Energetic and biomechanical constraints on animal migration distance.
ABSTRACT Animal migration is one of the great wonders of nature, but the factors that determine how far migrants travel remain poorly understood. We present a new quantitative model of animal migration and use it to describe the maximum migration distance of walking, swimming and flying migrants. The model combines biomechanics and metabolic scaling to show how maximum migration distance is constrained by body size for each mode of travel. The model also indicates that the number of body lengths travelled by walking and swimming migrants should be approximately invariant of body size. Data from over 200 species of migratory birds, mammals, fish, and invertebrates support the central conclusion of the model - that body size drives variation in maximum migration distance among species through its effects on metabolism and the cost of locomotion. The model provides a new tool to enhance general understanding of the ecology and evolution of migration.
- SourceAvailable from: Graeme C Hays[Show abstract] [Hide abstract]
ABSTRACT: The last 20 years have been exciting times for scientists working with charismatic marine mega-fauna. Here recent advances are reviewed. There have been advances in both data gathering and data-analysis techniques that have allowed new insights into the physiological and behavioural ecology of free-ranging mega-faunal species; some marine mega-faunal species have now become model organisms for cutting edge approaches to identify the underlying mathematical properties of animal search patterns and hence the underlying behavioural processes (e.g. Levy flight versus Brownian motion); the implications of climate change have started to become more apparent with extended time-series of animal movements, abundance and performance; conservation issues have become integrated into marine planning and have resulted in the advent of extended networks of marine protected areas (MPAs) as well as large MPAs that span many 100,000 km2; and collaborative cross-disciplinary teams have started to reveal the importance of ocean currents in animal dispersal, the ontogeny of migration and population genetic structure. Looking to the future, increased data availability (e.g. through data sharing) will likely allow more holistic across-taxa analyses to become routine.Journal of Experimental Marine Biology and Ecology 01/2014; 450:1–5. · 2.26 Impact Factor
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ABSTRACT: Many studies have shown plant species' dispersal distances to be strongly related to life-history traits, but how well different traits can predict dispersal distances is not yet known. We used cross-validation techniques and a global data set (576 plant species) to measure the predictive power of simple plant traits to estimate species' maximum dispersal distances. Including dispersal syndrome (wind, animal, ant, ballistic, and no special syndrome), growth form (tree, shrub, herb), seed mass, seed release height, and terminal velocity in different combinations as explanatory variables we constructed models to explain variation in measured maximum dispersal distances and evaluated their power to predict maximum dispersal distances. Predictions are more accurate, but also limited to a particular set of species, if data on more specific traits, such as terminal velocity, are available. The best model (R2 = 0.60) included dispersal syndrome, growth form, and terminal velocity as fixed effects. Reasonable predictions of maximum dispersal distance (R2 = 0.53) are also possible when using only the simplest and most commonly measured traits; dispersal syndrome and growth form together with species taxonomy data. We provide a function (dispeRsal) to be run in the software package R. This enables researchers to estimate maximum dispersal distances with confidence intervals for plant species using measured traits as predictors. Easily obtainable trait data, such as dispersal syndrome (inferred from seed morphology) and growth form, enable predictions to be made for a large number of species.Ecology 02/2014; 95(2):505-13. · 5.18 Impact Factor
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ABSTRACT: Aerobic activity levels increase with body temperature across vertebrates. Differences in these levels, from highly active to sedentary, are reflected in their ecology and behavior. Yet, the changes in the cardiovascular system that allow for greater oxygen supply at higher temperatures, and thus greater aerobic activity, remain unclear. Here we show that the total volume of red blood cells in the body increases exponentially with temperature across vertebrates, after controlling for effects of body size and taxonomy. These changes are accompanied by increases in relative heart mass, an indicator of aerobic activity. The results point to one way vertebrates may increase oxygen supply to meet the demands of greater activity at higher temperatures.PeerJ. 01/2014; 2:e346.
