Vol.3, No.4, 314-320 (2013) Open Journal of Animal Sciences
http://dx.doi.org/10.4236/ojas.2013.34047
Relating models of activity metabolism to the
metabolic efficiency of steady swimming
Anthony Papadopoulos
Natural Sciences Department, St. Petersburg College, Tarpon Springs, USA;
Received 2 September 2013; revised 5 October 2013; accepted 17 October 2013
Copyright © 2013 Anthony Papadopoulos. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Power-law () and exponential power-law
() functional forms model activity metabo-
lism (U) for fully submerged swimming ani-
mals, and are special cases of the power-law
polynomial equation,
c
abU+
qUc
ae
M






c
ψ
c
UqUc
abU ψ
Ma abUψ
ae
-1 if =1
=1+ =if
+

in which U is the observed total metabolic
rate measured at an observed steady swimming
speed, . The relationship between the meta-
bolic efficiency of steady swimming and the
exponents of U is addressed in this paper to
establish the use of (for c) and (for
) as optimal efficiencies for comparing the
hydrodynamic and muscle metabolic efficien-
cies among fully submerged animals that en-
gage in steady swimming activities. The meta-
bolic efficiency of steady swimming is trans-
formed into its ideal form (
M
U
M
c-1 >1
ˆ
U
ψ-1
ψ>1
η
) from which
the optimal hydrodynamic efficiency ()
and the optimal muscle metabolic efficiency
(
ˆh=
η c-1
ˆ=
m
η
-1
ψ
) are derived. These optimal efficien-
cies are therefore ideal metabolic efficiencies
measured at different optimal steady speeds.
Subsequently, linear () and exponential
() models are approximations with diver-
gent optimal muscle metabolic efficiencies
( and , respectively), but with a
similar optimal hydrodynamic efficiency (
+ abU
qU
ae
ˆm
η 1ˆm η 1ˆ
η
h

1).
Keywords: Activity Metabolism; Exponent;
Metabolic Efficiency; Power Law; Steady Swimming
1. INTRODUCTION
The activity metabolism (U
M
) of a swimming animal
is the observed rate of total metabolic energy measured
at an observed steady swimming speed, . The func-
tional forms that model U
U
M
for a fully submerged
animal are exponential () and power law ()
[1-7]. Both forms, however, are special cases of the
power-law polynomial equation [8],
qU
ae c abU

1if 1
1
if
c
c
UqU c
abU
MaabU
ae
 



 



(1)
in which the coefficients (a, b, c, and

) are hydrody-
namical and physiological descriptors (see [7,8] for de-
tails). Note:
11
and .bqa




An important question that has not been addressed in
the literature is how the swim-speed exponent (c) and the
metabolic exponent (

) analytically relate to efficien-
cies. Experimental evidence suggests that c in 1
U
M


is a useful coefficient for comparing the swimming effi-
ciency among different animals [9-14]. Papadopoulos [8]
proposes that c in Equation (1) relates inversely to the
propulsive (or hydrodynamic) efficiency of steady swim-
ming and that ψ relates inversely to the metabolic con-
version (or muscle metabolic) efficiency of steady swim-
ming. But there is no analytical formulation that explic-
itly shows how c or ψ relates to efficiency. Although
there is experimental evidence that suggests an inverse
association between c and swimming efficiency, the de-
finitive correspondence should be derived analytically
from theory. And this also applies to the metabolic ex-
ponent, ψ.
This paper addresses the analytical relationship be-
tween the metabolic efficiency of steady swimming and
the exponents of Equation (1). The analysis confirms that
1
c

(for c) is the optimal hydrodynamic efficiency
of an animal in its ideal steady swimming state and that
1
1


(for 1

) is the optimal muscle metabolic effi-
ciency of an animal in its ideal steady swimming state.

