# 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.

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Page 1

LETTER

Energetic and biomechanical constraints on animal migration

distance

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,

USA

*Correspondence: E-mail:

amhein@ufl.edu

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.

Keywords

Allometry, biomechanics, dispersal, ecomechanics, ecophysiology, energetics, migration, movement ecology,

scaling, spatial ecology.

Ecology Letters (2011)

INTRODUCTION

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.

MODEL DEVELOPMENT

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

Page 2

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

M0,

Yi¼

Z

M0

M0ð1?f Þ

?vðM;bÞ c

PmtnðMÞ þ PlocðM;bÞdM

ð1Þ

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

sum,PN

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

YT¼ E

X

N

i¼1

Yi

#

¼ N Yi

ð2Þ

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

migrants

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

Pwalk¼ cgM

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

migration.

The power required for swimming can be described by the resistive

model,

Lc

v

ð3Þ

0

(Alexander

Pswim¼ dAbv2:8

L0:2

b

ð4Þ

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

ratio Ptot⁄v.

Power required for flight near minimum power speed can be

described by the equation

h

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

/ M1=3

Substituting functions for Yi(section 1 Appendix S1) into eqn (2)

yields expressions for the expected maximum migration distances of

walking

Pfly¼ ð1 þ jÞ uM2L?2

wv?1þ /Abv3

f

i

ð5Þ

w, where Awis

0 , surface areas / M2=3

0 ).

YT¼ y0M0:34

swimming,

0

;

ð6Þ

YN

Y1

Y2

Y3

...

Body Mass (M0)

a

Mode of

locomotion

c

Morphology (β)

b

Pmtn and Ploc

d

Mass loss as fuel

is used

e

Yi

f

Yt=

N

i=1

Yi

A

B

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

Page 3

YT¼ y0p?0:64

0

M0:3

0

ð7Þ

and flying

YT¼ y0lnp0þ k1M0:42

p0þ k2M0:42

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).

0

0

??

ð8Þ

Model predictions

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

flying migrants.

0 , body length / M1=3

0 ) such that the

0 =M1=3

0

/ M0

0. Thus, after normalising

0

indicates

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

(see Results).

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

accountfordifferences inp0

Mnorm¼ M0:3

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).

according

p?1

0

totheequations

0p?0:64

0

and Mnorm¼ M0:42

0

respectively. To test our

LetterConstraints on animal migration distance 3

? 2011 Blackwell Publishing Ltd/CNRS

Page 4

RESULTS

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

[0.36,0.49]).

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

(passerines:

c1= )0.63,

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.

(a)

(b)

(c)

109

107

105

103

100

10–2

102

104

102

100

Body mass (kg)

106

10–7

10–5

10–3

1010–1

104

10–2

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

Page 5

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.

2009;Riveros&Enquist2011).However,sensitivityanalysisshowsthat

the shape of our predicted relationships and the agreement between

predictionsanddataarelargelyinsensitivetochangesinthevalueofthe

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.

DISCUSSION

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

Page 6

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.

ACKNOWLEDGEMENTS

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.

DGE-0802270.

AUTHORSHIP

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

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SUPPORTING INFORMATION

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