Energetic and biomechanical constraints on animal migration
Andrew M. Hein,1*Chen Hou1,2,3
and James F. Gillooly1
1Department of Biology, University
of Florida, Gainesville, FL 32611, USA
2Department of Systems and
Computational Biology, Albert
Einstein College of Medicine, Bronx,
NY 10461, USA
3Department of Biological Science,
Missouri University of Science and
Technology, Rolla, Missouri 65409,
Animal migration is one of the great wonders of nature, but the factors that determine how far migrants travel
remain poorly understood. We present a new quantitative model of animal migration and use it to describe the
maximum migration distance of walking, swimming and flying migrants. The model combines biomechanics
and metabolic scaling to show how maximum migration distance is constrained by body size for each mode of
travel. The model also indicates that the number of body lengths travelled by walking and swimming migrants
should be approximately invariant of body size. Data from over 200 species of migratory birds, mammals, fish,
and invertebrates support the central conclusion of the model – that body size drives variation in maximum
migration distance among species through its effects on metabolism and the cost of locomotion. The model
provides a new tool to enhance general understanding of the ecology and evolution of migration.
Allometry, biomechanics, dispersal, ecomechanics, ecophysiology, energetics, migration, movement ecology,
scaling, spatial ecology.
Ecology Letters (2011)
Each year, diverse species from around the planet set out on
migrations ranging from a few to thousands of kilometres in length
(Dingle 1996; Egevang et al. 2010; Hedenstro ¨m 2010). Biologists have
long hypothesised that this variation in migration distance among
species might be governed by differences in basic species character-
istics such as morphology and body size (Dixon 1892). Although
much progress has been made in understanding how these charac-
teristics are related to the mechanics of locomotion and to the
migratory capabilities of individual species (e.g. Pennycuick & Battley
2003; Alexander 2005), success in understanding variation in
migration distance among species has been limited. This is because
current models often require detailed information on the morphology
and behaviour of migrants (e.g. Alerstam & Hedenstro ¨m 1998;
Pennycuick & Battley 2003). This requirement has precluded a
quantitative analysis to determine the extent to which shared
functional characteristics such as body size could be responsible for
observed variation in migration distances among species. As a result,
the need for general theory and cross-species analyses of migration
has been strongly emphasised in recent years (Bauer et al. 2009;
Milner-Gulland et al. 2011).
Herein, we present a model to describe constraints on animal
migration distance. Our model expands on past approaches (Alexan-
der 1998; Hedenstro ¨m 2003; Pennycuick 2008) by incorporating (1)
the body mass-dependence of the cost of locomotion, (2) dynamic
changes in the body masses of migrants as they utilise stored fuel and
(3) scaling of morphological characteristics and maintenance meta-
bolism among migrants of different body masses. In contrast to past
approaches, the model assumes that the number of refuelling stops
made by migrants is unknown and may vary substantially among
species. This facilitates prediction of statistical patterns of migration
distance among species, even when the details of migratory behaviour
of individual species are unknown.
We treat migration as a process in which a migrant travels a distance
of Yt(km) by breaking the journey into a series of N legs of length Yi
(i 2 1, 2, ..., N, N ‡ 1, Fig. 1A). Describing variation in migration
distance among species, thus, requires describing the processes that
determine Yi, while accounting for among-species variation in N. To
accomplish this, we begin by making four simplifying assumptions
(see Appendix S1 in Supporting Information for detailed derivation
and alternative assumptions). We assume (1) that the total rate of
energy use by a migrating animal, Ptot(W), is the sum of the rate of
energy use for general maintenance, Pmtn, and that required for
locomotion, Ploc(i.e. Ptot= Pmtn+ Ploc= ) dG⁄dt, where G = Joules
of stored fuel energy), (2) that migrants using a particular mode of
locomotion are geometrically similar, such that linear morphological
characteristics (e.g. lengths of appendages) are proportional to M1 ⁄ 3
and surface areas are proportional to M2 ⁄ 3(where M is body mass
(kg), Peters 1983), (3) that migrant metabolism provides the power
required for locomotion and (4) that the number of refuelling stops
made by individuals of each species is independent of body mass.
