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(Index, Outline)

**Allometric law** (or power-law) describes the relationship between the body parts or processes within or among living organisms, usually expressed in power-law form:

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

Such **Allometric functions** ( are mathematical equations derived from the the study of allometry,

For example in Body size scaling relationships between a physiological quantity (such as the respiration rate, or the maximum reproduction rate) of organisms and their body size (frequently taken to be body weight)can be expressed mathematically in terms of a curve on a graph.

Many characteristics, ranging from brain size and heart rate to life span and population density, change consistently with body size. These relationships normally fit a simple power function. The use of logarithms makes the equation easier to visualize. The exponent becomes the slope of a straight line when the logarithm of the variable (say respiration rate) is plotted against the logarithm of body mass.

The terms **isometry**, **positive allometry**, and **negative allometry** are used in relation to the slope of the line. For example if heart rate, varies proportionally to body mass this is isometry; positive allometry is where the larger animals have proportionatly higher heart rates; while negative allometry is where larger animals have proportionately lower heart rate..

## ExamplesEdit

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.M^{\frac 1 6} m.s^{-1} $

## See alsoEdit

## ReferencesEdit

- 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