Two interacting populations of prey and predators, with populations $u(t), v(t)$ respectively, obey the evolution equations (with all parameters positive)

$$\begin{aligned} &\frac{d u}{d t}=u\left(\mu_{1}-\alpha_{1} v-\delta u\right) \\ &\frac{d v}{d t}=v\left(-\mu_{2}+\alpha_{2} u\right)-\epsilon \end{aligned}$$

Give an explanation in terms of population dynamics of each of the terms in these equations.

Show that if $\alpha_{2} \mu_{1}>\delta \mu_{2}$ there are two non-trivial fixed points with $u, v \neq 0$, provided $\epsilon$ is sufficiently small. Find the trace and determinant of the Jacobian in terms of $u, v$ and show that, when $\delta$ and $\epsilon$ are very small, the fixed point with $u \approx \mu_{1} / \delta$, $v \approx \epsilon \delta / \mu_{1} \alpha_{2}$ is always unstable.