Distance travelled on a single migratory leg
During any given leg of a migration, the rate of change in migration
distance per unit change in body mass can be expressed as
dYi=dM ¼ ðdYi=dtÞðdtc=dGÞ ¼ ?vc=ðPmtnþ PlocÞ;
where v is travel speed (m s)1) and c is the energy density of stored
fuel (Joules kg)1). The distance travelled on a particular leg can be
Ecology Letters, (2011)doi: 10.1111/j.1461-0248.2011.01714.x
? 2011 Blackwell Publishing Ltd/CNRS
obtained by integrating this expression from initial mass at the
beginning of the leg, M0 (kg), to final mass after all fuel energy
has been used, M0(1 ) f ), where f is the ratio of initial fuel mass to
PmtnðMÞ þ PlocðM;bÞdM
Here, v, Pmtnand Plochave been rewritten to show their dependence on
body mass and on a small set of morphological traits, b (lengths and
surface areas, e.g. wingspan, body cross-sectional area), which
determine the energetic cost of locomotion. This formulation allows
for changes in speed and rate of energy use, as the migrant loses
stored fuel mass.
Equation (1) can be used to predict how Yivaries among species by
specifying appropriate functions for v(M, b), Pmtn(M ) and Ploc(M, b).
We assume that Pmtnscales with body mass as Pmtn= p0M3 ⁄ 4,both
within and among individuals, where p0is a normalisation constant
that varies by taxon (Kleiber 1932; Hemmingsen 1960). Biomechanics
theory provides a means of expressing Plocand v as functions of M
and b for migrants using a particular mode of locomotion (see below).
Generalising to multi-leg migrations
Total distance travelled over the course of migration is given by the
species. To account for variation in N among species, we treat N as a
random quantity with mean, N. We treat Yias fixed for a given
species because we are interested in maximum migration distance.
Iterated expectation shows that the expected distance travelled over N
migratory legs is
i¼1Yi, where N is the number of migratory legs travelled by a
given species (Fig. 1A). N is unknown for the majority of migratory
¼ N Yi
where the operator, E, denotes the expected value (Rice 1995).
Eqn (2) shows that YTis proportional to Yi, which is given by eqn (1).
Parameterizing the model for walking, swimming and flying
The model developed above is general and applies to migrants using
any mode of locomotion. Herein, we parameterize the model for the
three dominant modes of migratory locomotion (walking, swimming
and flight) by using standard models of locomotion to describe the Ploc
and v terms in eqn (1) (biomechanical models described in detail in
Appendix S1). For walking migrants, Ploccan be described by
where Lcis stride length (m), v is walking speed (m s)1), c is a cost
coefficient (J N)1) and g is the acceleration due to gravity (m s)2,
Kram & Taylor 1990) The only morphological variable in eqn (2) is Lc,
which is proportional to leg length (Alexander & Jayes 1983). We
assume that walking migrants travel at speeds, v / M0:1
1998) and that they maintain these speeds over the course of
The power required for swimming can be described by the resistive
where d is a dimensionless cost coefficient, Abis body cross-sectional
area (m2), Lbis body length (m) and v is swimming speed (m s)1,
Alexander 2003). The set of relevant morphological variables, b, is Ab
and Lb. We assume that migrants swim at speeds that minimise the
Power required for flight near minimum power speed can be
described by the equation
where j is a dimensionless profile power coefficient, u and / are cost
coefficients (section 1.4 Appendix S1), Abis body cross-sectional area
(m2), Lwis wingspan (m) and j is proportional to Aw=L2
wing area (Pennycuick 2008). The set of relevant morphological
variables, b, is therefore Ab, Lwand Aw. We assume flying migrants
travel at speeds that minimise Pfly⁄vs(Pennycuick 2008).
