14 On an isolated island some rabbits have been accidently introduced.
In order to eliminate them, conservationists have introduced some birds of prey.
At time \(t\) years \(( t \geq 0 )\) there are \(x\) rabbits and \(y\) birds of prey.
At time \(t = 0\) there are 1755 rabbits and 30 birds of prey.
When \(t > 0\) it is assumed that:
- the rabbits will reproduce at a rate of \(a \%\) per year
- each bird of prey will kill, on average, \(b\) rabbits per year
- the death rate of the birds of prey is \(c\) birds per year
- the number of birds of prey will increase at a rate of \(d \%\) of the rabbit population per year.
This system is represented by the coupled differential equations:
$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4 x - 13 y \\
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.01 x - 1.95
\end{aligned}$$
14
- State the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\)
[0pt]
[2 marks]
14 - Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\)