomic order (Table 2), and the dotted lines represent
LOWESS fits to the data. The solid curves represent predictions made by the all-mammals scaling
fonnula (Table 1).
SILVA-SCALING OF BODY LENGTH
February 1998
3.0
r------------,
3.5
27
r------------,
Lagomorpha
tdarsuplalla
3.0
2.5
2.5
Taxon
...... Low •••
_.. _. AU mammol.
2.0 ' - - - - - " - - - - - - '
o
-1
4.0
r----------,
2.0
,
1.5
'
:
··
r
0"1
C
3.5
"-:'-:'~""
>-
2
r----------,
,o
3.5
o
a
o
o
Taxon
....•• Law •••
. _. All mammal.
OJ
3.0'-_ _ _ _" -_ _ _ _-'
2
0"1
o
0
o .'
-0
o
-1
/
Q)
-l
-2
Plnnlpedla
PerIssodactyla
..c
.....
'---_"-_"-_-L_-'--_-'
-3
4.0
Low •••
All mammal.
4
3
3.5
- - Toxan
...... Low ...
• .. _. All mammal.
3.0 '---'-_-'--_ _-.1.._ _--'
1
2
3
4
3.5
Primates
.
r------------,
Rodentia
,
,
3.0
,
• 0
3.0
2.5
2.5
2.0
o
-
T"xa"
Law •••
All mammal.
2.0 '--_"-_"-_-L_-'--_-'
-1
2
3
-2
o
- - Taxon
.•.•.. Low •••
_.. _. All mammal.
1.5 ~-~-L--~-L--~__:
-3
-2
-1
0
2
3
log10 Body Mass (kg)
FIG. 4.-Relationships between the logarithm of body length (mm) and the logarithm of body mass
(kg) of Lagomorpha, Marsupialia, Perissodactyla, Pinnipedia, Primates, and Rodentia. The solid lines
are simple log-log linear fits to the data of each taxonomic order (Table 2), and the dotted lines
represent LOWESS fits to the data. The solid curves represent predictions made by the all-mammals
scaling fonnula (Table 1).
28
JOURNAL OF MAMMALOGY
cantly from both elastic- and geometric-similitude models. For the Felidae, the geometric model could not be rejected. Within
Xenarthra, the regression slope of Mynnecophagidae was steeper than that for Dasypodidae but also was based on a smaller
sample size (Table 2). Based on confidence
intervals of the regression slopes, the geometric-similitude model could be accepted
for Dasypodidae, whereas elastic and geometric models were rejected for Myrmecophagidae (Table 2). For Primates, the scaling exponent differed between families of
simians and prosimians. When 123 primates species (Table 2) were sorted as simians (Callithricidae, Cebidae, and Cercopithecidae) or prosimians (Lorisidae), the regression slope for simians (slope = 0.289;
r = 0.96; n = 97) was significantly different (P < 0.05) from that of prosimians
(slope ~ 0.348; r ~ 0.97; n ~ 26). Based
on the confidence intervals, neither scaling
model fit simians; whereas the geometric
model could not be rejected for prosimians.
Within rodents, most families complied
with the geometric-similitude model. The
regression slope for Hetereromyidae was
steeper than that of other families but also
was based on a small sample size. For Echimyidae, however, the allometric exponent
was not significantly different from the one
expected from the elastic-similitude model.
Log-log relationships between body length
and body mass were· similar across most
families of bats. There were no significant
differences between families of Megachiroptera (Pteropodidae, n = 23) and Microchiroptera (all other bats, n = 456, Table
2). When all microchiropteran species were
pooled, the regression slope (slope = 0.309;
CI ~ 0.296--D.323) was not significantly
different from that of rnegachiropteran species (Pteropodidae, Table 2).
DISCUSSION
This study is the most comprehensive
analysis of the scaling of body length in
marrunals. It covers a broad range of taxa
and the entire range of size in marrunals.
Vol. 79, No.1
Body mass is highly correlated with body
length whether marrunals are pooled or
grouped by type of locomotion or taxon.
Results presented provide several equations
that can be used for predicting body length
in mammals. Note, however, that the regression for all mammals ignores non-linearity of the length: mass relationship and
scaling differences among and within tax0nomic orders.
