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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.

$\frac{d x}{d t} = 2 x$

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:

$\frac{d x}{d t} = x (2 - 0.01 y)$

$\frac{d y}{d t} = y (0.0002 x - 0.8)$

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

$\int \frac{1}{x} d x = \int 2 d t$ (M1)

$\ln x = 2 t + c$

$x = A e^{2 t}$ (A1)

$x (0) = 100 \Rightarrow A = 100$ (M1)

$x = 100 e^{2 t}$ (A1)

$x (1) = 739$ A1

Note: Accept $738$ for the final A1.

[5 marks]

a.

$t_{n + 1} = t_{n} + 0.25$ (A1)

Note: This may be inferred from a correct $t$ column, where this is seen.

$x_{n + 1} = x_{n} + 0.25 x_{n} (2 - 0.01 y_{n})$ (A1)

$y_{n + 1} = y_{n} + 0.25 y_{n} (0.0002 x_{n} - 0.8)$ (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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SPM.3.AHL.TZ0.2f:
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).
SPM.3.AHL.TZ0.2a.i:
Find the equilibrium population of brown squirrels suggested by this model.
SPM.3.AHL.TZ0.2d.i:
Write down the expressions for ${x_{n + 1}}$ and ${y_{n + 1}}$ that the conservationists will use.
SPM.3.AHL.TZ0.2d.iv:
Use the same method to find the long-term populations of squirrels when the initial populations are $x = 400$ , $y = 100$ .
SPM.3.AHL.TZ0.2a.ii:
Explain why the population of squirrels is increasing for values of $x$ less than this value.
16N.1.AHL.TZ0.H_11c:
Show that the function $f$ has a local maximum value when $x = \frac{{3\pi }}{4}$ .
16N.1.AHL.TZ0.H_11d:
Find the $x$ -coordinate of the point of inflexion of the graph of $f$ .
16N.1.AHL.TZ0.H_11e:
Sketch the graph of $f$ , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.
SPM.3.AHL.TZ0.2c.ii:
Write down the general solution of $\frac{{{\text{d}}y}}{{{\text{d}}t}} = 3y$ .
SPM.3.AHL.TZ0.2b.ii:
Find the other three equilibrium points.
16N.1.AHL.TZ0.H_11h:
Find the value $\kappa$ for $x = \frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.
16N.1.AHL.TZ0.H_11g:
Find the value of the curvature of the graph of $f$ at the local maximum point.
16N.1.AHL.TZ0.H_11f:
Find the area of the region enclosed by the graph of $f$ and the $x$ -axis.
SPM.3.AHL.TZ0.2d.ii:
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SPM.3.AHL.TZ0.2e:
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) $x = 1500$ , $y = 1000$
SPM.3.AHL.TZ0.2c.i:
By using separation of variables, show that the general solution of $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2x$ is $x = A{{\text{e}}^{2t}}$ .
16N.1.AHL.TZ0.H_11b:
Show that $\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x$ .
SPM.3.AHL.TZ0.2c.iii:
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.
SPM.3.AHL.TZ0.2b.i:
Verify that $x = 800$ , $y = 600$ is an equilibrium point.
16N.1.AHL.TZ0.H_11a:
Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
21M.2.AHL.TZ1.7c.i:
point $A$ .
21M.2.AHL.TZ1.7c.ii:
point $B$ .
21M.2.AHL.TZ1.7b:
(i) the population of rabbits 1 year after the foxes were introduced.

(ii) the population of foxes 1 year after the foxes were introduced.
21M.3.AHL.TZ2.2e.i:
Write down expressions for $M_{n + 1}$ and $S_{n + 1}$ in terms of $M_{n}$ and $S_{n}$ .
21M.3.AHL.TZ2.2e.ii:
Use Euler’s method to find an estimate for the mackerel population density after one year.
EXN.2.AHL.TZ0.4b:
Show that $r \approx \frac{N (n + 1) - N (n)}{N (n)}$
EXN.2.AHL.TZ0.4c:
Hence find three approximations for the value of $r$ .
21N.3.AHL.TZ0.2a.i:
Find the equation of the regression line of $h$ on $t$ .
21N.3.AHL.TZ0.2a.ii:
Interpret the meaning of parameter $a$ in the context of the model.
21N.3.AHL.TZ0.2a.iii:
Suggest why Eva’s use of the linear regression equation in this way could be unreliable.
21N.3.AHL.TZ0.2b.i:
Find the equation of the least squares quadratic regression curve.
21N.3.AHL.TZ0.2b.ii:
Find the value of $k$ .
21N.3.AHL.TZ0.2b.iii:
Hence, write down a suitable domain for Eva’s function $h (t) = p t^{2} + q t + r$ .
21N.3.AHL.TZ0.2c:
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21N.3.AHL.TZ0.2d:
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21N.3.AHL.TZ0.2e:
Use the general solution from part (d) and the initial condition $h (0) = 3.2$ to predict the value of $T$ .
21N.3.AHL.TZ0.2f:
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21N.3.AHL.TZ0.2g.ii:
Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .

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Topic 5—Calculus