Least-cost swimming speeds and transportation costs in some pelagic estuarine fishes

Fisheries Research, 1 (1981/1982) 117--127

117

Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

LEAST-COST SWIMMING SPEEDS AND T R A N S P O R T A T I O N COSTS IN SOME P ELAGI C E S T U A R I N E FISHES

JOHN

M. W A K E M A N

I and D O N A L D

E. W O H L S C H L A G

2

'Zoology Department, Louisiana Tech University, Ruston, L A 71272 (U.S.A.) 2 The University of Texas Marine Science Institute,Port Aransas Laboratory, Port Aransas, T X 78373 (U.S.A.)

(Accepted 14 September 1981)

ABSTRACT

Wakeman, J.M. and Wohlschlag, D.E., 1982. Least-cost swimming speeds and transportation costs in some pelagic estuarine fishes. Fish. R es., 1:117--127. A model, based on hydrodynamic theory, predicting least-cost swimming speed and cost of transport in mg 02. kg- 1. km- 1 in relation to body length for pelagic estuarine fishes, indicated that least-cost swimming speeds in length-specific terms are less for larger fish than for smaller fish. Predicted least-cost swimming speeds ranged from 1.3 lengths per second for a 50-cm fish to 4.7 lengths per second for a 10-cm fish. The predicted minimum cost o f transport for a fish 30 c m in b o d y length was 110 mg O2 "kg-~ "km -~ Predictions from the model were consistent with empirically derived estimates for least-cost swimming speeds and cost of transport values for four estuarine species.

INTRODUCTION A q u a n t i t y t e r m e d " cos t o f t r a n s p o r t " (defined as the energy expenditure required to transport a unit of mass 1 km) has been used by a n u m b e r of authors (Weis-Fogh, 1952; Tucker, 1970; Schmidt-Nielsen, 1972) to compare the effectiveness o f transport processes among different animals that may vary considerably in size or in m ode of l ocom o t i on. Such comparisons indicate t h a t l o c o m o t i o n is a more efficient process in swimming animals than in flying or running animals (Schmidt-Nielsen, 1972). Cost o f transport, CT, can be mathematically expressed as CT = P m / M . U

(1)

where Pm is the total p o w e r o u t p u t f r o m metabolic processes, M is b o d y mass and U is swimming speed. For aerobic organisms, CT can be conveniently expressed in terms of mg 02" k g - l . k m - l . In general, there is a least-cost velocity, Uopt, at which CT is minimum (Tucker, 1968; Pinshow et al., 1977), an observation t hat m a y be o f importance in understanding the swimming speeds used by migrating or foraging

118

fishes (Weihs, 1973, 1975; Webb, 1975; Ware, 1978). The minimum cost of transport usually decreases as body mass increases (Schmidt-Nielsen, 1972). Because it provides a measure of transport efficiency (Tucker, 1970), minimum CT may be of value in comparing the relative effectiveness of different swimming modes observed among fishes. The various swimming modes of fishes have been reviewed by Lindsey {1978). In this paper, we have developed a model expressing CT as a function of swimming speed which can be used to predict least-cost swimming speed for a "typical" estuarine fish. Least-cost swimming speeds and accompanying CT values obtained from the model are compared with empirically derived estimates of Vopt and CT for several estuarine species. DEVELOPMENT OF THE MODEL Assuming that elapsed time is relatively unimportant for a migrating species, selection should favor individuals that swim at a velocity, Uopt, where energy output is minimized. Vopt can be theoretically derived by the following argument. Both velocity, U (cm. s-l), and work, W (erg), are related to time, t, by t = D/U

(2)

t = W ~Pro

(3)

where D is distance in cm, Pin is total metabolic power in erg. s -1 , and U is velocity in cm" s -1 . Eliminating t in Equations (2) and (3) (4)

W = Pro'D~ V.

Pm consists of two main components: propulsive power output, Pp, required to overcome hydrodynamic drag, and resting metabolic power, Ps, required to maintain the basic metabolic machinery of the organism at rest. Thus Equation (4) may be written as W = (D.Pp + D-Ps) / U.

