## 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: * Bret Victor, [Interactive exploration of a dynamical system](http://vimeo.com/23839605/). You can run a very appealing version on your web browser, and download the source code, here: * JSXGraph team, [Carps and pikes](http://jsxgraph.uni-bayreuth.de/showcase/carpsandpikes.html). ## 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]]. ## References ## See also [[logistic equation]]. * Wikipedia, [Lotka-Volterra equation](http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation) and [Competitive Lotka-Volterra equations](http://en.wikipedia.org/wiki/Competitive_Lotka%E2%80%93Volterra_equations). * A review (10 years old) in chapter 2 of [Lee Worden's dissertation](http://leeworden.net/pubs/thesis.pdf). category:mathematical methods [[!redirects Lotka-Volterra equations]] [[!redirects Predator-Prey equations]] [[!redirects predator-prey equations]] [[!redirects predator-prey]] [[!redirects Lotka-Volterra]]