= Solution
At $\mu=-1/4$, the two branches meet at $(1/2,1/4)$. The <eigenvalues> there are $0$ and $1/2-\lambda$. For $\lambda\ne1/2$, the nonzero quadratic term and transverse stability identify a <saddle-node bifurcation>. If $\lambda>1/2$, it creates a <sink equilibrium> and a <saddle equilibrium> as $\mu$ increases; if $\lambda<1/2$, it creates a <source equilibrium> and a <saddle equilibrium>. At $\lambda=1/2$, the zero <eigenvalue> is double and the ordinary saddle-node description needs the two-parameter unfolding.
At $\mu=0$, the origin and lower branch cross. Since $s_-=-\mu+O(\mu^2)$, the branches cross transversely in the $(\mu,u)$ diagram. The origin changes from a <sink equilibrium> to a <saddle equilibrium> as $\mu$ increases through zero; the lower branch changes from a positive-$u$ <saddle equilibrium> to a negative-$u$ <sink equilibrium>. This is a <transcritical bifurcation>. A local centre-manifold calculation also gives $\dot u=\mu u+u^2+\cdots$, because $v=O(u^2)$ and the transverse <eigenvalue> is $-\lambda<0$.
The figure uses a representative $\lambda>1/2$; the upper branch's additional Hopf change is indicated. For $\lambda<1/2$ the upper branch is unstable everywhere on its existence interval. The exact <trace> criterion in part (a) specifies stability for every fixed positive $\lambda$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-56-quadratic-bifurcations.png]
{title=Equilibrium branches and the two-parameter diagram of the quadratic-product flow}
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