Question 13(a) | BetaMath

Question 13(a)

In an experiment, the population of insects, $P(t)$ , was modelled by the logistic differential equation $\frac{dP}{dt} = P(2000 - P)$ where $t$ is the time in days after the beginning of the experiment.

The diagram shows a direction field for this differential equation, with the point $S$ representing the initial population.

The direction field for the differential equation

i. Explain why the graph of the solution that passes through the point $S$ cannot also pass through the point $T$ .

ii. On the diagram provided on page 1 of the Question 13 Writing Booklet, clearly sketch the graph of the solution that passes through the point $S$ .

iii. Find the predicted value of the population, $P(t)$ , at which the rate of growth of the population is largest.

Solution

For part (i):

Time is always increasing, so the evolution of this dynamic system is represented by movement to the right. For all points above the line $P = 2000$ the direction field points right and downwards, so there is no way for a solution which passes through any point at or below the line $P = 2000$ to reach the point $T$ . Since $S$ is below the line $P = 2000$ , the solution that passes through $S$ cannot pass through $T$ .

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