I'm trying to find the critical points and bifurcation points of the following ODE:
$\frac{\mathrm{d}x}{\mathrm{d}t} = \beta{x} - \frac{x}{1 + x}$
I know that there are two critical points, $x = 0$ and $x = \frac{1}{\beta} - 1$. Also, the bifurcation diagram looks like (ignore the line at $x = -1$, it's an artifact of my Matlab code):
I know that there's a transcritical bifurcation at $x = 0$ with $\beta = 1$. However, when $\beta = 0$, there's only one critical point as it's an asymptote for $\frac{1}{\beta} - 1$.
My question is: is this a bifurcation point? If so, what type would it be, and what would the bifurcation point be (the $x$ value)?