I have the following non-dimensional model:
$\dot{u}=u(1-u)-X\frac{u}{\alpha+u}$
Now, assuming $\alpha>1$ and $X>0$ are constants, I want to show a bifurcation happens locally near the zero state when $X=\alpha$. I have tried setting the equation equal to zero to see if the equilibrium states change, but I dont seem to get anywehre with it. I was also thinking of maybe using taylor expansion of the denominator of the fractional term, but also havent gotten far using that. Any ideas how this can be shown?
Thanks