A general scaling law reveals why the largest animals are not the fastest

Abstract

Speed is the fundamental constraint on animal movement, yet there is no general consensus on the determinants of maximum speed itself. Here, we provide a general scaling model of maximum speed with body mass, which holds across locomotion modes, ecosystem types and taxonomic groups. In contrast to traditional power-law scaling, we predict a hump-shaped relationship resulting from a finite acceleration time for animals, which explains why the largest animals are not the fastest. This model is strongly supported by extensive empirical data (474 species, with body masses ranging from 30 μg to 100 tonnes) from terrestrial as well as aquatic ecosystems. Our approach unravels a fundamental constraint on the upper limit of animal movement, thus enabling a better understanding of realized movement patterns in nature and their multifold ecological consequences. Maximum speed could scale with body mass, but the largest animals are not actually the fastest. A general scaling law explains this with a hump-shaped relationship due to a finite limit on acceleration time.

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Articles
DOI: 10.1038/s41559-017-0241-4
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
1 EcoNetLab, German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Leipzig 04103, Germany. 2 EcoNetLab, Friedrich Schiller University
Jena, Dornburger Strasse 159, 07743 Jena, Germany. 3 Department of Ecology and Evolutionary Biology, Yale University, New Haven, CT 06511, USA.
4 Department of Life Sciences, Imperial College London, Silwood Park, Ascot SL5 7QN, UK. *e-mail: myriam.hirt@idiv.de
Movement is one of the most fundamental processes of life.
The individual survival of mobile organisms depends
on their ability to reach resources and mating partners,
escape predators, and switch between habitat patches or breed-
ing and wintering grounds. By creating and sustaining individual
home ranges1 and meta-communities2, movement also profoundly
affects the ability of animals to cope with changes in land use and in
climate3. Additionally, movement determines encounter rates and
thus the strength of species interactions4, which is an important
factor influencing ecosystem stability5. Accordingly, a generalized
and predictive understanding of animal movement is crucial6,7.
A fundamental constraint on movement is maximum speed.
The realized movement depends on ecological factors such as land-
scape structure, habitat quality or sociality, but the range within
which this realized movement occurs meets its upper limit at max-
imum movement speed. Similar to many physiological and eco-
logical parameters, movement speed of animals is often thought to
follow a power-law relationship with body mass8–10. However, sci-
entists have always struggled with the fact that, in running animals,
the largest are not the fastest11–14. In nature, the fastest running or
swimming animals such as cheetahs or marlins are of intermediate
size, indicating that a hump-shaped pattern may be more realis-
tic. There have been numerous attempts to describe this phenom-
enon11–13,15,16. Although biomechanical and morphological models
have been tailored to explain this within taxonomic groups14,16–18, a
general mechanistic model predicting the large-scale pattern (over
the full body-mass range) across all taxonomic groups and ecosys-
tem types is still lacking. Here, we fill this void with a maximum-
speed model based on the concept that animals are limited in their
time for maximum acceleration because of restrictions on the
quickly available energy. Consequently, acceleration time becomes
the critical factor determining the maximum speed of animals.
In the following, we first develop the maximum-speed model (in
equations that are illustrated in the conceptual Fig. 1), test the
model predictions employing a global database and eventually
illustrate its applications to advance a more general understanding
of animal movement.
Results
Model development. Consistent with prior models8,10, we start
with a power-law scaling of theoretical maximum speed vmax(theor) of
animals with body mass M:
=vaM(1
)
b
max(theor)
During acceleration, the speed of an animal over time t
saturates19–21 (Fig.1a, solid lines) approaching vmax(theor) (Fig.1a,
dotted lines):
=−
−
vt v() (1 e) (2
)
kt
max(theor)
The acceleration constant k describes how fast an animal
reaches vmax(theor). In analogy to Newton’s second law, the accelera-
tion k should scale relative to the ratio between maximum force,
F, and body mass, M: that is, k ~ F/M. Knowing that maximum
muscle force roughly scales with body mass as F ~ Md, this yields a
general power-law scaling of k with body mass M:
=−
kcM(3)
d1
with constants c and d. As the allometric exponent d of the muscle
force falls within the range 0.75 to 0.94 (refs. 14,22,23), the overall
exponent (d − 1) should be negative, implying that larger animals
need more time to accelerate to the same speed than smaller ones
(see conceptual Fig.1a; colour code exemplifies four animals of
different size). Note that this general scaling relationship also
allows for the special cases of a constant acceleration across species
or a linear relationship with body mass.
