Editing Competitive Lotka–Volterra equations (section)

===Two species===
Given two populations, {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions.  Thus the competitive Lotka–Volterra equations are:
<math display="block">\begin{align}
{dx_1 \over dt} &= r_1 x_1\left(1-\left({x_1+\alpha_{12}x_2 \over K_1}\right) \right) \\[0.5ex]
{dx_2 \over dt} &= r_2 x_2\left(1-\left({x_2+\alpha_{21}x_1 \over K_2}\right) \right).
\end{align}</math>

Here, {{math|''α''<sub>12</sub>}} represents the effect species 2 has on the population of species 1 and {{math|''α''<sub>21</sub>}} represents the effect species 1 has on the population of species 2.  These values do not have to be equal.  Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all ''α''-values are positive.  Also, note that each species can have its own growth rate and carrying capacity. A complete classification of this dynamics, even for all sign patterns of above coefficients, is available,<ref>{{cite journal | last=Bomze | first=Immanuel M. |author-link=Immanuel Bomze| title=Lotka-Volterra equation and replicator dynamics: A two-dimensional classification | journal=Biological Cybernetics | publisher=Springer Science and Business Media LLC | volume=48 | issue=3 | year=1983 | issn=0340-1200 | doi=10.1007/bf00318088 | pages=201–211| s2cid=206774680 }}</ref><ref>{{cite journal | last=Bomze | first=Immanuel M. |author-link=Immanuel Bomze| title=Lotka-Volterra equation and replicator dynamics: new issues in classification | journal=Biological Cybernetics | publisher=Springer Science and Business Media LLC | volume=72 | issue=5 | year=1995 | issn=0340-1200 | doi=10.1007/bf00201420 | pages=447–453| s2cid=18754189 }}</ref> which is based upon equivalence to the 3-type [[replicator equation]].