A population with variable growth and harvesting is modelled by the equation

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

where $r$ and $E$ are positive constants.

Given that $r>1$, show that a non-zero steady state exists if $0<E<E_{m}(r)$, where $E_{m}(r)$ is to be determined.

Show using a cobweb diagram that, if $E<E_{m}(r)$, a non-zero steady state may be attained only if the initial population $u_{0}$ satisfies $\alpha<u_{0}<\beta$, where $\alpha$ should be determined explicitly and $\beta$ should be specified as a root of an algebraic equation.

With reference to the cobweb diagram, give an additional criterion that implies that $\alpha<u_{0}<\beta$ is a sufficient condition, as well as a necessary condition, for convergence to a non-zero steady state.