7 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 an integer denoted by $N _ { t }$. The initial number of breeding pairs is given by $N _ { 0 }$.

An initial discrete population model is proposed for $N _ { t }$.

$$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$
\begin{enumerate}[label=(\roman*)]
\item (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .\\
(b) Show that $N _ { t + 1 } - N _ { t } < 150 - N _ { t }$ when $N _ { t }$ lies between 0 and 150 .\\
(c) Hence find the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when $N _ { 0 } \in ( 0,150 )$.

An alternative discrete population model is proposed for $N _ { t }$.

$$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
\item (a) Given that $N _ { 0 } = 8$, find the value of $N _ { 4 }$ for each of the two models.\\
(b) Which of the two models gives values for $N _ { t }$ with the more appropriate level of precision?
\end{enumerate}

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