Allometric Scaling and Central Source Systems

Allometric scaling relations abound in nature. Examples include the power law relating the metabolic rate of animals and plants to their masses and the power law describing the dependence of the size of the drainage basin of a river on the total amount of water contained in that river. The exponent is of the form $D/D+1\mathrm{}$, where $D$ is the dimension of the system. We show that this scaling exponent is simply a consequence of the source distribution of the systems considered and requires no further assumptions. To demonstrate the wide range of validity of the result we present a simple experiment that shows the predicted behavior in one dimension.