Spruce Budworm bifurcation diagram

Question

The ODE that appears in the Spruce Budworm Population problem is the following: $$\frac{dN}{dt}=r_PN\left(1-\frac{N}{K}\right) -p(N) \quad \quad \mbox{where} \quad \quad p(N)=\frac{BN^2}{A^2+N^2}$$ where $A$, $B$ are constants. After some variable changes (you can check them on pages $4$ and $5$ of the document liked above) we get to the following ODE $$\frac{du}{dτ} = ru \left(1 −\frac{u}{q}\right)−\frac{u^2}{1+u^2}$$ The bifurcation diagram, where we plot the parametric curves $r=\frac{2u^3}{(1+u^2)^2}$, $q=\frac{2u^3}{u^2-1}$ is this one. I want to know how can I find the coordinates of the point $A$ in the graph linked. I already know their values but I want to know a method to manually find their values. Thank you for your help

Answer 1

The cusp is only possible when neither $r$ nor $q$ provides a local coordinates, i.e., $r'(u)=q'(u)=0$. You just have to solve it and get $u=0,\pm\sqrt3$ and you must go back and chck indeed that is a cusp, not a problem of silly parametrization like $y=x=t^2$ at $t=0$.