= Positive trapping construction for the Leslie-Gower model
{title2=$P=ue^{-v}$}
A rectangular upper bound $u,v\leq M$, $M>1$, is inward pointing for the <Leslie-Gower predator-prey model>. To avoid the singular edge $u=0$, use $P=ue^{-v}$ and the inequality $\dot P/P\geq1-u[1+(a/d+b)^2/(4b)]$, obtained by completing a square. Choose $\varepsilon e^M$ sufficiently small and $0<\eta<\varepsilon$. Then $u\leq M$, $\eta\leq v\leq M$, $P\geq\varepsilon$ is a compact forward-invariant set in the positive quadrant. Removing an outward-flow neighborhood of a repelling equilibrium leaves a trapping region without equilibria, so the <Poincaré-Bendixson theorem> yields a periodic omega-limit orbit.
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