Specialist2024 NHT Exam 1 · Q9Tech-free6 marks

Logistic model of a kangaroo population

In a population, the number of kangaroos, $P$ , is modelled by the logistic differential equation

$\frac{dP}{dt} = 0.05P\left(10 - \frac{P}{50}\right),$

Find $\frac{d^2P}{dt^2}$ in terms of $P$ .

1 mark

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Part of the graph of $P$ against $t$ is shown below. The graph contains a single point of inflection, labelled $A$ .

In the context of the growth of the kangaroo population, what does the point of inflection at $A$ represent?

Graph of P against t for t from 0 to 6, starting at P = 100 when t = 0 and rising to about 410 at t = 6. The curve is concave up and then concave down, changing at a point of inflection labelled A near t ≈ 2.8, P = 250.

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Solve the logistic differential equation to show that $t = 2\log_e\left(\frac{4P}{500 - P}\right)$ .

3 marks

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Hence, or otherwise, find the coordinates of point $A$ .

1 mark

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