A delay model for a population of size $N_{t}$ at discrete time $t$ is given by

$$N_{t+1}=\max \left\{\left(r-N_{t-1}^{2}\right) N_{t}, 0\right\}$$

Show that for $r>1$ there is a non-trivial equilibrium, and analyse its stability. Show that, as $r$ is increased from 1 , the equilibrium loses stability at $r=3 / 2$ and find the approximate periodicity close to equilibrium at this point.