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Date May 2021 Marks available 3 Reference code 21M.3.AHL.TZ2.2
Level Additional Higher Level Paper Paper 3 Time zone Time zone 2
Command term Write down Question number 2 Adapted from N/A

Question

Alessia is an ecologist working for Mediterranean fishing authorities. She is interested in whether the mackerel population density is likely to fall below 5000 mackerel per km3, as this is the minimum value required for sustainable fishing. She believes that the primary factor affecting the mackerel population is the interaction of mackerel with sharks, their main predator.

The population densities of mackerel (M thousands per km3) and sharks (S per km3) in the Mediterranean Sea are modelled by the coupled differential equations:

dMdt=αM-βMS

dSdt=γMS-δS

where t is measured in years, and α, β, γ and δ are parameters.

This model assumes that no other factors affect the mackerel or shark population densities.

The term αM models the population growth rate of the mackerel in the absence of sharks.
The term βMS models the death rate of the mackerel due to being eaten by sharks.

Suggest similar interpretations for the following terms.

An equilibrium point is a set of values of M and S , such that dMdt=0 and dSdt=0.

Given that both species are present at the equilibrium point,

The equilibrium point found in part (b) gives the average values of M and S over time.

Use the model to predict how the following events would affect the average value of M. Justify your answers.

To estimate the value of α, Alessia considers a situation where there are no sharks and the initial mackerel population density is M0.

Based on additional observations, it is believed that

α=0.549,

β=0.236,

γ=0.244,

δ=1.39.

Alessia decides to use Euler’s method to estimate future mackerel and shark population densities. The initial population densities are estimated to be M0=5.7 and S0=2. She uses a step length of 0.1 years.

Alessia will use her model to estimate whether the mackerel population density is likely to fall below the minimum value required for sustainable fishing, 5000 per km3, during the first nine years.

γMS

[1]
a.i.

δS

[1]
a.ii.

show that, at the equilibrium point, the value of the mackerel population density is δγ;

[3]
b.i.

find the value of the shark population density at the equilibrium point.

[2]
b.ii.

Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of γ is halved. No other parameter changes.

[2]
c.i.

Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of α is doubled. No other parameter changes.

[2]
c.ii.

Write down the differential equation for M that models this situation.

[1]
d.i.

Show that the expression for the mackerel population density after t years is M=M0eαt

[4]
d.ii.

Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that α=0.549 to three significant figures.

[3]
d.iii.

Write down expressions for Mn+1 and Sn+1 in terms of Mn and Sn.

[3]
e.i.

Use Euler’s method to find an estimate for the mackerel population density after one year.

[2]
e.ii.

Use Euler’s method to sketch the trajectory of the phase portrait, for 4M7 and 1.5S3, over the first nine years.

[3]
f.i.

Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years.

[2]
f.ii.

State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid.

[2]
f.iii.

Markscheme

population growth rate / birth rate of sharks (due to eating mackerel)            A1


[1 mark]

a.i.

(net) death rate of sharks          A1


[1 mark]

a.ii.

γMS-δS=0          A1

since S0          R1


Note: Accept S>0.


getting to given answer without further error by either cancelling or factorizing           A1

M=δγ           AG


[3 marks]

b.i.

dMdt=0

αM-βMS=0          (M1)

(since M0)  S=αβ          A1


[2 marks]

b.ii.

Meq=δγδ12γ=2Meq                      M1

Note: Accept equivalent in words.


Doubles          A1


Note: Do not accept “increases”.


[2 marks]

c.i.

Meq=δγ is not dependent on α                      R1


Note:
Award R0 for any contextual argument.


no change         A1


Note: Do not award R0A1.


[2 marks]

c.ii.

dMdt=αM                    A1


[1 mark]

d.i.

1MdM=αdt                  M1


Note: Award M1 is for an attempt to separate variables. This means getting to the point fMdM=gtdt where the integral can be seen or implied by further work.


lnM=αt+c                  A1


Note: Accept lnM. Condone missing constant of integration for this mark.


M=keαt

when t=0, M0=k                 M1


Note: Award M1 for a clear attempt at using initial conditions to find a constant of integration. Only possible if the constant of integration exists. t=0 or “initially” or similar must be seen. Substitution may appear earlier, following the integration.


initial conditions and all other manipulations correct and clearly communicated to get to the final answer                   A1

M=M0eαt                  AG


[4 marks]

d.ii.

M=3M0 seen anywhere                (A1)

substituting t=2, M=3M0 into equation M=M0eαt                (M1)

3M0=M0e2α

α=12ln3  OR  0.549306                  A1


Note: The A1 requires either the exact answer or an answer to at least 4 sf.


0.549                 AG

 

[3 marks]

d.iii.

an attempt to set up one recursive equation                (M1)


Note: Must include two given parameters and Mn and Sn and Mn+1 or Sn+1 for the (M1) to be awarded.


Mn+1=Mn+0.10.549Mn-0.236MnSn                 A1

Sn+1=Sn+0.10.244MnSn-1.39Sn                 A1

 

[3 marks]

e.i.

EITHER
6.12  (6.11609)             A2


OR
6120  (6116.09) (mackerel per km3)             A2

 

[2 marks]

e.ii.

spiral or closed loop shape         A1

approximately 1.25 rotations (can only be awarded if a spiral)         A1

correct shape, in approximately correct position (centred at approx. (5.5, 2.5))         A1


Note: Award A0A0A0 for any plot of S or M against t.

[3 marks]

f.i.

EITHER

approximate minimum is (5.07223) 5.07 (which is greater than 5)           A1


OR

the line M=5 clearly labelled on their phase portrait           A1


THEN

(the density will not fall below 5000) hence sufficient for sustainable fishing           A1


Note: Do not award A0A1. Only if the minimum point is labelled on the sketch then a statement here that “the mackerel population is always above 5000 would be sufficient. Accept the value 5.07 seen within a table of values.

[2 marks]

f.ii.

Any two from:                           A1A1

• Current values / parameters are only an estimate,

• The Euler method is only an approximate method / choosing h=0.1 might be too large.

• There might be random variation / the model has no stochastic component

• Conditions / parameters might change over the nine years,

• A discrete system is being approximated by a continuous system,

Allow any other sensible critique.


If a candidate identifies factors which the model ignores, award A1 per factor identified. These factors could include:

• Other predators

• Seasonality

• Temperature

• The effect of fishing

• Environmental catastrophe

• Migration


Note: Do not allow:
             “You cannot have 5.07 mackerel”.
             It is only a model (as this is too vague).
             Some factors have been ignored (without specifically identifying the factors).
             Values do not always follow the equation / model. (as this is too vague).

 

[2 marks]

f.iii.

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]
d.i.
[N/A]
d.ii.
[N/A]
d.iii.
[N/A]
e.i.
[N/A]
e.ii.
[N/A]
f.i.
[N/A]
f.ii.
[N/A]
f.iii.

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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