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320 315
These optimal efficiencies can be used to compare the
hydrodynamic and muscle metabolic efficiencies among
fully submerged animals that engage in steady swimming
activities. Furthermore, the optimal muscle metabolic
efficiency (1

) determines the functional form of U
M
.
2. RESULTS
The metabolic efficiency of steady swimming can be
expressed as
U
U
U
P
M
  (2)
where U is the rate of useful metabolic energy re-
quired to swim at U, and U
P
M
is the rate of total
metabolic energy generated by swimming at U. For
convenience, let U be the rate of useful energy re-
quired to overcome the hydrodynamic drag. Then U is
the drag power, which is proportional to the product of
the power coefficient (
P
P
p
C) and cubed: U
3
Up
PCU
where
p
C for fully submerged animals that engage in
either sustained steady swimming or burst steady swim-
ming has been experimentally confirmed to be a power
law with respect to the Reynolds number (Re) [15-17]:
Rek
p
C   (3)
Since Re is proportional to ,
U
Re U
and that U is a constituent of U
P
M
, the drag power
() has the following identity (see [7], Appendix 1):
U
P
.
c
U
PbU (4)
In addition to the experimental confirmation of Equa-
tion (3), power laws conform to standard hydrodynamic
laws in which the logarithm of U and the logarithm of
are linearly related [7,12,15,16]:
P
U
loglog log
U
PbcU
By substituting Equations (1) and (4) into Equation (2),
U

then has a stationary value (or a maximum) meas-
ured at a particular steady speed (see Figure 1): as
increases from 0
U
0

, U

increases and then reaches
its maximum, after which U

decreases asymptotically
towards zero (Figure 1). The steady speed, at which
U

is maximized, can be determined by 'U

(differen-
tiating Equation (2) with respect to ), U'0
U


(eq-
uating 'U

to zero), and then solving for : U

10
,
11
c
mc
a
a
U
b 









 

0
.b
(5)
Equation (5), which is an optimal U, is also the
steady speed at which the metabolic cost of conversion
Figure 1. An example of the actual metabolic efficiency (Equa-
tion (2)) plotted with respect to the steady swimming speed (U).
The values of the coefficients used to construct this example
are as follows: a = 0.05, b = 0.5, c = 2.5, and

= 1.9. Note
that U

initially increases with increasing U and then reaches
a maximum at Umc, after which U

decreases asymptotically
towards zero.
(U
H
),
1U
UU
U
M
HP 


is minimized. Swimming at mc U thus optimizes the
metabolic efficiency of the muscles used for steady
swimming.
Substituting mc for U in Equation (2) yields the
maximum metabolic efficiency of steady swimming:
U

1
11
.
1
1
c
mc
mc
mc
bU
aabU











 (6)
Note that Equation (6) is exclusively dependent on the
metabolic exponent, ψ: as ψ increases, m

decreases.
This equation, however, is not applicable for comparing
the optimal muscle metabolic efficiency among different
animals due to the fact that the equation is not a ho-
mogenous function of ψ: comparing any two different
values of m

, in which ψ can vary only with respect to
different animals, is not evenly weighted. Let 1

and
21



be two different values of ψ for two different
animals. Then


1m

 and

2m


 are evenly com-
parable if




11mm

   such that the weight,

, is equivalent to the coefficient, λ. And that is not the
case with Equation (6). Subsequently, homogenizing m

would ensure that any two of its different values is
evenly comparable, which would be applicable for com-
paring the optimal muscle metabolic efficiency among

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320
316
different animals. This can be accomplished by trans-
forming Equation (1) such that only one functional form
is associated with metabolic efficiency. By raising the
left-hand and right-hand side of Equation (1) to the 1


power and multiplying that result by a

(in which
1
1



 ), the solution becomes
1
ˆUU U
M
aMa P
 
. (7)
Equation (7) is an important variable: ˆU
M
is the
ideal total metabolic rate because it is the sum of a (the
standard metabolic rate) and U, where a (=0 Pˆ
M
) de-
scribes the minimum metabolic rate required to sustain
physiological maintenance [2,3], while


c
U
P
bU
describes the rate of useful metabolic energy required to
overcome the hydrodynamic drag. Since Equation (5)
must be satisfied, the inequality,
ˆUU
M
M
implies that an actual animal could never achieve ˆU
M
but only in its ideal steady swimming state, that is, only
as the limit of U
M
as m

approaches 1. By substitut-
ing ˆU
M
for U
M
in Equation (2), the ideal metabolic
efficiency of steady swimming,
ˆˆ
U
U
U
P
M
  (8)
can thus be compared with U

