# The Azimuth Project 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:

## Details

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

${d x_1 \over d t} = r_1x_1\left(1-\left({x_1+\alpha_{12}x_2 \over K_1}\right) \right)$
${d x_2 \over d t} = r_2x_2\left(1-\left({x_2+\alpha_{21}x_1 \over K_2}\right) \right)$

Here $x_i$ is the population of the $i$th species ($i = 1,2$), with equilibrium population $K_i$ in the absence of the other species, and growth rate $r_i$. The constant $\alpha_{12}$ represents the effect species 2 has on the population of species 1, while $\alpha_{21}$ 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, $\alpha_{12}$ should be negative and $\alpha_{21}$ should be positive. To describe competition, $\alpha_{12}$ and $\alpha_{21}$ should be positive.

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