(5)

Assuming that maintenance costs are relatively independent of swimming speed, Ps can be represented by the standard metabolic rate. Pp, however, is related to both velocity, U, and drag, F (dyn), by Pp = F . U / e s

(6)

where e s is the efficiency with which chemical energy is converted to propulsive energy. Drag, F, for a swimming fish.can be estimated from hydrodynamic theory (Smit, 1965) as f

= ~p'A'U2"(1.2

C)

where p is density (g" cm-3), A is wetted surface area of the fish (cm 2) and

(7)

119

1.2 C is the drag coefficient, C (dimensionless), adjusted for body form of a "typical" pelagic fish (Bainbridge, 1961). Assuming that turbulent boundary layer conditions generally predominate for swimming fishes (Webb, 1978), C = 0.072 ( L . U / v ) -°'~

(8)

where L is body length (cm) and v is kinematic viscosity in stokes (Webb, 1975). Combining Equations (7) and (8) and simplifying yields F = k ' U 1'8

(9)

where k = 0.043 p . A O , / L ) °'2 . Substitution of Equation (9) for F in Equation (6) yields Pp = k . U 2"s/e s = K . U 2"s

(10)

where K is equal to k / e s. Equation (5) can now be rewritten as W = (D.K.

U 2"8 + D . P s )

/ U.

(11)

W is minimized when the slope of function (11) is equal to zero. Differentiating W with respect to U, dW/dU

=

( 1 . 8 D . K . U 2"8 - D . P s ) / U 2

(12)

and setting the numerator of (12) equal to zero, Uopt = ( P s / 1 . S K )

l/2"s

(13)

where Uopt is the velocity in cm-s -1 at which W is minimized. Swimming speeds calculated from Equation (13) can generally be more conveniently expressed as kin. h -1 by multiplying the result by a conversion factor of 0.036. PREDICTIONS F R O M T H E M O D E L

Appropriate values can be assigned to the various components of the above model as follows: density, p, of estuarine waters at 28°C is approximately 1.02 g-cm -3 (Bialek, 1966); kinematic viscosity, v, under these conditions is 0.0082 stokes (Myers, 1969); and surface area, .4, of a "typical" pelagic fish is approximately 0.4 L 2 (Bainbridge, 1961). Swimming efficiency, es, is the product of locomotory muscle efficiency and propulsor efficiency (both of which are dependent on velocity), and although e s has been shown to increase with increased sustained swimming speeds (Webb, 1971), the theoretical analysis is simplified by assuming e s to be 0.25 (Webb, 1975) over a range of sustained speeds. Note that values for p and u are dependent on temperature and salinity. The only other component for which an estimate is required is Ps (repre-

120

senting standard metabolic rate). Ps in erg. s -1 as a function of body length, L (cm), for a " t y p i c a l " estuarine fish can be estimated as Ps = 39167 (0.015 L2"4).

(14)

The rationale for this estimate is based on the premise that standard metabolism in fishes is typically proportional to weight raised to the 0.8 power (Winberg, 1956). Since weight is proportional to L 3 , standard metabolism is expected to be proportional to L 2"4. A value of 0.015 was arbitrarily assigned to the coefficient of proportionality because it yields values for oxygen uptake rates (mg 0 2 ' kg -1" h -~) which are close to those experimentally determined for several estuarine species in this study (below). Based on a conversion factor of 1.41 times l 0 s erg per mg 02 (Brett, 1973), mg 02 "h -~ is converted to erg. s -~ by multiplying by 39167. Inserting the above values in Equation (13), Uop t was plotted as a function of b o d y length for a typical fish at 28°C (Fig. 1). For convenience, swimming speeds were expressed as km • h -1 . Least-cost speeds increased as body length increased, ranging from 1.7 k m - h -~ for a 10-cm fish (4.7 L .s -1) to 2.4 km .h -~ for a 50-cm fish (1.3 L's-~). While Uop t for a 50-cm fish was close to the one length per second optimal speed predicted by Weihs (1973), theoretical leastcost specific speeds (L. s -~) increased as length decreased. Conclusions similar to these results were also reached by Weihs (1977) and Ware (1978). Uopt = ( P / 1 8 K )

1/28

3.0

E 25. a~

2D

E 0 15

10

10

20

30 Body length

40

50

( cm )

Fig. 1. Theoretical least-cost swimming speed (Vopt) as a function of body length for a "typical" estuarine fish at 28°C. Vopt is defined as the velocity at which cost of transport is at the minimum value. Derivation o f the equation is explained in the text.

