Consider the system of predator-prey equations

$$\begin{aligned} &\frac{d N_{1}}{d t}=-\epsilon_{1} N_{1}+\alpha N_{1} N_{2} \\ &\frac{d N_{2}}{d t}=\epsilon_{2} N_{2}-\alpha N_{1} N_{2} \end{aligned}$$

where $\epsilon_{1}, \epsilon_{2}$ and $\alpha$ are positive constants.

(i) Determine the non-zero fixed point $\left(N_{1}^{*}, N_{2}^{*}\right)$ of this system.

(ii) Show that the system can be written in the form

$$\frac{d x_{i}}{d t}=\sum_{j=1}^{2} K_{i j} \frac{\partial H}{\partial x_{j}}, \quad i=1,2$$

where $x_{i}=\log \left(N_{i} / N_{i}^{*}\right)$ and a suitable $2 \times 2$ antisymmetric matrix $K_{i j}$ and scalar function $H\left(x_{1}, x_{2}\right)$ are to be identified.

(iii) Hence, or otherwise, show that $H$ is constant on solutions of the predator-prey equations.