N2022 P1 Q12

Scientists are interested in the population of a particular species. They attempt to model the population \(P\) at time \(t\) days using a differential equation. Initially the population is observed to be \(50\) and after \(10\) days the population is \(100\).

The first model the scientists use assumes that the rate of change of the population is proportional to the population.

1. Write down a differential equation for this model and solve it for \(P\) in terms of \(t\).[5]

To allow for constraints on population growth, the model is refined to

\(\frac{\mathrm{d}P}{\mathrm{d}t}=\lambda P\left( 500-P \right)\)

where \(\lambda \) is a constant.

2. Solve this differential equation to find \(P\) in terms of \(t\).[6]
3. Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.[2]