Competition model

In a biological competition model, increasing one population reduces the growth of another. A two-species example is $\dot{u}=ru(1-u-v)$, $\dot{v}=v(1-\beta u)$, with both cross-effects negative. This asymmetric example has self-limitation only in $u$; disappearance of $u$ therefore permits unbounded growth of $v$. For $\beta >1$ its positive coexistence equilibrium is a saddle whose stable manifold separates exclusion outcomes.