= Logistic predator-prey model
{title2=$\dot x=x(a-bx-cy),\quad\dot y=y(dx-e)$}
This <Lotka-Volterra predator-prey model> includes self-limitation of the prey through the term $-bx^2$, with positive constants $a,b,c,d,e$. A positive coexistence <equilibrium point> exists when $a>be/d$, at $x_*=e/d$, $y_*=(a-bx_*)/c$. The <Lyapunov function>
$$
V=d[x-x_*-x_*\log(x/x_*)]
+c[y-y_*-y_*\log(y/y_*)]
$$
satisfies $\dot V=-bd(x-x_*)^2$: its predator-prey cross terms cancel. Its sublevel sets are compact in the positive quadrant. The only invariant subset of $\{\dot V=0\}$ is the coexistence <equilibrium point>, because staying on $x=x_*$ requires $y=y_*$. The <LaSalle invariance principle> therefore proves convergence to coexistence for every positive initial state.
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