Substituting eqns (3–5), corresponding migration speeds and the
mass-dependence of maintenance metabolism into eqn (1) allows Yito
be expressed as a function of initial mass M0, p0and b for each mode
of locomotion. In each of the biomechanical models described above,
the power required for locomotion depends, in part, on a set of
morphological lengths and areas, b, that do not change as the migrant
uses stored fuel to power migration. The dependence of Yion b can
be eliminated by expressing morphological variables in terms of
M0 based on the assumption of geometric similarity (i.e. lengths
Substituting functions for Yi(section 1 Appendix S1) into eqn (2)
yields expressions for the expected maximum migration distances of
Pfly¼ ð1 þ jÞ uM2L?2
w, where Awis
0 , surface areas / M2=3
Body Mass (M0)
Pmtn and Ploc
Mass loss as fuel
Figure 1 (A) Total migration distance is the sum of the distances travelled on each
of N migratory legs. (B) Migration distance on a single migratory leg. Body mass (a),
morphology (b) and mode of locomotion (c) govern the rate at which a migrant
uses stored fuel energy (d). This rate changes as migrant loses fuel mass (e), and
determines the maximum distance covered during a single leg (f, eqn (1)). The
relationship between a and b is governed by the mass-dependence of morphology.
Total rate of energy use (d) is determined by the mass-dependence of maintenance
metabolism and by the biomechanics of locomotion (eqns 3–5).
2 A. M. Hein, C. Hou and J. F. GilloolyLetter
? 2011 Blackwell Publishing Ltd/CNRS
YT¼ y0lnp0þ k1M0:42
migrants. Herein, y0is a proportionality constant that varies by mode
of locomotion, and k1and k2are empirical constants. Differences in
the functional forms of eqns (6–8) are caused by differences in the
way Plocdepends on mass in walking, swimming and flying migrants.
In the case of eqn (8), the predicted relationship does not follow a
simple power function in M0. This is because the cost of flight
increases more rapidly with increasing body mass than does the cost
of walking or swimming. The variable, p0, does not appear in the final
form of the equation for walking migrants because here we only
consider the distance travelled by walking mammals, for which p0is
roughly constant (White et al. 2009).
The exponents of the mass terms in eqns (6–8) describe how
maximum migration distance changes as a function of M0and reflect
the mass-dependence of maintenance and locomotory metabolism.
The constant, y0, describes effects of mass-independent factors, such
as the number of migratory legs, that affect the absolute distances
travelled by migrants but do not affect the scaling of migration
distance with body mass. The metabolic normalisation constant, p0,
and the morphological constants k1and k2can be estimated from
empirical measurements (see Materials and Methods).
The framework described here uses body mass (Fig. 1B box a),
morphology (Fig. 1B box b) and mode of locomotion (Fig. 1B box c)
to determine migratory speed, and the metabolic costs of locomotory
and maintenance metabolism (Fig. 1B box d). Equation (1) ensures
that changes in speed and metabolism as the migrant uses stored fuel
(Fig. 1B box e) are explicitly incorporated into the prediction of Yi
(Fig. 1B box f).
Equations (6–8) make several quantitative predictions that can be
tested against data. First, each equation predicts that, after normalising
for p0, a single curve can be used to describe expected maximum
migration distance (in km) as a function of M0for species using each
mode of locomotion. Second, each equation predicts how the number
of body lengths travelled – a measure of relative distance (Alerstam
et al. 2003) – varies with body mass. Migration distance and body
length scale similarly with mass in walking and swimming animals (i.e.
YTroughly proportional to M1=3
number of body lengths travelled during migration, Ybl, is described by
Ybl= YT⁄(body length) / M1=3
for differences in p0, the number of body lengths travelled by walking
and swimming animals should be approximately invariant with respect
to M0. In flying animals, however, dividing eqn (8) by M1=3
that Yblshould decrease with increasing mass for all but the smallest
0 , body length / M1=3
0 ) such that the
0. Thus, after normalising
MATERIALS AND METHODS
To evaluate the model, published measurements of maximum
migration distances of terrestrial mammals, fish, marine mammals
and flying insects and birds were collected. Data from studies that met
five criteria were included in the analysis: (1) reported movements
could be considered to-and-fro migration or one-way migration
(Dingle & Drake 2007), (2) individuals were directly tracked by mark-
recapture, telemetry or other means, groups of individuals were
tracked by repeated observation over the course of migration, or a
reliable estimate of distance travelled could otherwise be established,
(3) maximum travel distances, maps, tracks or other information that
allowed direct calculation of minimum estimates of the distances
travelled by individual animals were reported, (4) there did not exist
strong but indirect evidence from other studies (e.g. sightings of
unmarked individuals, stable isotope data) suggesting that the
maximum reported migration distance was substantially shorter than
true maximum migration distance and (5) in the case of flying species,
studies reported migration distances of species that rely, at least
partially, on flapping flight. The fifth criterion was imposed because
the biomechanical model of flight used to derive our predictions
applies most directly to flapping flight. Migration distance and body
mass data were included from a large dataset (Elphick 1995) for which
all of the selection criteria could not be verified for all species.