The global-allometric exponent of the relationship between body length and body
mass is 0.359. While this value suggests
that neither the geometric- nor the elasticsimilitude models accurately describe the
morphological design of mammals, it is in
better agreement with the geometric similarity model than with the elastic one. Analyses presented here also upholds expectations of Economos (1983) concerning the
non-linearity of the log-log relationship between body length and body mass. Results
show that the linear relationship does not
accurately fit data of the smallest «0.0 I
kg) and largest (> 100 kg) mammals. Differences in scaling exponents over the
range of body mass indicate that no single
similarity model can represent morphological design of mammals. Small manunals
tend to increase their body length with body
mass faster than the common linear relationship would predict, while large mammals increase their body length less rapidly,
approaching the theoretical values expected
from geometric or elastic similitude. Analyses also indicate that taxonomic groups
composed of predominantly small or large
species should have different scaling exponents. One explanation for these findings
might be a physiological restriction associated with the surface law that causes small
mammals to be heavier at a given body
length than a common linear relationship
predicts (McMahon, 1973; Peters, 1983).
This reduces the ratio of surface area to
body mass and may help reduce heat loss
in small manunals (Peters, 1983; Scholander, 1955). On the other hand, differences
in the scaling exponent between species of
February 1998
SILVA-SCALING OF BODY LENGTH
small and large mammals lend general support to Economos' (1984) prediction that
elastic forces in a gravitational field may
surpass the safety factor in the mechanical
s'trength of mammalian bones in large
mammals; thus, they show an exponent significantly smaller than the one found for
smaller ones.
Similar scaling values are found when all
mammals are pooled, and when marine and
terrestrial mammals are analyzed separately, although the average marine mammal is
about twice as long and weighs over three
times more than its terrestrial counterpart.
However, when wholly-marine species are
compared with terrestrial ones, body length
increases 19% faster in terrestrial forms.
These findings lend general support to studies suggesting that wholly-marine mammals
conform better with the geometric similarity because they do not have problems with
gravity (Economos, 1983, 1984), although
other factors also may explain differences
between terrestrial and marine species. One
explanation might be that marine mammals
need to be more massive than terrestrial
forms to reduce heat loss. As water is abetter thermal conductor than air, aquatic
mammals may have greater problems in
temperature regulation than terrestrial
forms, resulting in selection for larger body
mass rather than body length (Wolff and
Guthrie, 1985). It is also possible, however,
that these results derive from the strong correlation between taxonomy and means of
locomotion. For example, all volant mammals belong to the order Chiroptera, but no
species of the order Cetacea can walk. The
body shape and length: mass relationship
may therefore reflect some constraints of
swimming producing a streamline body in
marine mammals (R. Channell, pers.
comm.).
Significant differences also were found
when marine mammals that venture onto
land (Pinnipedia) were compared to cetaceans; the geometric model seems to be appropriate only for cetaceans. Pinnipeds may
be less modified fOIl aquatic life than are
29
wholly aquatic cetaceans. There is, however, considerable variability in Pinnipedia.
Scaling of body length is lower for Otariidae than for Phocidae, and only for Phocidae, the geometric model cannot be rejected. Obviously, the reason for the difference between phocids and otariids is not
due to their means of locomotion; apart
from the fact that otariids use forelimbs and
phocids hind limbs, locomotion is relatively
similar. Results suggest that for a given increment in body mass, body length increases by a smaller amount in otariids than in
phocids. This may be due to different ranges of body mass covered by these two families and nonlinearity of the log-log length:
mass relationship. On average, phocids are
heavier and larger than otariids (Table 2).
These differences also may result from different ecological adaptations related to reproduction. Although both female phocids
and otariids exhibit high levels of maternal
investment and milk fat, fat, and water deposition is more important in phocids than
otariids (Boness and Bowen, 1996; Iverson
et al., 1993; Riedman and Ortiz, 1979).
These different maternal strategies result in
different patterns of mass change in females, which tend to lose considerable mass
particularly when lactating (Boness and
Bowen, 1996). Thus, relationships between
body length and body mass may reflect different energetic strategies of females pinnipeds. Under this scenario, data for pinnipeds used in this analysis would need to
be unequally represented by females. Unfortunately, sources used only sporadically
reported sex of -individuals. Additional
studies are necessary to test if sexual differences in pinnipeds affect the scaling exponent of body length in this group. Therefore, it is difficult to decide if differences
between pinnipeds and cetaceans are due to
different means of locomotion, different
ecological adaptations, or both.
