Curvature in metabolic scaling

For more than three-quarters of a century it has been assumed that basal metabolic rate increases as body mass raised to some power p . However, there is no broad consensus regarding the value of p : whereas many studies have asserted that p is 3/4 ("Kleiber's law"), some have argued that it is 2/3, and others have found that it varies depending on factors like environment and taxonomy. Here we show that the relationship between mass and metabolic rate has convex curvature on a logarithmic scale, and is therefore not a pure power law, even after accounting for body temperature. This finding has several consequences. First, it provides an explanation for the puzzling variability in estimates of p , settling a long-standing debate. Second, it constitutes a stringent test for theories of metabolic scaling. A widely debated model based on vascular system architecture fails this test, and we suggest modifications that could bring it into compliance with the observed curvature. We extensively characterize a large class of such modifications, both for finite masses and asymptotically. Detailed exploration of such models demonstrates that they, in general, possess limiting asymptotic slopes and so behave like power laws for animals of very large mass. This appears to be inconsistent with the observed data, for which the slope of the scaling relation seems to grow without bound. Third, it raises the intriguing question of whether the scaling relation limits body size.