Including these data did not qualitatively affect our conclusions
We estimated the constants k1and k2in eqn (8) using empirical
studies of the morphology of flying insects and birds; however, the
general form of eqn (8) and the resulting predictions are not strongly
affected by variation in the empirical values used to estimate k1and k2
(section 2.2 Appendix S1). Empirical estimates of p0were used in eqns
(7–8) (Appendix S1). Body mass data were used to estimate body
lengths based on allometric equations (swimming mammals: Econo-
mos 1983; others: Peters 1983). Body lengths were used to convert
migration distance (km) into units of body lengths.
To evaluate our first prediction, we fitted eqns (6–8) to migration
distance data from walking (n = 33), swimming (n = 32) and flying
migrants (n = 141). Eqns (6) and (7) were fitted to log10-transformed
distance and body mass data using ordinary least squares. Eqn (8) was
fitted to log10-transformed distance and body mass data using non-
linear least squares (Gauss–Newton algorithm). Equations (6–8) have
the general form: YT= y0h(M0d,p0), where h is a known function, y0is
a constant, and d is a scaling exponent. For each equation, two models
were fitted: a model in which y0was fitted as a free parameter, but d
was set to the predicted value (i.e. d = 0.34, 0.3, 0.42; for walking,
swimming, and flying migrants respectively), and a model in which
both y0and d were fitted. Model r2values reported below are based on
the former method. The latter method was used to generate 95%
profile confidence intervals for the d parameter. Prior to fitting, body
mass values of swimming and flying animals were normalised to
second prediction – that the number of body lengths travelled was
invariant of mass in walking and swimming migrants, but decreased
with mass in flying migrants – we fitted log10-transformed migra-
tion distance (in body lengths) as a function of log10-transformed
body mass (kg) using a quadratic regression of the form,
log10ðYblÞ ¼ c0þ c1log10ðYblÞ þ c2log10ðYblÞ2, where ciare regres-
sion coefficients (Venebles & Ripley 1999). Species were separated
based on mode of locomotion and by taxonomic groups differing in
p0(i.e. walking mammals, fish, marine mammals, flying insects and
passerine and non-passerine birds were fitted separately). Statistical
analyses were implemented using the nlme package (Pinheiro et al.
2009) in R (2010).