Equation (11) was utilized to predict the energy required for a 30-cm fish to swim a distance of 1 km as a function of swimming speed (Fig. 2). This length was chosen because it represents the approximate average length of

121

the fishes for which CT values were empirically determined (below). The plot indicates a minimum value for work at 2.1 k m ' h -x (about 2 L-s-~), at which velocity oxygen consumption is 38 mg 02" km -~. The average weight of a "typical" fish of this length is observed to be about 350 g. On this basis, the predicted energetic cost can be converted to approximate cost of transport terms by dividing by 0.35, yielding a minimum cost of transport of 110 mg 02" kg -1" km -~. This prediction agrees quite well with Schmidt-Nielsen's (1972) report that cost of transport in fishes of this size range is generally a b o u t 0.5 kcal. kg - l . km -I (150 mg 02" kg -~" km-1). 120-

E • 80.

v

M

~40-

20 ¸

~.b

2:o

3'0

4'0

SwiFning speed

( k m h .I )

5'0

Fig. 2. Theoretical energetic cost for an estuarine fish, 30 cm in body length, to swim a distance of 1 km at 28°C as a function of s w i m m i n g speed in km -h -~ . Minimum energetic cost occurs at 2.1 k m . h -~ (2 L.s-~).

Values assigned to the various components of the model are subject to some estimation errors, but with the exception of the estimation for Ps, A, and es, these errors would be minimal. Reasonable variations in the values of p, r, and A have little effect on final predictions, while an estimate of 0.20 for e s (rather than 0.25) decreases numerical values for U o p t by less than 8%. In the application of the model to specific estuarine species, the most important variable in Ps. The sensitivity of the model to variations in Ps was examined b y comparing predicted Vop t values obtained using the original Ps estimate with values obtained when Ps was both halved and doubled (Table I), and similarly the effect of varying Ps on energetic cost is shown in Table II. These tables show that both [Top t and W decrease as standard metabolic rate

122 TABLE I Effects of changes in the estimate of resting metabolism (Ps) on theoretical least-cost swimming speeds (Uopt) for various sized fish. Columns contain predicted values for Uopt based on the original estimate for Ps (see text), and predicted Uopt when the Ps estimate is doubled or halved

P$

Total body length of fish (cm)

estimate Ps 2(Ps) 0.5(Ps)

10

30

50

47.1 c m . s -1 60.3 cm.s -1 36.8 cm.s -~

59.6 cm.s -1 76.4 c m . s -~ 46.5 cm.s -~

66.5 cm~s -1 85.2 cm.s -~ 51.9 cm,s -~

TABLE II Effects of changes in resting metabolism (Ps) estimate on theoretical oxygen requirements for various sized fishes to swim a distance of 1 km at least-cost velocity. Columns show values for oxygen consumption based on the original estimate of Ps and when the Ps estimate is doubled or halved

Ps

Total body length of fish (cm)

estimate Ps 2(Ps) 0.5(Ps)

