10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations:
\(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\)
\(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\)
where \(t \geq 0\) and \(k\) is a positive constant.
- On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
- Give the equations of any asymptotes.
- What does the model predict about the long term population of
- grey squirrels
- red squirrels?
Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels. - Comment on the validity of the model given by the equations, giving a reason for your answer.
- Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
- When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).