Whereas prolonged high speeds are related to the maximum
aerobic metabolism, maximum burst speeds are linked to anaero-
bic capacity24,25. For maximum aerobic speed, ‘slow twitch’ fibres
are needed, which are highly efficient at using oxygen for gener-
ating adenosine triphosphate (ATP) to fuel muscle contractions.
Thus, they produce energy more slowly but for a long period of
time before they become fatigued, and they allow for continuous,
A general scaling law reveals why the largest
animals are not the fastest
Myriam R. Hirt1,2*, Walter Jetz1,3,4, Björn C. Rall1,2 and Ulrich Brose1,2
Speed is the fundamental constraint on animal movement, yet there is no general consensus on the determinants of maxi-
mum speed itself. Here, we provide a general scaling model of maximum speed with body mass, which holds across locomo-
tion modes, ecosystem types and taxonomic groups. In contrast to traditional power-law scaling, we predict a hump-shaped
relationship resulting from a finite acceleration time for animals, which explains why the largest animals are not the fastest.
This model is strongly supported by extensive empirical data (474 species, with body masses ranging from 30 μ g to
100 tonnes) from terrestrial as well as aquatic ecosystems. Our approach unravels a fundamental constraint on the upper
limit of animal movement, thus enabling a better understanding of realized movement patterns in nature and their multifold
ecological consequences.
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extended muscle contractions. In contrast, maximum anaerobic
speed is fuelled by a special type of ‘fast twitch’ fibres, which use
ATP from the ATP storage of the fibre until it is depleted. Thus,
they produce energy more quickly but also become fatigued very
rapidly and only allow for short bursts of speed. As our maximum-
speed model is based on this maximum anaerobic capacity, the
critical time τ available for maximum acceleration is limited by
the amount of fast twitch fibre and their energy storage capacity.
This storage capacity is correlated with the amount of muscle tissue
mass, which is directly linked to body mass. Thus, similar to the
muscle tissue mass, τ should follow a power law:
τ=fM (4)
g
where the allometric exponent g should fall in the range 0.76
to 1.27 documented for the allometric scaling of muscle tissue
mass26–29. This power law implies that larger animals should have
more time for acceleration (dashed lines in conceptual Fig.1b, c).
However, the power-law relationship of the critical time τ in our
model allows for a negative or positive scaling of energy availabil-
ity with body mass as well as the lack of a relationship (constant
energy availability across body masses (f = 0)). Although we have
included power-law relationships of k and τ (equations (3) and (4))
in our model, these scaling assumptions are not strictly necessary.
Instead, our only critical assumptions are that acceleration over
time follows a saturation curve (equation (1)) and that the time
available for anaerobic acceleration is limited.
Within the critical time τ, after which the energy available for
acceleration is depleted, the animal reaches its realized maximum
speed vmax (points in Fig.1c), which may be lower than the theo-
retical maximum speed (Fig.1a, dotted lines). Combining equa-
tions (1)–(4) with t = τ yields
=−
−
−+
vaM(1 e)
bcfM
max
dg1
which
simplifies to
=−
−
vaM(1 e) (5
)
bhM
max
i
where i = d − 1 + g and h = cf. This equation predicts a hump-
shaped relationship between realized maximum speed and body
mass (conceptual Fig.1d).
The limiting term
−
−
1e
hM
i
represents the fraction of the
theoretical maximum speed that is realized and is defined on
the interval]0;1[. For low body masses, this term is close to
1 and the realized maximum speed approximates the theoreti-
cal maximum. With increasing body masses, this term decreases
and reduces the realized maximum speed. Put simply, small
to intermediately sized animals accelerate quickly and have
enough time to reach their theoretical maximum speed, whereas
large animals are limited in acceleration time and run out of
readily mobilizable energy before being able to reach their
theoretically possible maximum. Therefore, they have a lower
realized maximum speed than predicted by a power-law scaling
relationship.
