= Entropy-shaped Lyapunov function for weak competition
{title2=$V=\sum_i w_i[u_i-k-k\log(u_i/k)]$}
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.
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