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: enter image description here 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: enter image description here

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}]]

enter image description here

  • 2
    $\begingroup$ What would you say the main problem in OP's code was? The restricted range of u? $\endgroup$ – Chris K Apr 7 '18 at 15:14
  • $\begingroup$ 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. $\endgroup$ – Meagain Apr 8 '18 at 7:31

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