and Mnorm¼ M0:42
respectively. To test our
LetterConstraints on animal migration distance 3
? 2011 Blackwell Publishing Ltd/CNRS
Model predictions were evaluated using extensive data on maximum
migration distances of animals from around the world (n = 206
species, Data S1). Consistent with our first prediction, maximum
migration distance (km) varies systematically with body mass for
walking, swimming and flying migrants (Fig. 2: r2= 0.57, 0.65, 0.19;
for walking, swimming, and flying species respectively). The solid lines
show predicted migration distance based on eqns (6–8). There is a
tight correspondence between predicted relationships (solid lines) and
fitted models that treat both y0 and scaling exponents as free
parameters (dashed lines and 95% confidence bands). In the case of
walking and swimming animals, the data support model predictions of
linear relationships in log-log space, with observed scaling exponents
close to those predicted by eqns (6) and (7) (walking: pre-
dicted = 0.34, observed = 0.36 95%CI [0.25,0.48]; swimming: pre-
dicted = 0.3, observed = 0.34 [0.28,0.41]). In the case of flying
animals, data support the prediction that the relationship is non-linear
in log-log space reflecting the rapidly rising cost of flight with
increasing mass (Fig. 2c). Again, the observed mass exponent is close
to that predicted by eqn (8) (predicted = 0.42, observed = 0.43
Consistent with our second prediction, the number of body lengths
travelled by swimming and walking animals is independent of body
mass (Fig. 3). On average, walking mammals travel 1.5 · 105body
lengths (Fig. 3a). The slope and curvature terms in the quadratic
regression model does not differ from zero in walking mammals
(n = 33, P > 0.22) indicating that the number of body lengths
travelled is uncorrelated with body mass in this group. Swimming
animals travel an average of 1.7 · 106body lengths in a one-way
migratory journey. The mean distance travelled by fish (triangles in
Fig. 3b) exceeds that travelled by swimming mammals (squares in
Fig. 3b) by a factor of 4 (fish: 2.1 · 106body lengths; marine
mammals: 5.3 · 105body lengths, see Discussion), but the number of
body lengths travelled is independent of mass in each of these groups
(slope and curvature does not differ from zero, fish: n = 20, P > 0.38;
swimming mammals: n = 12, P > 0.43). In flying migrants, the
number of body lengths migrated declines clearly with increasing body
mass (Fig. 3c). In non-passerine birds (n = 80), coefficients of linear
and quadratic terms were both negative, and significantly different
from zero (c1= )0.59, c2= )0.19, P < 2.2 · 10)5). In passerine
birds (n = 45) and flying insects (n = 16), the c1term was negative
anddistinguishable from zero
P = 5.4 · 10)5; insects: c1= )0.16, P = 0.034). Results for flying
Figure 2 Maximum migration distance as a function of normalised body mass for (a) walking mammals, (b) swimming fish and marine mammals and (c) flying birds and
insects. Solid lines are predicted curves based on fits of eqns (6–8) to data with y0fitted as a free parameter. Dashed lines and confidence bands represent best fit curves and
95% confidence intervals from linear (a, b) or non-linear regression (c) with y0and the mass scaling exponent fitted as free parameters. In panel (a), body mass is M0(kg).
In panels (b) and (c), body mass is normalised according to the equations Mnorm= M00.3p)0.64and Mnorm= M00.42p0)1, respectively, to correct for differences in p0among
groups. Data on walking animals are from mammals only and are therefore not corrected for p0.
Body mass (kg)
Body lengths traveled
Figure 3 Number of body lengths travelled during migration by (a) walking mammals, (b) swimming fish (triangles) and mammals (squares) and (c) flying insects (triangles),
passerine birds (squares) and non-passerine birds (diamonds). Lines denote mean number of body lengths travelled by species using each mode of locomotion.
4 A. M. Hein, C. Hou and J. F. Gillooly Letter
? 2011 Blackwell Publishing Ltd/CNRS
migrants, confirm our prediction that larger flying migrants generally
travel fewer body lengths over the course of migration. The number
of body lengths travelled decreases with increasing mass such that the
smallest insects and birds travel around 1.4 · 108body lengths
whereas the largest birds travel around 5.2 · 106body lengths. In
other words, the number of body lengths covered by moths,
dragonflies and hummingbirds is roughly 25-times that travelled by
the largest ducks and geese.
A sensitivity analysis indicates that the agreement between model
predictions and data are robust to deviations from geometric similarity
and changes in the values of morphological and biomechanical
parameters used to derive eqns (6–8) (see section 2.2 Appendix S1
and Table S2). In particular, the value of the exponent in metabolic
scaling relationships has been a topic of much debate, with different
authors reporting different exponents depending on the particular
dataset and taxon studied and the method of analysis (e.g. White et al.
the shape of our predicted relationships and the agreement between
metabolic scaling exponent assumed (Appendix S1). Including data
from Elphick (1995) did not significantly change the estimate of the
mass exponent (0.36 95% CI [0.26,0.43] without data from Elphick
(1995), 0.43 [0.36,0.48] with data from Elphick (1995)). Including data
from Elphick (1995) decreased the model r2from 0.37 to 0.19.