(see Figure 2). Notice
that since ˆU
M
(for any 1

) is less than U
M
, ˆU

must be greater than U

(Figure 2).
Unlike Equation (2), Equation (8) is expressed as one
functional form, the result of which yield homogeneous
functions that are suitable for comparing the optimal
efficiencies among different animals. In particular, sub-
stituting for U in Equation (8) yields
mc
U
1
ˆ.
c
mc
mc
mc
bU
abU



 (9)
Equation (9) is the optimal muscle metabolic effi-
ciency of an animal in its ideal steady swimming state
(see Figure 2). Specifically, ˆm

is the ideal metabolic
efficiency measured at the steady speed—mc —at
which
U
U

is maximized (or U
H
is minimized). As a
result, ˆm

can be used to compare the optimal muscle
metabolic efficiency among different animals that engage
in steady swimming activities.
The next part to this analysis is to determine the opti-
mal hydrodynamic efficiency of an animal in its ideal
steady swimming state. The product of the hydrodynamic
efficiency and muscle metabolic efficiency yields the
overall (or energetic) efficiency of steady swimming
[3,9]. An important variable that accounts for the ener-
getic efficiency of steady swimming is the metabolic cost
of transport (U
F
) [18-20]:
Figure 2. An example of the metabolic efficiency plotted with
respect to the steady swimming speed (U). The solid line is the
actual efficiency, U

(= Equation (2)), whereas the dotted line
is the ideal efficiency, ˆU

(= Equation (8)). The values of the
coefficients used to construct this example are as follows: a =
0.05, b = 0.5, c = 2.5, and

= 1.9. The optimal metabolic
efficiency of the muscles used for steady swimming is the ideal
efficiency measured at Umc. Note that ˆU

continually in-
creases asymptotically towards 1.
.
U
U
M
FU  (10)
Equation (10) describes the actual total metabolic en-
ergy generated per unit of distance traveled [18-20], and
has, like U
H
, a minimum measured at a particular .
By differentiating U
U
F
with respect to U, equating U
F

to zero, and then solving for U, the steady speed that
minimizes Equation (10) is (see [8])

10
,0
11
c
mt
a
a
U
bc c










 

.b
(11)
Equation (11), like Equation (5), is an optimal U.
Thus, swimming at mt
U optimizes the energetic effi-
ciency of steady swimming. It should be noted that Equa-
tion (11) is traditionally derived by equating 1
U
F



to
zero and then solving for (see [21]), though ψ is, in
fact, a coefficient and thus should not be constrained to
any value [8]. Also, Weihs’ [22] optimal cruising speed,
in which
U
p
C is assumed not to vary with (or Re
3cc


), is a special case of Equation (11). Such
simplifying assumptions are not relevant to this paper.
Substituting for U in Equation (8) yields
mt
U
1
ˆ
c
mt
c
mt
bU
c abU



 (12)

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320 317
Equation (12) is the optimal energetic efficiency of an
animal in its ideal steady swimming state. Specifically,
ˆ

 is the ideal metabolic efficiency measured at the
steady speed—mt —at which U U
F
is minimized.
Therefore, ˆ

 can be used to compare the optimal en-
ergetic efficiency among different animals that engage in
steady swimming activities. Since the product of the hy-
drodynamic efficiency and muscle metabolic efficiency
yields the energetic efficiency of steady swimming (see
[3,9]), the optimal hydrodynamic efficiency (ˆh

) of an
animal in its ideal steady swimming state can thus be
determined by the ratio of ˆ

 to ˆm

:
ˆ1
ˆˆ
h
mc




where c is the swim-speed exponent. Alternatively, ˆh

can be derived by differentiating the ideal cost of trans-
port (ˆU
F
),
ˆ
ˆU
U
M
FU 
with respect to U, equating ˆ
F
 to zero, solving for
,
U

10
ˆ,
11
c
mt
a
a
U
bc c





 





