Scaling of Jumping Performance in Anuran Amphibians

Abstract

-Theories on the scaling of saltatory locomotion were examined using data of anuran jump performance. We explored whether take-off speed in geometrically similar animals would be the same regardless of body mass (an energy output model), or whether it would scale with mass0?11 (a power output model). Two frog species in central Panama, Eleutherodactylus fitzingeri (powerful jumpers, 0.14-8.65 g) and Bufo typhonius (slow hoppers; 0.16-11.60 g), were used for morphometric and take-off potential measurements. Like those of other species, our frog species showed isometric body shape in general in terms of muscle mass, hindlimb length, and interilial width. Scaling exponents for take-off speed were 0.16-0.17 for the two species and were not statistically deviated from those predicted by the power output approach. This conclusion would be supported by the fact that burst-type locomotion relies on the rate of energy release, not the total amount of energy released, by jumping muscles. A theory on scaling of saltatory locomotion postulates that geometrically similar animals exhibit the same jump capacity (height, distance, speed) regardless of size, because the energy output per muscle mass required for takeoff is essentially the same among these animals (Hill, 1950). This theory, however, has been challenged by empirical tests on various anuran species where results showed that jump distance scaled with body mass to the power of 0.2-0.3 (Zug, 1978; Emerson, 1978; Miller et al., 1993). Deviations of the empirical results from theoretical predictions led us to consider an alternative model for the scaling problem. BennetClark (1977) and Gabriel (1984) emphasized that power output (rate of energy release), rather than energy availability, produced by jumping muscles would be the major factor determining the locomotory scaling in relatively small animals. The present study revisits the theoretical bases of the two approaches, and uses the power output approach to try to explain how jump capacity scales allometrically with body mass. Theories associated with take-off speed Energy output approach.-This theory is based on the concept that the specific work done by the jumping muscle is the same among isometric animals (Hill, 1950). At maximal jumping, all extensor muscles of frog hindlimbs contract simultaneously. The average muscle force (F,) and the length (1) by which the muscles shorten yield energy output (work) for the take-off. Because force is proportional to cross-sectional area (A) and muscle shortening to the initial * To whom correspondence should be addressed. muscle length (L), the work (E) should be proportional to muscle mass and thus to body mass (m). E = F 1 cx A L x m, and so E/m = constant. Eq. (1) This work is converted into kinetic energy that is used by the frog to create its takeoff velocity (v). E = mv2/2, or E/m O v2 = constant. Eq. (2) Jump distance (D) can be defined with (v), takeoff angle (0), and the gravitational constant (g) as follows. D = v2sin20 /g. Eq. (3) If take-off angle is assumed to be similar among the animals, combining the above equations gives D oc E/m = constant. Eq. (4) Equations (2) and (4) confirm that take-off capacity by isometric animals is the same regardless of size. Power output approach.-This theory is based on the concept that the jump power of a whole animal (P,) is proportional to mechanical power (Pm) of its extensor muscles (Lutz and Rome, 1994). Because power is a product of force multiplied by velocity, Pi c Pm, and thus F, v x F,, V, Eq. (5) where Fj is the ground reaction force exerted by the hindlimbs at take-off and V is the average shortening velocity of extensors. Fi is given by Fi = mv'/2s, or F,i mv2/LH. Eq. (6) Here, s and LH are the acceleration distance and This content downloaded from 157.55.39.123 on Mon, 18 Jul 2016 06:01:23 UTC All use subject to http://about.jstor.org/terms SCALING OF ANURAN TAKE-OFF PERFORMANCE the hindlimb length, respectively. From equations (5) and (6), v oc (Fm V)/(mv2/LH), or v3 cc (F, V)/(m/LH). Eq. (7) As Fm is proportional to muscle cross-sectional area and V is proportional to muscle length, Pm is proportional to muscle mass and thus body mass (m). Equation (7) can be re-written as