Allometric law

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.


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}

See also


  • 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
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