0
, b 
(13)
and then substituting for U in Equation (8):
ˆmt
U
ˆ1
ˆ.
ˆ
c
mt
hc
mt
bU
c
abU
 
 (14)
As a result, Equation (14) is the ideal metabolic effi-
ciency measured at the steady speed——at which
ˆmt
U
ˆU
F
is minimized, and can thus be used to compare the
optimal hydrodynamic efficiency among different ani-
mals that engage in steady swimming activities.
3. DISCUSSION
Steady swimming is observed in animals engaging in
ecologically important activities such as competing for
limited resources, seeking favorable abiotic conditions,
and migration [23-27]. During such activities, an animal
swims at an optimal steady speed to minimize its meta-
bolic cost. For example, during migration, an animal
maximizes its distance per unit of total metabolic energy
by swimming at the steady speed (mt
U) at which the
metabolic cost of transport (U
F
) is minimized [22,28].
Thus maximizing the distance per unit of total metabolic
energy is essential for optimizing the steady swimming
performance during migration. Equally essential, how-
ever, is optimizing the steady swimming performance
during activities in which maximizing distance is not
essential. For example, while competing for limited re-
sources within a microhabitat, an animal maximizes its
useful metabolic energy per unit of total metabolic en-
ergy by swimming at the steady speed (mc U) at which
the metabolic cost of conversion (U
H
) is minimized.
Hence, an animal can optimize its steady swimming per-
formance for different activities by swimming at
(the steady speed at which
mt
U
U
F
is minimized) or mc U
(the steady speed at which U
H
is minimized). But there
is a fundamental difference between these two objectives:
U
H
, unlike U
F
, is dimensionless; the inverse of U
H
is,
in fact, the metabolic efficiency of steady swimming,
U

(= Equation (2); see Figure 1). A transformation of
U

is then formulated in order to yield optimal efficien-
cies that are homogenous functions of c and

of dif-
ferent animals. This is particularly important when com-
paring the optimal efficiencies among different animals.
This transformation (see Equation (7)) yields the ideal
metabolic efficiency of steady swimming, ˆU

(= Equa-
tion (8); see Figure 2), for which

can be any value
greater than 1. The ideal steady swimming state of an
animal can thus be interpreted as a special case of Equa-
tion (2) in which the maximum metabolic efficiency, m

(= Equation (6)), approaches 1.
Substituting mt (which is the steady speed that op-
timizes the energetic efficiency of steady swimming) for
in
U
UˆU

yields the optimal energetic efficiency of an
animal in its ideal steady swimming state and is identical
with Equation (12):
1
ˆ,1c
c


,


where c and

are the exponents in Equation (1), which
describes the activity metabolism (U
M
) of a fully sub-
merged animal engaged in steady swimming. The steady
speed that optimizes the metabolic efficiency of the mus-
cles used for steady swimming—mc U—is then substi-
tuted for in
UˆU

to yield the optimal muscle meta-
bolic efficiency of an animal in its ideal steady swim-
ming state (see Figure 2),
1
ˆ,1
m


,


which is identical with Equation (9). Since the energetic
efficiency is the product of the hydrodynamic and muscle
metabolic efficiencies [3,9], the optimal hydrodynamic
efficiency (ˆh

) is simply the ratio of ˆ

 to ˆm

:
ˆ1
ˆ,1
ˆ
h
m
c
c


,


which can also be derived by substituting into ˆU

the
steady speed () that optimizes the hydrodynamic
efficiency of an animal in its ideal steady swimming state
(see Equations (13) and (14)). The optimal efficiencies,
ˆmt
U
ˆh

and ˆm

, can thus be used to compare the hydrody-

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320
318
namic and muscle metabolic efficiencies among fully
submerged animals that engage in steady swimming ac-
tivities. Furthermore, the fact that c and

are inde-
pendent of the scale of U
M
and U further validates
the use of ˆh

and ˆm

as ideal efficiencies for com-
paring the optimal hydrodynamic and optimal muscle
metabolic efficiencies among different animals.
It is important to note that neither c nor

can equal
exactly 1; otherwise, ˆh

or ˆm

is undefined (see
Equations (5) and (13)). As a result, the first-degree
power-law functional form of U
M
can be only ap-
proximated:
ˆ 1
Um
 when
c
bUMa
Also, since the value of

is estimated by fitting
Equation (1) to activity metabolism,

cannot approach
infinity, because infinity is not a number. The curve-fit
estimate of