10

30

50

3.4 mg 02 5.3 mg O 2 2.2 mg 02

37.8 mg 02 59.5 mg 02 24.4 mg 02

116.4 mg 02 181.9 mg 02 74.6 mg 02

is d e c r e a s e d . T a b l e I indicates t h a t halving the Ps e s t i m a t e for a given sized fish r e d u c e s t h e p r e d i c t e d least-cost s w i m m i n g s p e e d in c m . s -1 b y a b o u t 22%. T a b l e I I indicates t h a t halving t h e Ps e s t i m a t e r e d u c e s t h e t h e o r e t i c a l m i n i m u m e n e r g e t i c c o s t b y 3 5%. EXPERIMENTAL MATERIALS AND METHODS Using p r e v i o u s l y d e s c r i b e d t e c h n i q u e s (Wohlschlag a n d W a k e m a n , 1 9 7 8 ) , o x y g e n c o n s u m p t i o n r a t e s at various s w i m m i n g speeds at 28°C w e r e determ i n e d f o r f o u r e s t u a r i n e species in a B l a z k a - t y p e r e s p i r o m e t e r . Species s t u d i e d w e r e s p o t t e d s e a t r o u t (Cynoscion nebulosus), red d r u m (Sciaenops ocellata), s h e e p s h e a d (Archosargus probatocephalus) and s t r i p e d b u r r f i s h (Chilomycterus schoepfi). E x p e r i m e n t a l fish averaged a b o u t 30 c m in b o d y length. T e m p e r a t u r e a c c l i m a t e d ( 4 8 h) fish w e r e c o m p e l l e d t o s w i m in t h e 207-1 r e s p i r o m e t e r at speeds ranging f r o m zero t o m a x i m u m sustained speeds. M a x i m u m s u s t a i n e d s p e e d was d e f i n e d as t h e highest v e l o c i t y w h i c h a fish c o u l d m a i n t a i n f o r at least 5 m i n w i t h o u t burst-glide s w i m m i n g b e h a v i o u r ( W a k e m a n and Wohlschlag, 1979). E a c h o f t h e s e species c o u l d m a i n t a i n such

123 swimming speeds for several hours without evidence of fatigue. Experimental salinities were within the range normally encountered by these fishes in local estuaries (20--30 ppt). Oxygen consumption rates were converted to weight specific terms (mg 02" kg -1" h-'). Since preliminary plots indicated a linear relation between the loglo of oxygen uptake rate and swimming velocity in k m . h -1 (Fig. 3), least squares linear regressions were fitted to the data in this form. Red d r u m Temp. 28°C 300-

0~25o-

}j j/

o

--200+

Y = 214 + 015X ( r= 095 N=51) 150

CT = lOe(lobU

)l U

400 -T

E "T

~o o"

2O0-

~ I

loo

lb

2'.o Swimming

3:o speed

4'0

do

( k m • h -+ )

Fig. 3. Upper panel: experimentally determined values for weight-specific oxygen uptake rates of red drum at speeds ranging from zero to maximum sustained swimming speed, with fitted least squares regression. Lower panel: cost of transport (CT) as a function of swimming speed, a is the intercept from the regression in the upper panel and b is the

slope of the regression. The basic regression model was log

Y

= a + bvU

(15)

where Y is mg 02" kg -~" h - ' , by is the regression coefficient and a is the intercept representing the logarithm (base 10) of weight-specific resting oxygen uptake rate.

124 An empirical function describing cost of transport, CT, in relation to swimming speed was derived from Equation (15) as follows. Taking antilogs of (15), Y = 10 a (10bvU).

(16)

Since CT has the dimensions mg 02" kg -1" k m - ' , CT

=

Y/U

=

10 a ( 1 0 bvU)/U.

(17)

Assuming a constant resting metabolic rate, estimated least-cost speed can be obtained from Equation (17) by differentiating with respect to U, dCT/dU = 10 a (10 b~U) (bv.ln 1 0 . U - 1) / U 2

(18)

and setting this derivative equal to zero Uopt = 1 / bv.ln 10 = 1 / b v (2.3026).

(19)

Least-cost swimming speeds and minimum cost of transport values were obtained for each species from Equations (19) and (17) respectively, using empirically obtained estimates for by and a. RESULTS AND DISCUSSION Logarithms of oxygen uptake rate of swimming fishes increased linearly with increased swimming velocity. A plotted example is shown in Fig. 3. Least squares regression equations fitted to the data for each species are given in Table III. Empirical estimates of least-cost swimming speeds and minimum cost of transport values derived from the regression using Equations (19) and (17) are also shown in Table III. Using estimates for resting metabolic power, Ps, obtained from the intercept values in the regression equations, theoretical Uopt values (based on body lengths of 30 cm) were predicted for each species from the model (Equation (13)). Empirically estimated least-cost speeds agreed well with theoretical Vopt values (Table III). For all species, both theoretical and empirically derived Uopt values were considerably higher than the generally accepted estimate of 1 L. s-' based on Weihs' (1973) theoretical discussion. That discussion, however, did not take body length or Reynolds number into consideration, and in a more recent analysis of the influence of body size on preferred cruising speeds of aquatic organisms, Weihs (1977) concluded that Uopt is proportional to L °'3s-°'s°. Weihs (1973) noted that his predicted optimal speeds were lower than cruising speeds generally observed in migrating fishes or in laboratory studies. Empirically determined least-cost speeds for sockeye salmon were considerably higher than those predicted by the 1973 Weihs model (Webb, 1975}. Sharp and Dotson (1977} estimated the daylight cruising speed of albacore to be approximately 2 L. s -1 . Similar cruising speeds have been reported for tuna (Magnuson, 1970) while even higher specific speeds have been observed in migrating mullet (Peterson, 1976).

