I'm reading an article on internet. It states the following:
Consider the following extension of the celebrated Lotka-Volterra system $$x'= x(ax+by+c), y'= y(dx+ey+ f)$$ where all the parameters are real numbers. It appears in most texts books of mathematical ecology. By uniqueness of solutions it is clear that if it has periodic orbits then they do not intersect the coordinate axes. By making the change of variables $x → ±x, y → ±y$, if necessary, we can restrict our attention to the first quadrant U and prove that the system has no periodic orbit in it.
I don't see it as clear as the author. I've been trying to figure it out but I can't prove why.
Thanks for your time.
If a solution touches one of the coordinate axes, say $\{x=0\}$, then it touches one of the orbits of the ODE $\dot{y}=ey^2+fy$. By uniqueness, the given solution must lie entirely within $\{x=0\}$.