Lotka-Volterra inter-specific competition equations

The Lotka-Volterra inter-specific competition equations are a simple model of the population dynamics of two species competing for some common resource. The form is similar to the Lotka-Volterra equations for predation in that the equation for each species has one term describing the effect of the other species. In the equations for predation, the base population model is exponential. For the competition equations, the logistic equation was the basis.

The logistic population model, when used by ecologists often takes the following form:

<math>{dN \over dt} = rN({K-N \over K})<math>

Here N is the size of the population at a given time, r is inherent per-capita growth rate, and K is the carrying capacity. Given 2 populations, N₁ and N₂, with logistic dynamics, the Lotka-Volterra formulation adds an additional term to account for the species' interactions. Thus the Lotka-Volterra inter-specific competition equations are:

<math>{dN_1 \over dt} = rN_1({K-N_1-\alpha_{21}N_2 \over K})<math>

<math>{dN_2 \over dt} = rN_2({K-N_2-\alpha_{12}N_1 \over K})<math>

Here, <math>\alpha_{12}<math> represents the effect species 1 has on the population density of species 2 and <math>\alpha_{21}<math> represents the effect species 2 has on the population density of species 1.