Leslie-Gower predator-prey model

The system $\dot{u}=u(1-u)-auv/(u+d)$, $\dot{v}=bv(1-v/u)$ combines logistic prey growth, saturating predation and a predator carrying level proportional to prey density. Its unique positive equilibrium is $u=v=s$, where $(1-s)(s+d)=as$. Its Jacobian matrix has determinant $bs\sqrt{(1-a-d{)}^2+4d}/(s+d)>0$ and trace $s(1-d-2s)/(s+d)-b$. The positive trapping construction for the Leslie-Gower model gives a compact positive region and a limit cycle whenever this equilibrium is a repeller.

A rectangular upper bound $u,v\le M$, $M>1$, is inward pointing for the Leslie-Gower predator-prey model. To avoid the singular edge $u=0$, use $P=u{e}^{-v}$ and the inequality $\dot{P}/P\ge 1-u[1+(a/d+b{)}^2/(4b)]$, obtained by completing a square. Choose $\epsilon e^M$ sufficiently small and $0<\eta <\epsilon$. Then $u\le M$, $\eta \le v\le M$, $P\ge \epsilon$ 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.