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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: $\log y \sim a \log x \,\!$

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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 $V o p t$ of flying bodies (insects, birds, airplanes) and body mass $M$ in kg raised to the power $1 / 6$ :

$V_{opt} \sim 30.M^{\frac 1 6} m.s^{-1}$

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

Category: Mathematical biology