Determine long-term or comparative behaviour

A question is this type if and only if it asks you to analyse the particular solutions to compare the long-term behaviour of two quantities (e.g. which population dies out first, or whether one quantity dominates the other).

2 questions · Challenging +1.0

4.10h Coupled systems: simultaneous first order DEs
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OCR Further Pure Core 1 2018 December Q10
16 marks Standard +0.8
10 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. $$\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
    1. Find the general solution for \(x\).
    2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
  3. Find the particular solutions for \(x\) and \(y\).
  4. Determine whether the model predicts that the predators will die out before the prey.
OCR Further Pure Core 1 2021 June Q5
16 marks Challenging +1.2
5 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. \(\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x\)
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
    1. Find the general solution for \(x\).
    2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
  3. Find the particular solutions for \(x\) and \(y\).
  4. Determine whether the model predicts that the predators will die out before the prey.