I have the Lotka-Volterra equation
where $x$ and $y$ are non-negative. It is easy to see that the $x$- and $y$-axis are invariant sets. I can see from plots that a periodic orbit exists. Now I want to prove this by hand calculations according to the Poincare-Bendixon criterion.
So, is there any invariant set that is a closed and bounded trajectory and contains no equilibrium point, which I can find by hand calculations?