A non-constant periodic solution of a nonlinear dynamics system

Question

I had a problem with this exercise.

A generalization of a predator-prey system given by Brauer and Castillo-Chavez is

\begin{align*}
    dx/dt = x(1 - x/30 - y/(x+10))
    dy/dt = y(x/(x+10) - 1/3)
\end{align*}

I found that the fixed points are (0, 0), (30, 0), and (5, 12.5) and have saddle, saddle, source classification, respectively. The region characterized by 'x >= 0, y >= 0', and 'x + y <= 50' is a trapping region.

I am trying to figure out whether the system has a non-constant periodic solution. I tried to find its gradient, Lyapunov function and Dulac's criteria, but they didn't work.

I drew phase plane and I saw that there is no limit cycle, I expect to show that there is no limit cycle and thus no non-constant periodic solution.