Predict population extinction or event time

A question is this type if and only if it asks you to use the particular solution to determine when a population (or quantity) reaches zero or when two populations become equal, giving a specific time or year.

2 questions · Challenging +1.0

4.10h Coupled systems: simultaneous first order DEs
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Edexcel CP1 2024 June Q8
15 marks Challenging +1.2
  1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.
At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
  3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
  4. determine the value of \(T\), giving your answer to 2 decimal places.
  5. Suggest a limitation of the model.
Edexcel CP2 Specimen Q7
17 marks Standard +0.8
  1. At the start of the year 2000, a survey began of the number of foxes and rabbits on an island.
At time \(t\) years after the survey began, the number of foxes, \(f\), and the number of rabbits, \(r\), on the island are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } f } { \mathrm {~d} t } = 0.2 f + 0.1 r \\ & \frac { \mathrm {~d} r } { \mathrm {~d} t } = - 0.2 f + 0.4 r \end{aligned}$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } f } { \mathrm {~d} t ^ { 2 } } - 0.6 \frac { \mathrm {~d} f } { \mathrm {~d} t } + 0.1 f = 0\)
  2. Find a general solution for the number of foxes on the island at time \(t\) years.
  3. Hence find a general solution for the number of rabbits on the island at time \(t\) years. At the start of the year 2000 there were 6 foxes and 20 rabbits on the island.
    1. According to this model, in which year are the rabbits predicted to die out?
    2. According to this model, how many foxes will be on the island when the rabbits die out?
    3. Use your answers to parts (i) and (ii) to comment on the model.