Test of model predictions by empirical database. To test the
model predictions (Fig.1d), we compiled literature data on maxi-
mum speeds of running, flying and swimming animals includ-
ing not only mammals, fish and bird species but also reptiles,
molluscs and arthropods. Body masses of these species range
Acceleration to theoretical maximum speed
ab
cd
Speed
Speed
Time
Time
v
max1
v
max2
v
max3
v
max4
Acceleration curve
Theoretical maximum speed
Allometry of critical time for acceleration
log
10
(speed)
log
10
(body mass)
log
10
(body mass)
Critical time
Realized maximum speed
t
1
t
2
t
3
t
4
Critical time
Realized maximum speed
Allometry of realized maximum speed
log
10
(time)
t
1
t
3
t
4
t
2
Figure 1 | Concept of time-dependent and mass-dependent realized maximum speed of animals. a, Acceleration of animals follows a saturation curve
(solid lines) approaching the theoretical maximum speed (dotted lines) depending on body mass (colour code). b, The time available for acceleration
increases with body mass following a power law. c,d,This critical time determines the realized maximum speed (c), yielding a hump-shaped increase of
maximum speed with body mass (d).
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from 3 × 10−8 kg to 108,400 kg. Statistical comparison amongst
multiple models (see Methods) shows that the time-dependent
maximum-speed model is the most adequate (see Supplementary
Table3). Our model (Fig.2, parameter values in Supplementary
Table4) shows that the initial power-law increase of speed with
body mass is similar for running and flying animals (b = 0.26
and 0.24, respectively). However, flying animals are nearly six
times as fast as running ones (a = 143 and 26, respectively). For
swimming animals, the power-law increase in speed is steeper
(b = 0.36, Fig.2a). This is because water is 800 times as dense and
60 times as viscous as air30 (in which both flying and running
animals move). Small aquatic animals are slower than running
animals of the same body mass, whereas larger species approach
a similar speed to that of their running equivalents. This implies
that in water, body mass brings a greater benefit in gaining speed.
The second exponent is lower for flying animals (i = − 0.72) than
for running (i = − 0.6) and swimming ones (i = − 0.56). Future
research will need to disentangle the relative importance of
anaerobic and musculoskeletal constraints on movement speed
by measuring muscle force, muscle mass, body mass and maxi-
mum acceleration for the same species to narrow down this large
range of possible exponents. Furthermore, this may allow us to
address the systematic differences in the exponent i between the
locomotion modes as well as potential morphological side effects
(for example quadrupedal versus bipedal running, or soaring ver-
sus flapping flight).
Although the model provides strikingly strong fits with
observations (R2 = 0.893), some unexplained variation remains.
This might partially be explained by the fact that our data prob-
ably include not only maximum anaerobic speeds but also some
slightly slower maximum aerobic speeds. Moreover, we assessed
the robustness of our model by exploring this residual variation
with respect to taxonomy (arthropods, birds, fish, mammals, mol-
luscs, reptiles), primary diet (carnivore, herbivore, omnivore),
thermoregulation (ectotherm, endotherm) and locomotion mode
(flying, running, swimming). As taxonomy and thermoregulation
are highly correlated, we made a first model without taxonomy and
a second model without thermoregulation and compared them by
their Bayesian information criterion (BIC) values (see Methods
for details). According to this, the model including thermoregu-
lation instead of taxonomy is the most adequate (∆ BIC = 27.37).
In this model, the differences between the diet types were not
significant. In contrast, combinations of locomotion mode with
thermoregulation exhibited significant differences (Fig.3). In fly-
ing and running animals, endotherms generally tend to be faster
than ectotherms (Fig. 3a,b). Metabolic constraints may enable
endotherms to have higher activity levels than ectotherms at the
low to intermediate temperatures most commonly encountered
1
Speed (km h
–1
)
Flying
Running
Swimming
Birds
Arthropods
Mammals
10
–10
10
–6
10
–2
10 10
4
10
7
10
–10
10
–6
10
–2
10 10
4
10
7
1
0.01
0.1
10
100
1,000
ab
cd
0.01
0.1
10
100
1,000
Body mass (kg)
Birds
Arthropods
Mammals
Reptiles
Birds
Arthropods
Mammals
Reptiles
Fish
Molluscs
Figure 2 | Empirical data and time-dependent model fit for the allometric scaling of maximum speed. a, Comparison of scaling for the different
locomotion modes (flying, running, swimming). b–d,Taxonomic differences are illustrated separately for flying (b;n = 55), running (c;n = 458) and
swimming (d;n = 109) animals. Overall model fit: R2 = 0.893. The residual variation does not exhibit a signature of taxonomy (only a weak effect of
thermoregulation; see Methods).
