Consider the nonlinear Kolmogorov system $$\frac{dx}{dt}=xf(x,y) \\ \frac{dy}{dt}=yg(x,y)$$ where $f,g$ are continuously differentiable functions in $\mathbb{R}^2$. Then prove that the above system has either a stable equilibrium point or a stable limit cycle if the following $5$ conditions and $4$ requirements are satisfied $${(1) \ \frac{\partial f}{\partial y}<0 \\ (2) \ \ x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}<0 \\ (3) \ \frac{\partial g}{\partial y}<0 \\ (4) \ x\frac{\partial g}{\partial x}+y\frac{\partial g}{\partial y}>0 \\ (5) \ f(0,0)>0}$$ and $${(1) \ f(0,A)=0 ; A>0 \\ (2) \ f(B,0)=0 ; B>0 \\ (3) \ g(C,0)=0 ; C>0 \\ (4) \ B>C}$$
Even though it seems easy to see that the system may have a stable equilibrium point using the $5$ conditions above along with Routh-Hurwitz criteria, but the other case that if the system has an unstable equilibrium point, then it must have a stable limit cycle around that point seems a little difficult to prove. I have a strong reason to believe that it has something to do with Poincaré-Bendixson theorem, but I'm not so sure. Any idea to prove this is appreciated.