, however, can be a value much greater
than 1. And so, the exponential power-law functional
form of U
M
can be only approximated [8]:
ˆ
Um
 
wh
qU c
een 1 Ma
Notice that the functional form of U
M
(= Equation
(1)) depends on the value of ˆm

; this implies that the
power-law () and exponential power-law ()
models suggest different biology with regard to the
metabolic conversion (or muscle metabolic) efficiency of
steady swimming: a very high value of
c
abUqU c ae
ˆm

(i.e., when
ˆ1
m

) implies that the activity metabolism of an animal
is best modeled as a first-degree power-law polynomial,
whereas a very low value of ˆm

(i.e., when ˆ1
m

)
implies that the activity metabolism of an animal is best
modeled as an exponential power law. Of course, the
linear form,
ˆˆ
d
Uh
bU when 1 an1
m
Ma

  
which is a special case of , is approximated
when
c
Uab
ˆh

and ˆm

are both very high values. And the
exponential form,
ˆˆ
1
qU
Ma  
c
U
e when
ae
and
h
1 
m
which is a special case of , is approximated when
qU
ˆh

is a very high value and ˆm

is a very low value. In
essence, the linear form of U
M
suggests high hydro-
dynamic and high muscle metabolic efficiencies because
it has no or little curvature with respect to . The ex-
ponential form of U
U
M
, however, suggests that the two
efficiencies are compensatory: hydrodynamic efficiency
is very high, while muscle metabolic efficiency is very
low; a high hydrodynamic efficiency thus compensates
for a low muscle metabolic efficiency.
The metabolic efficiency of steady swimming (Equa-
tion (2)) has an intimate connection with activity me-
tabolism. To understand why this is so, consider the fol-
lowing equivalence of ˆU
M
:
1
ˆˆˆ
expd e,
ˆ
C
U
UU
h
UMM
U






 (15)
where


1ln 10C

 is the constant of integration,
which is determined by satisfying the condition 0
ˆ
M
a

.
The left-hand side of Equation (15) is exclusively in
terms of efficiency (ˆU

) and speed ()—two basic
terms in hydrodynamics. Moreover, as Papadopoulos [8]
noted, Equation (1) can be derived from Equation (15)
simply by multiplying m
U
ˆ

(the optimal muscle meta-
bolic efficiency) by ˆh

(the optimal hydrodynamic effi-
ciency) or replacing ˆh

with ˆˆ
ˆ
hm


 (the optimal
energetic efficiency):
2
ˆˆ
expd e,
ˆ
C
U
UU
UMM
U








 (16)
where the constant of integration, , is de-
termined by satisfying the condition 0

1
2lnCa
 


M
a. Thus,
Equation (16) is identical with Equation (1):

11
ˆ1c
UU
MaMaabU

 

Equations (15) and (16) show analytically the unique
relationship between activity metabolism and the meta-
bolic efficiency of steady swimming. And so, with regard
to efficiency, the difference between the ideal total
metabolic rate (= Equation (7)) and the actual total meta-
bolic rate (= Equation (1)) is the optimal muscle meta-
bolic efficiency, ˆm

; this coefficient is clearly important
to consider not only because it takes into account muscle
metabolic efficiency, but also because it determines the
functional form of U
M
[8].
4. CONCLUSIONS
A fully submerged animal that engages in steady
swimming has an ideal metabolic efficiency ( ˆU

) from
which the optimal efficiencies, ˆh

and ˆm

, are derived
(Equations (1)-(14)). The optimal hydrodynamic effi-
ciency (ˆh

) and the optimal muscle metabolic efficiency
(ˆm

) are thus ideal metabolic efficiencies measured at
the optimal steady speeds, U and mc
U, respectively
(see Figure 2). And from hydrodynamic principles (see
[3,9]), the product of
ˆmt
ˆh

and ˆm

represents the opti-
mal overall (or optimal energetic) efficiency (*
ˆ

) of an
animal in its ideal steady swimming state:
11
ˆˆˆ .
hm c

 


