Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 2 / a / Solution

The first equation factors as $\dot{u}=u(\mu +u-v)$ and the second gives $v=u^2$ at an equilibrium. HenceThe 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 $\mu =s^2-s$ givesThus $P_{-}$ 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.