N

52 51 58 49

Species

Sheepshead Red drum Spotted seatrout Striped burrfish

Log Log Log Log

Y Y Y Y

= = = =

2.150 2.137 2.309 1.880

Equation

÷ + + +

0.12 0.15 0.13 0.24

U U U U

0.89 0.95 0.88 0.86

r (L .s -~)

( k m - h -~) 3.3 2.7 3.0 1.7

Hopt

Uopt 3.6 2.9 3.3 1.8

Calculated

Calculated

106 128 165 113

Min. CT (mg O 5 -kg -1 . k i n - ' )

2.8 2.8 3.2 2.3

( L . s -t)

Uopt

Theoretical

Linear regression equations for log x0 mg O2" kg -~. h-1 ( Y ) as a function o f observed swimming velocity ( U ) in km-h-~ at 28°C. Estimates for least-cost swimming speed (Uopt) and cost o f transport (CT) in mg O2 -kg-1. km-1 were derived from the regression equations (see text). Theoretical Uopt was obtained from the model. Specific swimming speeds (L -s -~ ) are based on average body lengths of 30 cm

TABLE III

b~

126

The relation between cost of transport and velocity was U-shaped for each species. The plotted function for red drum is given as an example (Fig. 3, lower panel). Empirically derived CT-velocity curves showed good agreement with the theoretical curve in Fig. 2. A similar U-shaped relation has been observed for sockeye salmon (Brett, 1965). Minimum cost of transport values estimated for each species (Table III) agreed well with the theoretical prediction of 110 mg 02" kg -1" km -1 for a "typical" 30-cm fish (above). Tucker (1970) and Schmidt-Nielsen (1972) have reviewed transportation costs for running, flying and swimming animals. Based on an average oxycalorific equivalency of 4.8 kcal/1 (Brett, 1973), energetic costs reported by these authors can be converted to mg 02" kg -~" km -~ by multiplying by 1.43/0.0048. Schmidt-Nielsen (1972) reported CT values to be similar among a wide variety of fishes, being approximately 150 mg 02" kg -1" km -~ , while CT values for running animals of the same general size are generally higher than 2000 mg O2" kg -1" km -1 (Tucker, 1970). Minimum CT values obtained in this study (Table III) were similar to those reported for fishes by Schmidt-Nielsen (1972), supporting the conclusion that transport efficiency is high in swimming animals in comparison with terrestrial animals. This high transport efficiency, which is accomplished in spite of problems associated with the high viscosity and density of water, may be due in part to the fact that most fishes maintain nearly neutral buoyancy, with little energy expended to support body weight (Lindsey, 1978). Values for the regression coefficient, by (Table III), ranged from 0.12 to 0.24 and minimum CT values ranged from 106 to 165 mg 02" kg -1" km -1. The lowest by and CT values were obtained for sheepshead and may be related to morphological features and swimming characteristics in this species generally associated with increased hydromechanical efficiency (Lighthill, 1970; Webb, 1975). Such features include increased body depth enhanced by large median fins, a large caudal fin in conjunction with a relatively narrow-necked caudal peduncle, and concentration of the lateral movement of the propulsive wave in the caudal region of the body. The relatively high by value obtained for striped burrfish may be associated with this species' ostraciform swimming mode in which the body is held rigid while propulsive thrust is generated by oscillations of the caudal and pectoral fins. The high by value, however, was counterbalanced by a relatively low resting metabolic rate so that CT in this species was similar to that of the fishes which swam by means of propulsive body waves. ACKNOWLEDGMENTS

We gratefully acknowledge support for this study by the Texas Department of Water Resources. Research assistant Ronald Ilg aided in the capture and maintenance of fishes and in making metabolic determinations. REFERENCES Bainbridge, R., 1961. Problems of fish locomotion. Symp. Zool. Soc. Lond., 5: 13--32.