When observed migration distances are plotted against predictions of
eqns (6–8), points from all three groups cluster around a 1 : 1 line
(Fig. 4). The data shown in Fig. 4 suggest that variation in maximum
migration distances among species as distinct as Blue Whales
(Balaenoptera musculus), Wildebeest (Connochaetes taurinus) and Bar-tailed
Godwits (Limosa lapponica) appears to be driven, in part, by the basic
differences in metabolism, morphology and biomechanics described by
our model. The variation explained by the model reflects the influence
of constraints on energetics and biomechanics imposed by body mass.
There is a large body of work describing how morphology (Peters
1983; Alexander 2003), biomechanics (Alexander 2003, 2005) and basic
energetic properties such as maintenance metabolism (West et al. 1997;
Banavar et al. 2010) are linked to body mass. Our model extends results
of these studies by specifying how these quantities influence maximum
migration distance of diverse species, thereby linking body mass to
migration distance. Our results show that constraints imposed by body
mass are detectable in migration distance data, despite variation in
migration distance among species with similar body masses (i.e.
variation about predicted relationships shown in Figs 2 and 4).
Migration distance data highlight the important role of basic
differences in energetics in driving differences in migration distance
among taxa. For example, the number of body lengths travelled during
migration is independent of body mass within both swimming
mammals and fish; however, fish travels an average of four times the
number of body lengths travelled by swimming mammals. Equation
(7) shows that the distances travelled by these groups depend on the
metabolic normalisation constant, p0, which describes mass-indepen-
dent differences in the maintenance metabolic rates of fish and marine
mammals. In these groups, p0 differs by a factor of roughly 9.1
(p0» 3.9 W kg)3 ⁄ 4in marine mammals, p0» 0.43 W kg)3 ⁄ 4in fish,
see Appendix S1), whereas body length exhibits a similar relationship
with mass in both groups (l » 0.44M1 ⁄ 3) suggesting that the number
of body lengths migrated by fish is greater by a factor of
(9.1)0.64= 4.1, which is very close to the observed factor of 4. Thus,
the difference in the mean number of body lengths travelled by these
groups may be driven by basic differences in the cost of maintenance
metabolism. Data also reveal patterns that do not appear to be caused
by the energetic and biomechanical factors considered here. For
example, swimming is significantly less costly than flight in terms of
the energy required to travel a given distance (Weber 2009), yet
virtually all flying organisms travel distances that are as great or greater
than those travelled by most swimming species (Fig 4). Whether this
pattern is driven by differences in migratory behaviour or other
ecological or evolutionary factors remains unknown and will likely be
a fruitful area of future research.
It is worth noting that other hypotheses may provide alternative
explanations for some of the qualitative patterns observed in
migration distance data. For example, the model predicts that
migration distance (km) of larger flying species does not depend
strongly on mass. An increase in mass from 10)6kg to 10)3kg,
increases expected migration distance by a factor of more than 8,
whereas an increase in mass from 10)2kg to 10 kg increases expected
migration distance by a factor of less than 2. This occurs because the
energetic cost of flight increases rapidly with increasing mass to the
degree that the increasing fuel mass that can be carried by larger
migrants provides a diminishing increase in migration distance. An
alternative explanation for this observation is that many subtropical
and temperate habitats in the northern and southern hemispheres are
separated by 5 · 103km1· 104km and that many flying migrants
may not be under selection to migrate greater distances. In general, the
relationship between the distances travelled by migrants and the global
distribution of suitable migratory habitats is poorly known but may
ultimately influence the distances travelled by many species.