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in nature31. This pattern is reversed in aquatic systems, in which
endotherms (mammals and penguins) are significantly slower than
ectotherms (mainly fish, Fig.3c). We assume that this is due to the
transition undergone by aquatic endotherms from a terrestrial to
an aquatic lifestyle. Semi-aquatic endotherms are adapted to move-
ment in two different media, which reduces swimming efficiency
in comparison to wholly marine mammals: they have 2.4 × 105
times higher costs of transport32. But also, in marine mammals,
costs of transport are considerably higher than in fish of similar
size because they have higher energy expenditures for maintaining
their body temperature32. Thus, the effect of thermoregulation on
the allometric scaling of maximum speed depends on the locomo-
tion mode and the medium. Future research combining maximum
speed and ambient temperature data could provide a more detailed
analysis of temperature effects on maximum speed. Overall, the
significant effect of thermoregulation explained only ~4% of the
residual variation, suggesting that the vast majority of the varia-
tion in speed across locomotion modes, ecosystem types and taxo-
nomic groups is well explained by our maximum-speed model.
Discussion
Our findings help to solve one of the most challenging questions
in movement ecology over recent decades: why are the largest
animals not the fastest? Some studies have suggested a threshold
beyond which animals run more slowly than predicted by a power-
law relationship owing to biomechanical constraints13, thus imply-
ing that speed scaling depends on body-mass range11,12. Others
have invoked morphology, locomotion energetics and biomechan-
ics10–13,15,17,18 to suggest that the maximum speed of running animals
is constrained by the ability of muscles and bones to withstand the
stress of the locomotor force hitting the ground17,18,33. Size-related
increases in locomotor stress may thus be mitigated by taxon-
specific adaptations of bones, muscles and postures until eventu-
ally reaching limits at which larger body sizes come at the cost of
reduced speed17. As these biomechanical concepts were lacking
mechanistic predictions, the hump-shaped relationship between
maximum speed and body mass has often been characterized with
polynomial functions including linear and quadratic terms. We
have thus also used polynomials as the best available alternative
to compare against our model predictions. Although they offer a
flexible way to describe nonlinear patterns, we find that polyno-
mials do not predict the overall scaling relationship as accurately
as our general time-dependent maximum-speed model, which
provides the single most general capture of patterns and processes
across taxa and a larger body-mass range. Our speed predictions
are thereby derived from only two main species traits: body mass
and locomotion mode, which explain almost 90% (R2 = 0.893) of
the variation in maximum speed. This general approach allows a
species-level prediction of speed which is crucial for understand-
ing movement patterns, species interactions and animal space use.
However, our model allows prediction of the speed not only of
extant but also of extinct species. For example, palaeontologists
have long debated the potential running speeds of large birds34
and dinosaurs35,36 that roamed past ecosystems. The benchmark
of speed predictions is set by detailed morphological models35,36.
Interestingly, our maximum-speed model yields similar predic-
tions by only accounting for body mass and locomotion mode
(almost 80% of the morphological speed predictions are within
the confidence intervals of our model predictions; Table1). For
instance, in contrast to a power-law model, the morphological and
the time-dependent model predict lower speeds for Tyrannosaurus
compared with the much smaller Velociraptor. This is consistent
with theories claiming that Tyrannosaurus was very likely to have
been a slow runner37. A simple power-law model only yields rea-
sonable results for lower body masses (such as flightless birds),
whereas predictions for large species such as giant quadrupedal
dinosaurs are unrealistically high. In contrast, our time-dependent
model makes adequate predictions for small as well as large spe-
cies including extinct dinosaurs (Fig.4, green triangles). Note that
the highly accurate prediction of the dinosaur speeds is achieved
without free parameters as the model parameters are only obtained
by fits to data of extant species (Fig.2, and grey points in Fig.4).
Our model also allows inferences to be drawn about evolu-
tionary and ecological processes by analysing the deviations of
empirically measured speeds from the model predictions. Higher
maximum speeds than predicted indicate evolutionary pressure
on optimizing speed capacities that could, for instance, arise from
coevolution of pursuit predators and their prey.
