Basic functions: Power functions
Context 1: Kleiber's law and allometry between brain weight and body weight
Power functions you mostly encounter in biological or medical studies in which sizes of body parts are measured during growth and in which interrelation or relationships with other quantities are determined empirically. The English name for this area is allometry.
A well-known example is the Kleiber's law describing the relationship between body weight and energy requirements of mammals: the metabolic rate is proportional to \(m^\frac{3}{4}\), where \(m\) is the body weight of the animal. However, it is doubted whether a single formula for all mammals works. But when one restricts the relationship for example to the class of marsupials, then this law apply. This describes that larger animals comparatively spend less energy than smaller animals: the metabolic rate per kg of weight is proportional to \(\displaystyle \frac{m^\frac{3}{4}}{m}=m^{-\frac{1}{4}}\). Again this is a power law.
Related to the above example is the relationship between body weight and the weight of the brains of mammals. For primates, the following allometric connection is used (Martin, RD (1983) Human Brain Evolution in an Ecological Context New York. American Museum of Natural History) \[B=0.11482\,G^{0.75}\] where the body weight (\(G\)) and the weight of the brains (\(B\)) are in grams. For non-primates, a similar formula holds, but with a slightly different factor: \[B=0.05495\,G^{0.74}\]
Suppose that there is a power relation
\[y=c\cdot x^p\]
between magnitudes \(y\) and \(x\), then the coefficient \(p\) is called the allometric factor and one says that \(y\) scales as the \(p\)-th power of \(x\). A much used notation for this type of scaling is \(y\propto x^p\).
If \(p=1\), then \(y\) is directly proportional to \(x\) and there is a straight line relationship.
if \(p=-1\)), then \(y\) is inversily proportional to \(x\).