While model predictions are supported by data, there is substantial
unexplained variation in Figs 2 and 4. Investigating why particular
species deviate from predictions may be an effective way to identify
Figure 4 Observed and predicted migration distances for the walking, swimming
and flying animals shown in Fig. 2. Data from walking mammals (green circles),
swimming fish (blue triangles) and marine mammals (blue squares), and flying
insects (red triangles), passerine birds (red squares) and non-passerine birds (red
diamonds) are shown. Black points and illustrations show the well-studied migrants
Connochaetes taurinus (Wildebeest), Balaenoptera musculus (Blue Whale) and Limosa
lapponica (Bar-tailed Godwit). Solid line indicates 1 : 1 line.
LetterConstraints on animal migration distance 5
? 2011 Blackwell Publishing Ltd/CNRS
ecological and evolutionary factors that drive differences in migration
distance but are not currently included in our model. Our model
ignores variation in fuel and morphology of species with similar masses
and does not consider the possibility that some migrants may seek to
minimise the time spent migrating. Two additional factors, in
particular, are likely to contribute to observed residual variation. First,
differences in the number migratory legs among otherwise similar
species will lead to variation in migration distance among species as
indicated by eqn (2). Second, species that interact strongly with abiotic
currents during migration are likely to deviate from model predictions.
The lack of information regarding the type and number of refuelling
stops made by migratory species, and the lack of information about the
manner in which many flying and swimming migrants interact with
abiotic currents represents an important gap in current knowledge. In
the case of some well-studied species such as the arctic tern (Sterna
paradisaea), it is clear that these variables are important in facilitating
extremely long-distance migrations. Individuals of this species stop at
multiple highly productive foraging sites to refuel during migration
(Egevang et al. 2010). This species is also known to track global wind
systems thereby taking advantage of favourable air currents. In the case
of species that migrate against abiotic currents, migration distances
might be expected to be shorter than our model predicts. Indeed, many
of the swimming migrants that fall below the predicted line in Fig. 2,
are anadromous fish such as shad (Alosa sapidissima), alewife (Alosa
pseudoharengus) and river lamprey (Lampetra fluviatilis) that swim against
water currents during upriver migrations. Increased understanding of
the interactions between migrants and abiotic currents and the number
of migratory stopovers will allow for extensions of the model that
could further improve our understanding of the reasons for inter-
specific differences in migration distance. In its current form, the
model presented here provides a general expectation on maximum
migration distance, which can be seen as a metric against which the
distances travelled by particular species can be compared.
The body sizes of migratory animals vary by over 11 orders of
magnitude. The model presented here makes specific quantitative
predictions about how this variation in size drives patterns of
migration distance among species. It attributes differences in the
distances travelled by migrants to systematic differences in metabolism
and morphological traits that are tightly coupled to body size, and to
differences in the underlying mechanics of walking, swimming, and
flight. In doing so, it provides an analytically tractable framework for
studying the influence of energetics and biomechanics on migration
distance that is consistent with data on species ranging from the
smallest migratory insects to the largest whales.
We thank S. P. Vogel, D. J. Levey, T. Bohrmann, A. P. Allen and J. H.
Brown for insightful discussion and comments, and G. Blohm for
assistance with illustrations. AMH was supported by a National
Science Foundation Graduate Research Fellowship under Grant No.
A.M.H., C.H. and J.F.G. conceived the study; A.M.H., C.H. and J.F.G.
developed the model; A.M.H. compiled the data and performed
analyses; A.M.H., C.H. and J.F.G. wrote the paper.
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6 A. M. Hein, C. Hou and J. F. GilloolyLetter
? 2011 Blackwell Publishing Ltd/CNRS
Additional Supporting Information may be found in the online
version of this article:
Appendix S1 Model derivation, sensitivity and statistical analyses.
Data S1 Maximum migration distance and body mass data.
As a service to our authors and readers, this journal provides
supporting information supplied by the authors. Such materials are
peer-reviewed and may be re-organised for online delivery, but are not
copy-edited or typeset. Technical support issues arising from
supporting information (other than missing files) should be addressed
to the authors.
Editor, Marco Festa-Bianchet
Manuscript received 16 August 2011
First decision made 16 September 2011
Manuscript accepted 22 October 2011
LetterConstraints on animal migration distance 7
? 2011 Blackwell Publishing Ltd/CNRS