# The relationship between body mass and field metabolic rate among individual birds and mammals

**Allometry** is the study of the relationship of body size to shape,^{[1]} anatomy, physiology and finally behaviour,^{[2]} first outlined by Otto Snell in 1892,^{[3]} D’Arcy Thompson in 1917 in *On Growth and Form*^{[4]} and Julian Huxley in 1932.^{[5]} Allometry is a well-known study, particularly in statistical shape analysis for its theoretical developments, as well as in biology for practical applications to the differential growth rates of the parts of a living organism’s body. One application is in the study of variousinsect species (e.g., the Hercules Beetle), where a small change in overall body size can lead to an enormous and disproportionate increase in the dimensions of appendages such as legs, antennae, or horns. The relationship between the two measured quantities is often expressed as a power law:

- or in a logarithmic form:

where is the **scaling exponent** of the law. Methods for estimating this exponent from data use type 2 regressions such asmajor axis regression or reduced major axis regression as these account for the variation in both variables, contrary to least squares regression, which does not account for error variance in the independent variable (e.g., log body mass). Other methods include measurement error models and a particular kind of principal component analysis.

Allometry often studies shape differences in terms of ratios of the objects’ dimensions. Two objects of different size but common shape will have their dimensions in the same ratio. Take, for example, a biological object that grows as it matures. Its size changes with age but the shapes are similar. Studies of ontogenetic allometry often use lizards or snakes as model organisms because they lack parental care after birth or hatching and because they exhibit a large range of body size between the juvenile and adult stage. Lizards often exhibit allometric changes during their ontogeny.^{[6]}

In addition to studies that focus on growth, allometry also examines shape variation among individuals of a given age (and sex), which is referred to as static allometry. Comparisons of species are used to examine interspecific or evolutionary allometry (see also Phylogenetic comparative methods).

**Kleiber’s law**,^{[1]} named after Max Kleiber‘s biological work in the early 1930s, is the observation that, for the vast majority of animals, an animal’s metabolic rate scales to the ¾ power of the animal’s mass. Symbolically: if *q*_{0} is the animal’s metabolic rate, and *M* the animal’s mass, then Kleiber’s law states that *q*_{0} ~ *M*^{¾}. Thus a cat, having a mass 100 times that of a mouse, will have a metabolism roughly 32 times greater than that of a mouse. In plants, the exponent is close to 1.

The exponent for Kleiber’s law, which is called a power law, was a matter of dispute for many decades. It is still contested by a diminishing number as being ? rather than the more widely accepted ¾. Because the law concerned the capture, use, and loss of energy by a biological system, the system’s metabolic rate was, at first, taken to be ?, because energy was thought of mostly in terms of heat energy. Metabolic rate was expressed in energy per unit time, specifically calories per second. Two thirds expressed the relation of the square of the radius to the cube of the radius of a sphere, with the volume of the sphere increasing faster than the surface area, with increases in radius. This was purportedly the reason large creatures lived longer than small ones – that is, as they got bigger they lost less energy per unit volume through the surface, as radiated heat.

The problem with ? as an exponent was that it did not agree with a lot of the data. There were many exceptions, and the concept of metabolic rate itself was poorly defined and difficult to measure. It seemed to concern more than rate of heat generation and loss. Since what was being considered was not necessarily Euclidean geometry, the appropriateness of ? as an exponent was questioned. Kleiber himself came to favor ¾, and that is the number favored today by the foremost proponents of the law, despite that ¾ also does not agree with much of the data, and is also troubled with exceptions. Theoretical models presented by Geoffrey West, Brian Enquist, and James Brown,^{[6]} – known as the WBE model – purport to show how the ¾ observation can emerge from the constraint of how resources are distributed through hierarchical branching networks. Their understanding of an organism’s metabolic/respiratory chain is based entirely on blood-flow considerations. Their claims have been repeatedly criticized as mistaken, given that the role of fractal capillary branching is not demonstrated as fundamental to the exponent ¾; and that blood-flow claims severely limit the relevance of the equation to organisms greater than *e*^{?6} (? .0025) grams when the simultaneous claim is made that the equation is relevant over 27 orders of magnitude, extending from bacteria, which do not have hearts, to whales or forests.