Another question addressed in this study
is whether volant mammals are less massive
than non-volant forms of similar body
length. Analyses indicate that insectivores
30
JOURNAL OF MAMMALOGY
are heavier (ca. 25%) than bats of similar
length and body length increases ca. 5%
faster with increased body mass in Chiroptera than in Insectivora. Cristoffer (1991)
reached similar conclusions in an examination of bats and rodents from Nigeria and
Thailand. In addition, LOWESS analyses
show that the increase of body length is different between smaller «30 g) and larger
bats (>30 g). It is possible that the difference in the allometric exponent between
small and large bats helps large bats minimize wing loading. Constraints imposed by
flight on structural and physiological attributes of bats may become progressively
greater as body mass increases, so that body
length increases more rapidly than does
body mass (Cristoffer, 1991). This also suggests that the difference between the regression slope of the log-log relationship
between body length and body mass between volant and non-volant mammals depends on absolute body mass of the species
in the study. The conclusion regarding similarity between microchiropterans and macrochiropterans with respect to the allometric
exponent is limited by the fact that Megachiroptera only contains one family and Microchiroptera contains several. Alternatively, apparent similarity between microchiropterans and macrochiropterans may indicate that differences in Chiroptera result
from physiological or morphologically adaptations associated with flight rather than
taxonomy.
Ordinal scaling suggests several similarities among mammals belonging to different orders but covering a comparable range
of body mass. For most taxonomic orders,
the allometric exponent complies better
with the geometric-similitude model than
with the elastic one. The only clear exception is Perissodactyla. Reasons for this
trend are not clear, but they may reflect disproportionate contribution of rhinos (n = 6)
and tapirs (n = 6) in my analysis, both of
which possess a morphological design
clearly different from that exhibited by
horses and zebras. The same pattern is ob-
Vol. 79, No.1
served in Suidae (Artiodactyla, Table 2).
Uris may reflect similar effects of environment on functional design or morphology
of these three families.
Species of Artiodactyla scale similarly to
Pinnipedia and Carnivora. Lagomorpha and
Rodentia also exhibit a comparable scaling
exponent. At the family level, results indicate that scaling exponents also vary within
the same order. Moreover, results indicate
that scaling exponents derived for some
families deviate significantly from those detennined when all the species in an order
are pooled (Xenarthra, Primates and Carnivora, Table 2). In particular, variability
among families within Carnivora suggests
the great dissimilarity in their shape. Although there are some significant differences possibly due to small sample sizes,
relative constancy of exponents in families
in orders such as Rodentia may show that
species in this order are more uniform in
morphological design.,. Scaling differences
between orders and families might reflect,
therefore, greater heterogeneity across orders compared to within them. Thus, although all analyses agree on the importance
of absolute body mass, taxa of species surveyed also may have a significant influence
on the scaling exponent of the length: mass
relationship.
Scaling differences of body length across
and within taxonomic orders might reflect
some physiological, ecological, and evolutionary adaptations found in mammals.
Several studies suggest the role of phylogenetic history as an alternate explanation
for morphological design (Huey, 1987;
Lauder, 1991; Wainwright. 1991). For example, evidence from paleontological, morphological, and genetic data shows that pinnipeds evolved from a common carnivore
ancestor that was probably an ursid or a
mustelid (Amasson and Widegren, 1986;
Lento et aI., 1995; McLaren, 1960; Novacek. 1992; Tedford, 1976). It is possible,
therefore, that similarities found between
Carnivora and Pinnipedia lend general support to studies suggesting that these orders
February 1998
SIL VA-SCALING OF BODY LENGTH
are evolutionarily related. On the other
hand, the difference in the scaling exponent
between otariids and phocids may result
from recent ecological adaptations and not
from characteristics that they have inherited
from a common ancestor. Findings also
show that for a given increment in mass,
body length increases 45% faster in Canidae than in Felidae. This difference upholds
the idea that morphology influences ecology by limiting ability of an individual to
perfonn important tasks in its daily life (Alexander, 1988; James, 1991; Wainwright
1991). For example, effects of morphological variation in Carnivora may explain why
even though Felidae and Canidae are predators, each family uses different strategies
to capture prey and exploit different types
of prey (Earle, 1987; Eisenberg, 1981).
Morphological design and large size of Felidae may allow them to capture prey much
larger than themselves (Rosenzweig, 1966;
Vezina, 1985; Wilson, 1975).
ACKNOWLEDGMENTS
I am grateful to J. A. Downing who encouraged me to do this study. I also thank R. Peters,
F.-J. Lapointe, R. Channell, M. Prieto, and one
anonymous reviewer for their comments and
suggestions on the manuscript.
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Submitted 23 July 1996. Accepted 1 April 1997.
Associate Editor was Janet K. Braun.