= Competitive Lotka-Volterra model
{c}
{title2=$\dot u=u(a-u-Av),\quad\dot v=\rho v(b-v-Bu)$}
For $A,B,\rho>0$, these <Lotka-Volterra equations> describe two populations limited by their own density and depressed by the other population. The nonnegative quadrant is invariant. Positive <equilibria> are intersections of the non-axis <nullclines> $u+Av=a$ and $Bu+v=b$; their existence and stability distinguish stable coexistence, competitive exclusion, and <bistability>. Positive solutions have no periodic orbits: multiplying the vector field by the <Dulac function> $1/(uv)$ makes its divergence $-1/v-\rho/u<0$, so the <Bendixson-Dulac theorem> applies. Boundedness, boundary equilibria, and their stable manifolds must still be checked before concluding convergence to one particular equilibrium.
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