Allometric law

An allometric law describes the relationship between two attributes of living organisms, and is usually expressed as a power-law:

: $y\; propto\; x^\{a\}\; ,!$ or in a logarithmic form: $log\; y\; sim\; a\; log\; x\; ,!$

where $a$ is the scaling exponent of the law. Methods for estimating this exponent from data tend to involve a particular kind of principal component analysis.

Examples

Some examples of allometric laws:
* Kleiber's law, the proportionality between metabolic rate $q\_\{0\}$ and body mass $M$ raised to the power $3/4$: : $q\_\{0\}\; sim\; M^\{frac\; 3\; 4\}$
* the proportionality between breathing and heart beating times $t$ and body mass $M$ raised to the power $1/4$: : $t\; sim\; M^\{frac\; 1\; 4\}$
* mass transfer contact area $A$ and body mass $M$: : $A\; sim\; M^\{frac\; 7\; 8\}$
* the proportionality between the optimal cruising speed $V\_\{opt\}$ of flying bodies (insects, birds, airplanes) and body mass $M$ in "kg" raised to the power $1/6$:: $V\_\{opt\}\; sim\; 30\; cdot\; M^\{frac\; 1\; 6\}\; m\; cdot\; s^\{-1\}$

ee also

* allometry
* constructal law
* Rensch's rule
* scaling law
* square-cube law
* Metabolic theory of ecology

References

* A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000. ISBN 0-521-79388-2
* A. Bejan, Constructal theory of organization in nature: dendritic flows, allometric laws and flight, Design and Nature, CA Brebbia, L Sucharov & P Pascola (Editors). ISBN 1-85312-901-1