 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.