Application of the Exponential and Power Functions to the Study of Allometric Growth, with Particular Reference to Doryline Ants

Abstract

Studies of allometry, or relative growth, have used the power function, y = axb, where a and b are constants, almost exclusively to describe measurements made on some organ y in terms of measurements made on a reference organ x. For this type of application the exclusive use of the power function is questioned on both empirical and logical grounds, and use of the exponential function y = abx is suggested as an alternative. Data representing measurements on army-ant larvae are presented in which a better fit is obtained through use of the exponential function. Fictitious data are also presented to illustrate a situation where failure to consider using the exponential function could lead to a possible misapplication of the power function. Logically, it is demonstrated that the power and exponential functions can be differentiated on the basis of how x and y vary separately as a function of time. If both organs (x and y) are exponential functions of time, the allometric relationship is of the power form. If the organ y is exponential and the reference organ x is linear (as functions of time), the allometric relationship is exponential. Finally, it is suggested that the choice of a function to describe an allometric relationship should also be based, in part, on statistical criteria: What function provides an adequate and simple fit to the data? In particular, the power function should not be regarded as an inherent law of relative growth.