Date | May 2021 | Marks available | 3 | Reference code | 21M.2.AHL.TZ1.7 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
A biologist introduces 100 rabbits to an island and records the size of their population (x) over a period of time. The population growth of the rabbits can be approximately modelled by the following differential equation, where t is time measured in years.
dxdt=2x
A population of 100 foxes is introduced to the island when the population of rabbits has reached 1000. The subsequent population growth of rabbits and foxes, where y is the population of foxes at time t, can be approximately modelled by the coupled equations:
dxdt=x(2-0.01y)
dydt=y(0.0002x-0.8)
Use Euler’s method with a step size of 0.25, to find
The graph of the population sizes, according to this model, for the first 4 years after the foxes were introduced is shown below.
Describe the changes in the populations of rabbits and foxes for these 4 years at
Find the population of rabbits 1 year after they were introduced.
(i) the population of rabbits 1 year after the foxes were introduced.
(ii) the population of foxes 1 year after the foxes were introduced.
point A.
point B.
Find the non-zero equilibrium point for the populations of rabbits and foxes.
Markscheme
∫1xdx=∫2dt (M1)
ln x=2t+c
x=Ae2t (A1)
x(0)=100⇒A=100 (M1)
x=100e2t (A1)
x(1)=739 A1
Note: Accept 738 for the final A1.
[5 marks]
tn+1=tn+0.25 (A1)
Note: This may be inferred from a correct t column, where this is seen.
xn+1=xn+0.25xn (A1)
(A1)
(A1)
Note: Award A1 for whole line correct when or . The column may be omitted and implied by the correct and values. The formulas are implied by the correct and columns.
(i) ( OR ) A1
(ii) OR A1
[6 marks]
both populations are increasing A1
[1 mark]
rabbits are decreasing and foxes are increasing A1A1
[2 marks]
setting at least one to zero (M1)
A1A1
[3 marks]