Because many physiological and ecological processes such as
metabolism, growth and feeding rates depend on ambient temper-
ature (ectotherms) or body temperature (endotherms)38,39, it is not
surprising that some variables of movement speed and acceleration
also increase with temperature40. In our model, such a tempera-
ture dependence could be included as a Boltzmann factor in the
constant a (equation (5)). Sufficient ambient temperature mea-
surements at the point in time and space of the animals’ maximum
–0.6
–0.2
0.2
0.6
a
Maximum speed residuals
Ectotherm
Ectotherm Ectotherm
Endotherm
Endotherm
Endotherm
bc
Figure 3 | Effect of thermoregulation on the maximum speed of animals. These are the residuals of the relationship in Fig.2. a,b,In flying (a) and
running (b) animals, endotherms are generally faster than ectotherms. c,In swimming animals, this effect is reversed, with ectotherms being generally
faster than endotherms. Box plots show medians (horizontal line), an approximation of 95% confidence intervals suitable for comparing two medians
(notches), 25th and 75th percentiles (boxes), the most extreme values (whiskers), and outliers (dots).
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speed are currently lacking, but our model offers a framework to
include temperature effects formally in future work.
In ecological research, our maximum-speed model provides
a mechanistic understanding of the upper limit to animal move-
ment patterns during migration, dispersal or bridging habitat
patches. The travelling speed characterizing these movements
is the fraction of maximum speed that can be maintained over
longer periods of time. It would be interesting to analyse how
travel speed scales with body mass on the large body-mass
scale and whether it also follows a hump-shaped pattern. If so,
animals would use an approximately fixed percentage of their
maximum speed during travel. If, however, travel speed follows
a power-law relationship with body mass, large and small ani-
mals would use a higher proportion of their maximum speed
during travel than intermediately sized animals. This would
also affect different measurements of animal space use as well as
migration and dispersal distances. Although home ranges1,41 and
day ranges42 of animals have been shown to follow power-law
relationships with body mass, migration distances of flying ani-
mals, for example, follow a curvilinear relationship with body
mass43. Our new results call for mechanistic analyses of how the
hump-shaped scaling pattern of maximum speed could poten-
tially affect other movement parameters.
The integration of our model as a species-specific scale (“what
is physiologically possible”) with research on how this fraction
is modified by species traits and environmental parameters such
as landscape structure, resource availability and temperature
(“what is ecologically realized in nature”) could help to provide
a mechanistic understanding unifying physiological and ecologi-
cal constraints on animal movement. In addition to generalizing
our understanding across species traits and current landscape
characteristics, this integrated approach will aid the prediction
of how species-specific movement, and subsequently home
ranges nd meta-communities, may respond to ongoing landscape
fragmentation and environmental change. Thus, our approach
may act as a simple and powerful tool for predicting the natu-
ral boundaries of animal movement and help in gaining a more
unified understanding of the currently assessed movement data
across taxa and ecosystems6,7.
Methods
Data. We searched for published literature providing data on the maximum speeds
of running, ying and swimming animals by using the search terms “maximum
speed”, “escape speed” and “sprint speed”. From this list, we excluded publications
on (1) vertical speeds (mainly published for birds) to avoid side-eects of
gravitational acceleration that are not included in our model, or (2) the maxima
of normal speeds (including also dispersal and migration). is resulted in a data
set containing 622 data points for 474 species (see Supplementary Table1 for an
overview). Our data include laboratory and eld studies as well as meta-studies
(which are mainly eld studies but may also include a minor amount of laboratory
studies). For some data points, the study type could not be ascertained, and they
were marked as “unclear”. For an overview of the study type of our data, see
Supplementary Table2.
Model fitting. We fitted several models to these data: (1) the time-dependent
maximum-speed model (equation (5)), (2) three polynomial models (simple
10–9 10–7 10–5 10–3 10–1 101103105107
0.01
0.1
1
10
100
1,000
Speed (km h–1)
Body mass (kg)
Extant species
Dinosaurs (morphological calculations)
Model prediction (fitted to data from extant species)
Figure 4 | Predicting the maximum speed of extinct species with the time-
dependent model. The model prediction (grey line) is fitted to data of extant
species (grey circles) and extended to higher body masses. Speed data
for dinosaurs (green triangles) come from detailed morphological model
calculations (values in Table1) and were not used to obtain model parameters.
