The right plot to use for bifurcation diagram

I have on my hand the differential equation $\dot{u} = ru(1-\frac{u}{q})-\frac{u^2}{1+u^2}$ for which I need to analyse its bifurcations. I know that the bifurcations occurs at $q(u) = \frac{2u^3}{u^2-1}$ and $r(u) = \frac{2u^3}{(1+u^2)^2}$ but I have trouble plotting it. This question comes from the book Mathematical Biology: I. An Introduction,Third Edition by J.D.Murra on page 8-9 and the correct plot looks like this: According to the book this is meant to be given paraetrically by $q$ and $r$ defined above for $u \geq \sqrt{3}$. But if I go ahead and try plotting them using ParametricPlot it givens me the following nonsense: Code I used was as follows: ParametricPlot[{(2 x^3)/(x^2 - 1), (2 x^3)/(1 + x^2)^2}, {x, 3^(1/2), 4}].

Could somebody tell me what is going wrong and how do I fix this? Thanks in advance!

q[u_] := (2 u^3)/(u^2 - 1)
r[u_] := (2 u^3)/(1 + u^2)^2
Show[ParametricPlot[{q[u], r[u]}, {u, 0, 30},
AspectRatio -> 1/GoldenRatio, PlotRange -> {{4, 30}, {-0.1, 0.8}},
Ticks -> {{0, 0}, {0, 1/2}}, AxesOrigin -> {4, 0},
LabelStyle -> Directive[Black, Large],
AxesStyle -> Directive[Black, Thick], PlotStyle -> {Black, Thick},
AxesLabel -> {Style[q, Large, Thick], Style[r, Large, Thick]}],
Plot[1/2, {x, 2, 50}, PlotStyle -> {Black, Thick}]] • What would you say the main problem in OP's code was? The restricted range of u? – Chris K Apr 7 '18 at 15:14
• This is exactly what I needed! Thanks a lot! My problem was that when the graph is zoomed out sufficiently the cusp becomes impossible to detect. – Meagain Apr 8 '18 at 7:31