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Lotka-Volterra equation

Idea

The Lotka-Volterra equation is a simple model of predator-prey interactions. A variant describes two species competing for the same resources.

You can download some nice software for visualizing solutions of the Lotka-Volterra equation, and watch a video demonstration of it, at:

You can run a very appealing version on your web browser, and download the source code, here:

Details

The Lotka-Volterra equation is a actually a pair of coupled differential equations, similar to the logistic equation but more fancy:

dx 1dt=r 1x 1(1(x 1+α 12x 2K 1)){d x_1 \over d t} = r_1x_1\left(1-\left({x_1+\alpha_{12}x_2 \over K_1}\right) \right)
dx 2dt=r 2x 2(1(x 2+α 21x 1K 2)){d x_2 \over d t} = r_2x_2\left(1-\left({x_2+\alpha_{21}x_1 \over K_2}\right) \right)

Here x ix_i is the population of the iith species (i=1,2i = 1,2), with equilibrium population K iK_i in the absence of the other species, and growth rate r ir_i. The constant α 12\alpha_{12} represents the effect species 2 has on the population of species 1, while α 21\alpha_{21} represents the effect species 1 has on the population of species 2. These values do not have to be equal.

To describe predator-prey interactions where species 1 is the predator and species 2 is the prey, α 12\alpha_{12} should be negative and α 21\alpha_{21} should be positive. To describe competition, α 12\alpha_{12} and α 21\alpha_{21} should be positive.

The Lotka-Volterra equation can be derived as the rate equation of a stochastic Petri net.

References

See also logistic equation.