We are asked to analyze the one-dimensional system $\frac{dx}{dt}=f(x)=x-rx(1-x)=rx^2+(1-r)x$.
The system has a fixed point for all values of $r$ at $x^{\star}=0$, and the algebraic form shows the system undergoes a transcritical bifurcation at $r_{c}=1$ (alternatively, we could apply the tangency condition for standard bifurcations (saddle-node, transcritical, and pitchfork) to get the $r_{c}$ value).
A quick graphical analysis (or, perhaps a linear stability analysis) of the different cases let's us quickly sketch the associated dynamics near $x_{c}=0$ as $r$ varies (e.g. bifurcation diagram, vector flows, etc.). It is easy to see after doing this that the system indeed undergoes a transcritical bifurcation at $(x_{c},r_{c})=(0,1)$ (stabilities interchange). That's fine, I understand this well enough.
My question is this: there is clearly another bifurcation at $(x_{c},r_{c})=(0,0)$. For $r<0$ there is an unstable fixed point at the origin, and a positive stable fixed point which tends to $x=1$ as $r\to-\infty$ and to $+\infty$ as $r\to0$. Once $r=0$, there is only an unstable fixed point at the $x=0$. Once $r>0$, the stable fixed point that was at $+\infty$ becomes a stable point at $-\infty$ and tends to $0$ as $r\to1$, and this is of course the transcritical bifurcation.
So what is going on at $(x_{c},r_{c})=(0,0)$? The text we're using seems to completely ignore this bifurcation point, and simply plots it and makes no further comment about it. I looked at solutions to this very problem from other sources, and they also make no detailed analysis of this point. (See here at the very last problem, for example: http://www.personal.psu.edu/axm62/math449%20sheets/math449HW3soln.pdf).