127 Bialek, E.L., 1966. Handbook of Oceanographic Tables. U.S. Nay. Oceanogr. Off. Spec. Publ., SP-68,417 pp. Brett, J.R., 1965. The relation of size to the rate of oxygen consumption and sustained swimming speeds of sockeye salmon (Oncorhynchus nerka). J. Fish. Res. Board Can., 22: 1491--1501. Brett, J.R., 1973. Energy expenditure of sockeye salmon (Oncorhynchus nerka) during sustained performance. J. Fish. Res. Board Can., 30: 1799--1809. Lighthill, M.J., 1970. Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech., 44: 265--301. Lindsey, C.C., 1978. Form, function, and locomotory habits in fish. In: W.S. Hoar and D.J. Randall (Editors), Fish Physiology, Vol. VII. Academic Press, New York, NY. Magnuson, J.J., 1970. Hydrostatic equilibrium of Euthynnus affinis, a pelagic teleost without a gas bladder. Copeia, No. 1, pp. 56--85. Myers, J.J., 1969. Handbook of Ocean and Underwater Engineering. McGraw-Hill Book Co., New York, NY, 425 pp. Peterson, C.H., 1976. Cruising speed during migration of the striped mullet (Mugil cephalus L.): an evolutionary response to predation? Evolution, 30 (2): 393--396. Pinshow, B., Fedak, M.A. and Schmidt-Nielsen, K., 1977. Terrestrial locomotion in penguins: it costs more to waddle. Science, 195: 592--594. Schmidt-Nieisen, K., 1972. Locomotion: energy cost of swimming, flying and running. Science, 177: 222--228. Sharp, G.D. and Dotson, D.C., 1977. Energy for migration in albacore, Thunnus alalunga. Fish. Bull., U.S., 75(2): 447--450. Smit, H., 1965. Some experiments on the oxygen consumption of goldfish (Carassius auratus L.) in relation to swimming speed. Can. J. Zool., 43: 623--633. Tucker, V.A., 1968. Respiratory exchange and evaporative water loss in the flying budgerigar. J. Exp. Biol., 48: 67--87. Tucker, V.A., 1970. Energetic cost of locomotion in animals. Comp. Biochem. Physiol., 34A: 841--846. Wakeman, J.M. and Wohlschlag, D.E., 1979. Salinity stress and swimming performance of spotted seatrout. Proc. Annu. Conf. Southeast. Assoc. Fish Wildl. Agencies, 31: 357--361. Ware, D.M., 1978. Bioenergetics of pelagic fish: theoretical change in swimming speed and ration with body size. J. Fish. Res. Board Can., 35: 220--228. Webb, P.W., 1971. The swimming energetias of trout. II. Oxygen consumption and swimming efficiency. J. Exp. Biol., 55: 521--540. Webb, P.W., 1975. Hydrodynamics and energetics of fish propulsion. Fish. Res. Board Can., Bull. 190, 158 pp. Webb, P.W., 1978. Hydrodynamics: nonscombroid fish. In: W.S. Hoar and D.J. Randall (Editors), Fish Physiology. Vol. VII. Academic Press, New York, NY. Weihs, D., 1973. Optimal cruising speed for migrating fish. Nature, 245: 48--50. Weihs, D., 1975. An optimum swimming speed of fish based on feeding efficiency. Israel J. Technol., 13: 163--167. Weihs, D., 1977. Effects of size on sustained swimming speeds of aquatic organisms. In: T.J. Pedley (Editor), Scale Effects in Animal Locomotion. Academic Press, New York, NY. Weis-Fogh, T., 1952. Weight economy of flying insects. Trans. Ninth Int. Cong. Entomol., 1: 341--347. Winberg, G.G., 1956. Rate of metabolism and food requirements of fishes. Fish. Res. Board Can., Transl. Set. 194, 1960. Published in: Nauchnye Trudy Belorusskovo Gosudarstvennovo Universiteta imeni V. I. Lenina, Minsk (1956). Wohlschlag, D.E. and Wakeman, J.M., 1978. Salinity stresses, metabolic responses and distribution of the coastal spotted seatrout, Cynoscion nebulosus. Contrib. Mar. Sci. Univ. Tex., 21: 171--185.