2026 EJC P1 Q11

At a local prawn farming kelong, the prawn population, \(P\), at the time \(t\) months after the opening of the farm, is measured by Mr Xia, the prawn farmer.

1. Mr Xia proposes that the rate of change of \(P\) is proportional to \(P\). Write down a differential equation for this model.[1]

When the farm first opened, there were 2000 prawns. After 12 months, there are 3000 prawns.

2. Solve the differential equation in part (a). Comment on the suitability of this model.[5]

Mr Xia realises that he needs to take into consideration the size of his kelong, and proposes an alternative model

\(\frac{\mathrm{d}P}{\mathrm{d}t}=rP\left(1-\frac PN\right)\quad(*)\), where \(r\) is a positive constant, and \(N\) is a positive integer such that \(0<P<N\).

3. Show that \(\frac{\mathrm{d}^2P}{\mathrm{d}t^2}=r^2P\left(1-\frac PN\right)\left(1-\frac{2P}{N}\right)\).[2]
4. Hence show that if the rate of growth of prawn population is at a maximum, then \(P=\frac N2\). You do not need to verify that this value of \(P\) gives a maximum. Furthermore, find the corresponding rate of growth, leaving your answer in terms of \(N\).[3]

After some time, Mr Xia wants to sell some of the prawns every month at a constant rate of \(h\) prawns per month.

5. Modify the differential equation \((*)\) to take into account the monthly sale of prawns.[1]
6. What happens in the long run if \(h\) is larger than the rate of growth found in part (d)?[1]