Although the coefficients and
1
ˆhc 

1 ˆm



 are
inverses of the exponents in Equation (1) and thus remain
constant with respect to U for any animal, they can
indeed vary only with respect to different animals. Com-
paring ˆh

and ˆm

among different animals requires

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320 319
that these optimal efficiencies are homogenous functions
of c and

: ˆh

and ˆm

are, in fact, homogenous due
to the transformation of Equation (2) into Equation (8),
which is the ideal form of U

. Furthermore, since ˆh

and ˆm

are inverses of the exponents in Equation (1),
the different models of U
M
exclusively arise from the
different values of ˆh

and ˆm

.
5. ACKNOWLEDGEMENTS
I thank the reviewers for providing comments that improved this
manuscript. I was employed at Texas Tech University, where I inde-
pendently wrote this manuscript and derived all of the results. Support
for this work was provided in part by National Science Foundation
award DEB-0616942 to Sean H. Rice.
REFERENCES
[1] Ivlev, V.S. (1960) Active metabolic intensity in salmon
fry (Salmo salar L.) at various rates of activity. Salmon
and Trout Comm, Int Counc Explor Sea, Copenhagen,
213, 1-16.
[2] Brett, J.R. (1964) The respiratory metabolism and swim-
ming performance of young sockeye salmon. Journal of
the Fisheries Research Board of Canada, 21, 1183-1226.
http://dx.doi.org/10.1139/f64-103
[3] Webb, P.W. (1974) Hydrodynamics and energetics of fish
propulsion. Bulletin of the Fisheries Research Board of
Canada, 190, 109-119.
[4] O’Dor, R.K. and Webber, D.M. (1991) Invertebrate ath-
letes: Trade-offs between transport efficiency and power
density in cephalopod evolution. The Journal of Experi-
mental Biology, 160, 93-112.
[5] Hind, A.T. and Gurney, W.S.C. (1997) The metabolic cost
of swimming in marine homeotherms. The Journal of
Experimental Biology, 200, 531-542.
[6] Fish, F.E. (2000) Biomechanics and energetics in aquatic
and semiaquatic mammals: Platypus to whale. Physiolo-
gical and Biochemical Zoology, 73, 683-698.
http://dx.doi.org/10.1086/318108
[7] Papadopoulos, A. (2008) On the hydrodynamics-based
power-law function and its application in fish swimming
energetics. Transactions of the American Fisheries Soci-
ety, 137, 997-1006. http://dx.doi.org/10.1577/T07-116.1
[8] Papadopoulos, A. (2009) Hydrodynamics-based functional
forms of activity metabolism: A case for the power-law
polynomial function in animal swimming energetics. PLoS
ONE, 4, e4852.
http://dx.doi.org/10.1371/journal.pone.0004852
[9] Webb, P.W. (1993) Swimming. In: Evens, D.H., Ed., The
Physiology of Fishes, CRC Press, Boca Raton, 47-73.
[10] Wardle, C.S., Soofiani, N.M., O’Neill, F.G., Glass, C.W.
and Johnstone, A.D.F. (1996) Measurements of aerobic
metabolism of a school of horse mackerel at different
swimming speeds. Journal of Fish Biology, 49, 854-862.
h ttp :// d x. doi . org/1 0 .1111 / j. 1095-8649.1996.tb00084.x
[11] Pettersson, L.B. and Hedenström, A. (2000) Energetics,
cost reduction and functional consequences of fish mor-
phology. Proceedings of the Royal Society B, 267, 759-
764. http://dx.doi.org/10.1098/rspb.2000.1068
[12] Korsmeyer, K.E., Steffensen, J.F. and Herskin, J. (2002)
Energetics of median and paired fin swimming, body and
caudal fin swimming, and gait transition in parrotfish
(Scarus schlegeli) and triggerfish (Rhinecanthus aculea-
