I have this system of equations: \begin{equation} \dot{x}=(1-ux)x^{2/3}-x^{1/2}y \\ \dot{y}=p(x-1)y \\ x>0,y>0,0<u<1,p>1 \end{equation}
I am supposed to show that there are equilibrium points (which I've done: $(0,0),(1/u,0),(1,1-u)$) There is only one valid equilibrium point, $(1,1-u)$, an unstable focus when $p$ is large enough, given $u<1/2$. I want to find a positively invariant region that encloses the unstable focus. This can be done with a triangle using the lines $y=0,y=c(1-ux), y=dx-e$ for suitable choices of $c,d,e$ except for a neighborhood around the origin. Exclude the origin from your region by a curve that connects the first and second lines. I want to use Poincare-Bendixson theorem to show the existence of a periodic solution and for what parameter they exist/don't exist, and use this trapping region to do so. This question is originally from the book "Differential Dynamical Systems" by James D. Meiss. Below is an image of the phase-plane I made.
