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Date May Specimen paper Marks available 3 Reference code SPM.3.AHL.TZ0.2
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Sketch Question number 2 Adapted from N/A

Question

The number of brown squirrels, x , in an area of woodland can be modelled by the following differential equation.

d x d t = x 1000 ( 2000 x ) , where  x > 0

One year conservationists notice that some black squirrels are moving into the woodland. The two species of squirrel are in competition for the same food supplies. Let y be the number of black squirrels in the woodland.

Conservationists wish to predict the likely future populations of the two species of squirrels. Research from other areas indicates that when the two populations come into contact the growth can be modelled by the following differential equations, in which t is measured in tens of years.

d x d t = x 1000 ( 2000 x 2 y ) x y  ≥ 0

d y d t = y 1000 ( 3000 3 x y ) x y  ≥ 0

An equilibrium point for the populations occurs when both  d x d t = 0 and  d y d t = 0 .

When the two populations are small the model can be reduced to the linear system

d x d t = 2 x

d y d t = 3 y .

For larger populations, the conservationists decide to use Euler’s method to find the long‑term outcomes for the populations. They will use Euler’s method with a step length of 2 years ( t = 0.2 ).

Find the equilibrium population of brown squirrels suggested by this model.

[2]
a.i.

Explain why the population of squirrels is increasing for values of x less than this value.

[1]
a.ii.

Verify that  x = 800 y = 600  is an equilibrium point.

[2]
b.i.

Find the other three equilibrium points.

[4]
b.ii.

By using separation of variables, show that the general solution of  d x d t = 2 x is  x = A e 2 t .

[4]
c.i.

Write down the general solution of d y d t = 3 y .

[1]
c.ii.

If both populations contain 10 squirrels at t = 0 use the solutions to parts (c) (i) and (ii) to estimate the number of black and brown squirrels when t = 0.2 . Give your answers to the nearest whole numbers.

[2]
c.iii.

Write down the expressions for x n + 1 and y n + 1 that the conservationists will use.

[2]
d.i.

Given that the initial populations are x = 100 , y = 100 , find the populations of each species of squirrel when t = 1 .

[3]
d.ii.

Use further iterations of Euler’s method to find the long-term population for each species of squirrel from these initial values.

[1]
d.iii.

Use the same method to find the long-term populations of squirrels when the initial populations are  x = 400 y = 100 .

[1]
d.iv.

Use Euler’s method with step length 0.2 to sketch, on the same axes, the approximate trajectories for the populations with the following initial populations.

(i)      x = 1000 y = 1500

(ii)    x = 1500 y = 1000

[3]
e.

Given that the equilibrium point at (800, 600) is a saddle point, sketch the phase portrait for x ≥ 0 , y ≥ 0 on the same axes used in part (e).

[2]
f.

Markscheme

2000        (M1)A1

[2 marks]

a.i.

because the value of  d x d t is positive (for  x > 0 )        R1

[1 mark]

a.ii.

substitute  x = 800 y = 600  into both equations      M1

both equations equal 0       A1

hence an equilibrium point       AG

[3 marks]

b.i.

x = 0 y = 0       A1

x = 2000 y = 0 x = 0 y = 3000        M1A1A1

Note: Award M1 for an attempt at solving the system provided some values of x and y are found.

[4 marks]

b.ii.

1 x d x = 2 d t       M1

ln x = 2 t + c          A1A1

Note: Award A1 for RHS, A1 for LHS.

x = e c e 2 t         M1

x = A e 2 t   (where  A = e c )       AG

[4 marks]

c.i.

y = B e 3 t         A1

Note: Allow any letter for the constant term, including A .

[1 mark]

c.ii.

x = 15 y = 18        (M1)A1

[2 marks]

c.iii.

x n + 1 = x n + 0.2 x n 1000 ( 2000 x n 2 y n )

y n + 1 = y n + 0.2 y n 1000 ( 3000 3 x n y n )        M1A1

Note: Accept equivalent forms.

[2 marks]

d.i.

x = 319 y = 617      (M1)A1A1

[3 marks]

d.ii.

number of brown squirrels go down to 0, 
black squirrels to a population of 3000        A1

[1 mark]

d.iii.

number of brown squirrels go to 2000, 
number of black squirrels goes down to 0       A1

[1 mark]

d.iv.

(i) AND (ii)

     M1A1A1

[3 marks]

e.

    A1A1

Note: Award A1 for a trajectory beginning close to (0, 0) and going to (0, 3000) and A1 for a trajectory beginning close to (0, 0) and going to (2000, 0) in approximately the correct places.

[2 marks]

f.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
c.iii.
[N/A]
d.i.
[N/A]
d.ii.
[N/A]
d.iii.
[N/A]
d.iv.
[N/A]
e.
[N/A]
f.

Syllabus sections

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

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