15 An isolated island is populated by rabbits and foxes. At time \(t\) the number of rabbits is \(x\) and the number of foxes is \(y\).
It is assumed that:
- The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time.
- If there were no foxes present, the number of rabbits would increase by \(120 \%\) in a unit period of time.
- When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to \(110 \%\) of the current number of foxes.
- At time \(t = 0\), the number of foxes is 20 and the number of rabbits is 80 .
15
- Construct a mathematical model for the number of rabbits.
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15
- (ii) Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time.
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15 - Suggest one way in which the model that you have used for the number of rabbits could be refined.
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