A population model for two species is given by

$$\begin{aligned} &\frac{d N}{d t}=a N-b N P-k N^{2} \\ &\frac{d P}{d t}=-d P+c N P \end{aligned}$$

where $a, b, c, d$ and $k$ are positive parameters. Show that this may be rescaled to

$$\begin{aligned} &\frac{d u}{d \tau}=u(1-v-\beta u) \\ &\frac{d v}{d \tau}=-\alpha v(1-u) \end{aligned}$$

and give $\alpha$ and $\beta$ in terms of the original parameters.

For $\beta<1$ find all fixed points in $u \geqslant 0, v \geqslant 0$, and analyse their stability. Assuming that both populations are present initially, what does this suggest will be the long-term outcome?