- A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish.
At time \(t\) years, the number of white-clawed crayfish, \(w\), and the number of signal crayfish, \(s\), are modelled by the differential equations
$$\begin{aligned}
& \frac { \mathrm { d } w } { \mathrm {~d} t } = \frac { 5 } { 2 } ( w - s ) \\
& \frac { \mathrm { d } s } { \mathrm {~d} t } = \frac { 2 } { 5 } w - 90 \mathrm { e } ^ { - t }
\end{aligned}$$
- Show that
$$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} w } { \mathrm {~d} t } + 2 w = 450 \mathrm { e } ^ { - t }$$
- Find a general solution for the number of white-clawed crayfish at time \(t\) years.
- Find a general solution for the number of signal crayfish at time \(t\) years.
The model predicts that, at time \(T\) years, the population of white-clawed crayfish will have died out.
Given that \(w = 65\) and \(s = 85\) when \(t = 0\)
- find the value of \(T\), giving your answer to 3 decimal places.
- Suggest a limitation of the model.