Paper 2, Section I, C

The size of the population of ducks living on the pond of a certain Cambridge college is governed by the equation

$\frac{\mathrm{d} N}{\mathrm{~d} t}=\alpha N-N^{2}$

where $N=N(t)$ is the number of ducks at time $t$ and $\alpha$ is a positive constant. Given that $N(0)=2 \alpha$, find $N(t)$. What happens as $t \rightarrow \infty$ ?

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