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

$$N_{t+1}=\frac{r N_{t}}{1+b N_{t-1}^{2}}$$

where $t$ is discrete time, $r>1$ and $b>0$. Investigate the linear stability about the positive steady state $N^{*}$. Show that $r=2$ is a bifurcation value at which the steady state bifurcates to a periodic solution of period 6 .