Competition
Competition occurs when two populations use a shared limited resource, so that the presence of each population reduces the growth of the other.
A two-species competition model
\[\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1+\alpha_{12}N_2}{K_1}\right),\]\[\frac{dN_2}{dt}=r_2N_2\left(1-\frac{N_2+\alpha_{21}N_1}{K_2}\right).\]The coefficients \(\alpha_{12}\) and \(\alpha_{21}\) measure how strongly one species affects the other relative to within-species competition.
Possible outcomes
Depending on parameter values, one species may exclude the other, both may coexist, or the eventual winner may depend on the initial populations.
Key idea. Competition models convert the biological effect of shared limiting resources into coupled equations in which each species changes the other's per-capita growth rate.