Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced.

The population $x$, in thousands, of red squirrels is modelled by the equation
$$x = \frac{a}{1 + kt},$$
where $t$ is the time in years, and $a$ and $k$ are constants. When $t = 0$, $x = 2.5$.

\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{dx}{dt} = -\frac{kx^2}{a}$. [3]

\item Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate $a$ and $k$. [3]

\item What is the long-term population of red squirrels predicted by this model? [1]
\end{enumerate}

The population $y$, in thousands, of grey squirrels is modelled by the differential equation
$$\frac{dy}{dt} = 2y - y^2.$$

When $t = 0$, $y = 1$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Express $\frac{1}{2y - y^2}$ in partial fractions. [4]

\item Hence show by integration that $\ln\left(\frac{y}{2-y}\right) = 2t$.

Show that $y = \frac{2}{1 + e^{-2t}}$. [7]

\item What is the long-term population of grey squirrels predicted by this model? [1]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q3 [19]}}