Let us consider the system
\begin{align*} \frac{dx}{dt} &= -a \frac{xy}{1+x} - x + y\\ \frac{dy}{dt} &= -bx + cxy - d\frac{xy}{1+x} + ey \end{align*}
$(0,0)$ is one of the fixed-point of the above system. I have calculated the Jacobian matrix at $(0,0)$.
I was wondering if the above system can exhibit Hopf bifurcation?
I tried to proceed analytically using the Hurwitz criterion mentioned in the Wiki for the vdP oscillator.
Applying the method, I am getting the following condition for Hopf bifurcation $0<b<1$ and $e=1$. Is this correct? can this be visualized?
EDIT:
Is there any test/ method to check for the Hopf bifurcation in many parameter systems such as this one?