Table 1 | Maximum-speed predictions for extant and extinct flightless birds, and bipedal and quadrupedal dinosaurs
Taxa Body mass (kg) Speed (kmh–1)
Power law (95% CI) Morphological
models Time-dependent model
(95% CI)
Flightless birds
Dromaius (extant) 27. 2 40.92 (38.58–43.40) 47.88 57.62 (47.65–60.91)
Struthio (extant) 65.3 49.33 (46.27–52.59) 55.44 62.75 (46.71–66.03)
Patagornis (extinct) 45 45.56 (42.83–48.46) 50.40 61.34 (47.39–64.68)
Bipedal dinosaurs
Velociraptor 20 38.32 (36.19–40.58) 38.88 54.56 (46.89–57.82)
Allosaurus 1,400 94.87 (87.09–103.34) 33.84 40.78 (28.93–44.83)
Tyrannosaurus 6,000 129.41 (117.47–142.57) 28.8 27.05 (17.84–31.52)
Quadrupedal dinosaurs
Triceratops 8,478 139.32 (126.11–153.91) 26.4 24.36 (15.70–28.83)
Apatosaurus 27,869 179.59 (161.01–200.31) 12.3 16.75 (9.77–21.09)
Brachiosaurus 78,258 223.85 (199.00–251.80) 17. 6 11.99 (6.39–16.04)
Model predictions of a simple power law, morphological models and our time-dependent maximum-speed model are compared (references in Supplementary Table5). Confidence intervals (95% CI) are
given for the power law and time-dependent model.
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polynomial model without cofactor; polynomial model with taxon as cofactor
but without interaction term; and polynomial model with taxon as cofactor
with interaction term) with linear and quadratic terms, and (3) three power-law
models (simple power law without cofactor; power law with taxon as cofactor
but without interaction term; and power law with taxon as cofactor with
interaction term). For swimming animals, we excluded reptiles and arthropods
from the statistical analyses as they contained only one data point each
(see Supplementary Table1). The polynomial and power-law models were fitted
by the lm function, and the time-dependent model by the nls function in
R (version 3.2.3)44. The quality of the fits was compared according to the
Bayesian information criterion (BIC) that combines the maximized value of
the likelihood function with a penalty term for the number of parameters in
the model. The model with the lowest BIC is preferred, and the results of this
showed that the time-dependent maximum-speed model developed in the main
text provided the best fit in all cases (see Supplementary Table3). For flying
animals, the simple polynomial model performed second best, whereas for
running animals the polynomial model with taxon as cofactor with interaction
term and for swimming animals the power-law model with taxon as cofactor
with interaction term were second best (see Supplementary Table3). Overall,
the lower BIC values indicate that the time-dependent maximum-speed
model provides a fit to the data that is substantially superior to power-law
relationships, models with taxonomy as cofactor or (non-mechanistic but also
hump-shaped) polynomials. The fitted parameter values of the time-dependent
maximum-speed model for flying, running and swimming animals are given in
Supplementary Table4.
Residual variation analysis. We analysed the residuals of the time-dependent
maximum-speed model (Fig.2 of the main text) with respect to taxonomy
(arthropods, birds, fish, mammals, molluscs, reptiles), primary diet type
(carnivore, herbivore, omnivore), locomotion mode (flying, running, swimming)
and thermoregulation (ectotherm, endotherm) using linear models. As taxonomy
and thermoregulation are highly correlated, we made a first model without
taxonomy and a second model without thermoregulation:
Model 1: residuals ~ (thermoregulation + diet type) × locomotion mode
Model 2: residuals ~ (taxonomy + diet type) × locomotion mode
We compared the two models by means of BIC and carried out a further
mixed-effects model analysis on the superior model. This model included the
study type as a random factor influencing the intercept, which ensures that
differences among study types do not drive our statistical results. We acknowledge
that the direct inclusion of multiple covariates in the model-fitting process
would be preferable to residual analysis to avoid biased parameter estimates45.
However, this was impeded by the complexity of fitting the nonlinear model
with four free parameters (equation (5)), and our main goal was less to estimate
the exact parameters than to document the main variables affecting the
unexplained variation.
Data availability. The data supporting the findings of this study are available
within the Article and its SupplementaryInformation files.
Received: 8 November 2016; Accepted: 16 June 2017;
Published: xx xx xxxx
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Acknowledgements
M.R.H., W.J., B.C.R. and U.B. acknowledge the support of the German Centre for
integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig funded by the German
Research Foundation (FZT 118).
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Supplementary information is available for this paper at doi:10.1038/s41559-017-0241-4.
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Author contributions
M.R.H. and U.B. developed the model. M.R.H. gathered the data. M.R.H. and B.C.R. carried
out statistical analyses. W.J. was involved in study concept and data analyses. M.R.H. and
U.B. wrote the paper. All authors discussed the results and commented on the manuscript.
Competing interests
The authors declare no competing financial interests.
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