Volterra-Lotka equations
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The Volterra-Lotka equations, also known as the prey-predator equations, are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems.

The usual form of the equations is:

The equations were proposed by Vito Volterra and A. J. Lotka in the 1920s to characterise the dynamics of a biological system where:

y is the number of some predator (for example, foxes);

x is the number of its prey (for example, rabbits);

t represents the development of the two populations against time; and

a, b, c and d are parameters respresenting the interaction of the two species.

The prey are supposed to have unlimited food and to reproduce indefinitely unless subject to predation. The predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline.

The equations have periodic solutions which do not have a simple expression in terms of the usual trigonometric functions. However, an approximate linearised solution yields a simple harmonic motion with the population of predators leading that of prey by 90°.

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Bibliography

• E. R. Leigh (1968) The ecological role of Volterra's equations, in Some Mathematical Problems in Biology - a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.

The original article can be found at: http://en.wikipedia.org/wiki/Volterra-Lotka_equations.
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