Show that system is Transcritical bifurcation In what ways can you show that transcritical bifurcation occurs? For example take the system 
$$\dfrac{dx}{dt}=xr+2x^2 $$

Hints:
We allow $r$ to vary and it can be less, equal or greater than zero.
For these three ranges, we have the critical points:


*

*$r \lt 0 \rightarrow  -r x + 2x^2 = 0 \implies x = 0, x = \dfrac{r}{2}$

*$r = 0 \rightarrow 2x^2 = 0 \implies x = 0$

*$r \gt 0 \rightarrow r x + 2 x^2 = 0 \implies x = 0, x = -\dfrac{r}{2}$


Note that for all three choices of $r$, $x  = 0$ is a critical point.
Now, we can plot a phase portrait ($x' = rx + 2x^2, y' = -y$) for $r \lt 0, r = 0, r \gt 0$ to determine the stability of the critical points.



You should be able to determine the stability of each critical point and then draw the bifurcation diagram in the $rx-plane$. Here is a start. Add in the dashed lines and the stability information.

