= Solution
The first equation factors as $\dot u=u(\mu+u-v)$ and the second gives $v=u^2$ at an <equilibrium>. Hence
$$
\boxed{O=(0,0),\qquad P_\pm=(s_\pm,s_\pm^2),\qquad
s_\pm=\frac{1\pm\sqrt{1+4\mu}}2.}
$$
The two nonzero branches exist for $\mu\ge-1/4$, coincide at equality, and the lower branch coincides with the origin at $\mu=0$. For $-1/4<\mu<0$ both are in $u>0$; for $\mu>0$, $P_-$ has $u<0$ and $P_+$ has $u>0$.
At the origin, the <eigenvalues> are $\mu,-\lambda$. At any nonzero <equilibrium> $(s,s^2)$, using $\mu=s^2-s$ gives
$$
J_s=\begin{pmatrix}s&-s\\2\lambda s&-\lambda\end{pmatrix},
\quad \operatorname{tr}J_s=s-\lambda,\quad \det J_s=\lambda s(2s-1).
$$
Thus $P_-$ is a <saddle equilibrium> for $-1/4<\mu<0$, and a <sink equilibrium> for $\mu>0$ when it lies in the negative half-plane. The upper branch has positive <determinant> and is stable for $s_+<\lambda$, unstable for $s_+>\lambda$. These statements include the existence and stability information needed for the subsequent diagrams.
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