Entropy-shaped Lyapunov function for weak competition ($V={\sum }_i{w}_i[u_i-k-k\log (u_i/k)]$)

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.