Allometric law

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..

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