Competitive Lotka-Volterra model ($\dot{u}=u(a-u-Av),\dot{v}=\rho v(b-v-Bu)$)

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.

For $\dot{n}=n(1-n-m/2)$, $\dot{m}=qm(1-m-n/2)$, choose $k=2/3$ and weights $1,1/q$. Then $\dot{V}=-(n-k{)}^2-(m-k{)}^2-(n-k)(m-k)$, strictly negative away from coexistence. The function diverges at the positive-quadrant boundary and at infinity, keeping trajectories in compact interior level sets. This proves global convergence for positive initial states, while boundary-axis data retain their single-species dynamics.