tus). The Journal of Experimental Biology, 20 5, 1253-
1263.
[13] Behrens, J.W., Praebel, K. and Steffensen, J.F. (2006)
Swimming energetics of the Barents Sea capelin (Mallo-
tus villosus) during the spawning migration period. Jour-
nal of Experimental Marine Biology and Ecology, 331,
208-216. http://dx.doi.org/10.1016/j.jembe.2005.10.012
[14] Ohlberger, J., Staaks, G. and Holker, F. (2006) Swimming
efficiency and the influence of morphology on swimming
costs in fishes. Journal of Comparative Physiology B,
176, 17-25.
http://dx.doi.org/10.1007/s00360-005-0024-0
[15] Wu, T.Y. (1977) Introduction to the scaling of aquatic
animal locomotion. In: Pedley, T.J., Ed., Scale Effects in
Animal Locomotion, Academic Press, New York.
[16] Wu, T.Y. and Yates, G.T. (1978) A comparative mech-
anophysiological study of fish locomotion with implica-
tions for tuna-like swimming mode. In: Sharp, G.D. and
Dizon, A.E., Eds., Physiological Ecology of Tuna, Aca-
demic Press, New York.
[17] Fung, Y.C. (1990) Biomechanics: Motion, flow, stress,
and growth. Springer-Verlag, New York.
[18] Tucker, V.A. (1970) Energetic cost of locomotion in ani-
mals. Comparative Biochemistry and Physiology, 34, 841-
846. http://dx.doi.org/10.1016/0010-406X(70)91006-6
[19] Tucker, V.A. (1975) The energetic cost of moving about.
American Scientist, 63, 413-419.
[20] van Ginneken, V., Antonissen, E., Müller, U.K., Booms,
R., Eding, E., Verreth, J. and van den Thillart, G. (2005)
Eel migration to the Sargasso: Remarkably high swim-
ming efficiency and low energy costs. The Journal of Ex-
perimental Biology, 208, 1329-1335.
http://dx.doi.org/10.1242/jeb.01524
[21] Videler, J.J. and Nolet, B.A. (1990) Costs of swimming
measured at optimum speed: Scale effects, differences
between swimming styles, taxonomic groups, and sub-
merged and surface swimming. Comparative Biochemis-
try and Physiology, 97A, 91-99.
http://dx.doi.org/10.1016/0300-9629(90)90155-L
[22] Weihs, D. (1973) Optimal fish cruising speed. Nature,
245, 48-50. http://dx.doi.org/10.1038/245048a0
[23] Plaut, I. (2001) Critical swimming speed: Its ecological
relevance. Comparative Biochemistry and Physiology,
131A, 41-50.
[24] Blake, R.W. (2004) Fish functional design and swimming
performance. Journal of Fish Biology, 65, 1193-1222.
h ttp :// d x. doi . org/1 0 .1111 / j. 0022-1112.2004.00568.x
[25] Langerhans, R.B. (2009) Trade-off between steady and
unsteady swimming underlies predator-driven divergence
in Gambusia affinis. Journal of Evolutionary Biology, 22,

A. Papadopoulos / Open Journal of Animal Sciences 3 (2013) 314-320
320
1057-1075.
h ttp :// d x. doi . org/1 0 .1111 / j. 1420-9101.2009.01716.x
[26] Dougherty, E., River, G., Blob, R. and Wyneken, J. (2010)
Hydrodynamic stability in posthatchling loggerhead (Ca-
retta caretta) and green (Chelonia mydas) sea turtles.
Zool, 113, 158-167.
http://dx.doi.org/10.1016/j.zool.2009.10.001
[27] Fu, S.-J., Peng, Z., Cao, Z.-D., Peng, J.-L., He, X.-K., et
al. (2012) Habitat-specific locomotor variation among
Chinese hook snout carp (Opsariichthys bidens) along a
river. PLoS ONE, 7, e40791.
http://dx.doi.org/10.1371/journal.pone.0040791
[28] Brodersen, J., Nilsson, P.A., Ammitzbøll, J., Hansson,
L.A., Skov, C., et al. (2008) Optimal swimming speed in
head currents and effects on distance movement of winter
migrating fish. PLoS ONE, 3, e2156.
http://dx.doi.org/10.1371/journal.pone.0002156