In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, $t$ years after their introduction, is denoted by $N_t$. The initial number of breeding pairs is given by $N_0$.

An initial discrete population model is proposed for $N_t$.

Model I: $N_{t+1} = \frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)$

\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item For Model I, show that the steady state values of the number of breeding pairs are 0 and 150. [3]

\item Show that $N_{t+1} - N_t < 150 - N_t$ when $N_t$ lies between 0 and 150. [3]

\item Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when $N_0 \in (0, 150)$. [2]
\end{enumerate}

An alternative discrete population model is proposed for $N_t$.

Model II: $N_{t+1} = \text{INT}\left(\frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\right)$

\item Given that $N_0 = 8$, find the value of $N_4$ for each of the two models and give a reason why Model II may be considered more suitable. [3]
\end{enumerate}

\hfill \mbox{\textit{OCR Further Additional Pure 2017 Q7 [11]}}