These are the Lotka-Volterra equations for two species:

$$ \frac{dN_{1}}{dt} = (r_{1}N_{1}) \left( 1-\alpha_{11}N_{1} - \alpha_{12}N_{2} \right) $$

$$ \frac{dN_{2}}{dt} = (r_{2}N_{2}) \left( 1-\alpha_{22}N_{2} + \alpha_{21}N_{1} \right) $$

These equations calculate equilibrium values of N1 and N2:

$$ \frac{dN_{1}}{dt} = 0 \text{ when } N_{2} = - \frac{\alpha_{11}}{\alpha_{12}}N_{1} + \frac{1}{\alpha_{12}} $$

$$ \frac{dN_{2}}{dt} = 0 \text{ when } N_{2} = - \frac{\alpha_{21}}{\alpha_{22}}N_{1} + \frac{1}{\alpha_{22}} $$