Consider a model of a population $N_{\tau}$ in discrete time

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

where $r, b>0$ are constants and $\tau=1,2,3, \ldots$ Interpret the constants and 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, using a cobweb diagram, that the population $N_{\tau}$ is bounded as

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

and attains the bounds.