A discrete model for a population $N_{t}$ consists of

$$N_{t+1}=\frac{r N_{t}}{\left(1+b N_{t}\right)^{2}},$$

where $t$ is discrete time and $r, b>0$. What do $r$ and $b$ represent in this model? Show that for $r>1$ there is a stable fixed point.

Suppose the initial condition is $N_{1}=1 / b$, and that $r>4$. Show, with the help of a cobweb, that the population $N_{t}$ is bounded by

$$\frac{4 r^{2}}{(4+r)^{2} b} \leqslant N_{t} \leqslant \frac{r}{4 b}$$

and attains those bounds.