Global existence of solution of ODE - Gray-Scott model

I am dealing with the ODE version of Gray-Scott model:\begin{equation}\begin{split}\dot{x} &= -xy^2 + F(1-x)\\\dot{y} &= xy^2 - (F + k)y,\end{split}\end{equation}where $F>0$ and $k>0$ are parameters. I would like to prove global existence of solution. I tried to find bound for the solution using functions of type $V(x,y)= x^2 + y^2$ in hopes of finding such function that has non-positive derivative. I also tried elimination of time variable to investigate the solution curves.

Does anyone have an idea how to do it? I am not sure whether global solution exists for all initial conditions. Any help or references will be greatly appreciated.