13 A scientist is attempting to model the number of insects, \(N\), present in a colony at time \(t\) weeks. When \(t = 0\) there are 400 insects and when \(t = 1\) there are 440 insects.
- A scientist assumes that the rate of increase of the number of insects is inversely proportional to the number of insects present at time \(t\).
(a) Write down a differential equation to model this situation.
(b) Solve this differential equation to find \(N\) in terms of \(t\). - In a revised model it is assumed that \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ^ { 2 } } { 3988 \mathrm { e } ^ { 0.2 t } }\). Solve this differential equation to find \(N\) in terms of \(t\).
- Compare the long-term behaviour of the two models.