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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=x2-0.01y

dydt=y0.0002x-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.

[5]
a.

(i)   the population of rabbits 1 year after the foxes were introduced.

(ii)  the population of foxes 1 year after the foxes were introduced.

[6]
b.

point A.

[1]
c.i.

point B.

[2]
c.ii.

Find the non-zero equilibrium point for the populations of rabbits and foxes.

[3]
d.

Markscheme

1xdx=2dt         (M1)

lnx=2t+c

x=Ae2t         (A1)

x0=100A=100         (M1)

x=100e2t         (A1)

x1=739          A1


Note: Accept 738 for the final A1.


[5 marks]

a.

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 2-0.01yn         (A1)

yn+1=yn+0.25yn 0.0002xn-0.8         (A1)

         (A1)

 

Note: Award A1 for whole line correct when t=0.5 or t=0.75. The t column may be omitted and implied by the correct x and y values. The formulas are implied by the correct x and y columns.


(i)    2840   (2836  OR  2837)          A1

(ii)   58  OR  59          A1


[6 marks]

b.

both populations are increasing         A1


[1 mark]

c.i.

rabbits are decreasing and foxes are increasing        A1A1


[2 marks]

c.ii.

setting at least one DE to zero          (M1)

x=4000,  y=200        A1A1


[3 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
d.

Syllabus sections

Topic 5—Calculus » AHL 5.16—Eulers method for 1st order DEs
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