IBDP MAI : Topic 5 Calculus - AHL 5.11Definite and indefinite integration AI HL Paper 3
Question : Mackerel Population Dynamics [31 marks]
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: \(\frac{dM}{dt} = \alpha M – \beta MS\), \(\frac{dS}{dt} = \gamma MS – \delta S\), where t is measured in years, and \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are parameters.
a Question a [2 marks] – Interpretation of Terms
(a) Suggest similar interpretations for the following terms.
(i) \(\gamma MS\):
(ii) \(\delta S\):
Show Solution
Ans: 2. (a) (i) population growth rate / birth rate of sharks (due to eating mackerel)
(ii) (net) death rate of sharks
Detailed Solution: (i) \(\gamma MS\) represents the shark population increase due to predation on mackerel, as it’s positive in \(\frac{dS}{dt}\).
(ii) \(\delta S\) is the natural decrease in shark population (e.g., mortality), as it’s negative in \(\frac{dS}{dt}\).
b Question b [5 marks] – Equilibrium Point
(b) An equilibrium point is a set of values of \(M\) and \(S\), such that \(\frac{dM}{dt} = 0\) and \(\frac{dS}{dt} = 0\). Given that both species are present at the equilibrium point,
(i) show that, at the equilibrium point, the value of the mackerel population density is \(\frac{\delta}{\gamma}\):
(ii) find the value of the shark population density at the equilibrium point:
Show Solution
(b) (i) γ MS – δS = 0 since S ≠ 0 getting to given answer without further error by either cancelling or factorizing M = \(\frac{\delta}{\gamma}\)
(ii) \(\frac{dM}{dt} = 0
\alpha M – \beta MS = 0 (sin M \neq 0)
S = \frac{\alpha}{\beta}\)
Detailed Solution: (i) Set \(\frac{dS}{dt} = \gamma MS – \delta S = 0\),
factorize: \(S (\gamma M – \delta) = 0\).
Since \(S \neq 0\), \(\gamma M = \delta\),
So \(M = \frac{\delta}{\gamma}\).
(ii) Set \(\frac{dM}{dt} = \alpha M – \beta MS = 0\),
Factorize: \(M (\alpha – \beta S) = 0\).
Since \(M \neq 0\), \(\alpha = \beta S\),
So \(S = \frac{\alpha}{\beta}\).
c Question c [4 marks] – Equilibrium Changes
(c) 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.
(i) Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of \(\gamma\) is halved. No other parameter changes:
(ii) Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of \(\alpha\) is doubled. No other parameter changes:
Show Solution
(c) (i) \(M_{eq} = \frac{\delta}{\gamma} \Rightarrow \frac{\delta}{\frac{1}{2}\gamma} = 2M_{eq}\)
(ii) \(M_{eq} \frac{\delta}{\gamma}\) is not dependent on \(\alpha\) no change
Detailed Solution: (i) \(M_{eq} = \frac{\delta}{\gamma}\),
If \(\gamma\) halves, new \(M_{eq} = \frac{\delta}{\frac{\gamma}{2}} = 2 \frac{\delta}{\gamma} = 2M_{eq}\),
doubling \(M\).
(ii) \(M_{eq} = \frac{\delta}{\gamma}\) is independent of \(\alpha\), so doubling \(\alpha\) does not affect \(M_{eq}\).
d Question d [8 marks] – Mackerel Growth
(d) To estimate the value of \(\alpha\), Alessia considers a situation where there are no sharks and the initial mackerel population density is \(M_0\).
(i) Write down the differential equation for \(M\) that models this situation:
(ii) Show that the expression for the mackerel population density after \(t\) years is \(M = M_0 e^{\alpha t}\):
(iii) Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that \(\alpha = 0.549\) to three significant figures:
Show Solution
(d) (i) \(\frac{dM}{dt} = \alpha M\)
(ii) \(\int \frac{1}{M} dM = \int \alpha dt\) \(\ln |M| = \alpha t + c\) \(M = k e^{\alpha t}\) when \(t = 0\),
\(M = k\) initial conditions and all other manipulations correct and clearly communicated to get to the final answer \(M = M_0 e^{\alpha t}\)
(iii) \(M = 3M_0\) seen anywhere substituting \(t = 2\) \(M = 2M_0\) into equation
\(M = M_0 e^{\alpha t} = 3M_0 = M_0 e^{2\alpha}\) \(\alpha = \frac{1}{2} \ln 3\)
OR 0.549306… ≈ 0.549
Detailed Solution: (i) With \(S = 0\),
\(\frac{dM}{dt} = \alpha M\).
(ii) Separate variables: \(\frac{dM}{M} = \alpha dt\),
integrate: \(\ln M = \alpha t + c\),
\(M = e^{\alpha t + c} = k e^{\alpha t}\),
at \(t = 0\),
\(M = M_0\),
So \(k = M_0\),
hence \(M = M_0 e^{\alpha t}\).
(iii) At \(t = 2\), \(M = 3M_0\),
So \(3M_0 = M_0 e^{2\alpha}\),
\(e^{2\alpha} = 3\),
\(2\alpha = \ln 3\),
\(\alpha = \frac{\ln 3}{2} \approx 0.549306 \approx 0.549\).
e Question e [5 marks] – Euler’s Method
(e) Alessia decides to use Euler’s method to estimate future mackerel and shark population densities. The initial population densities are estimated to be \(M_0 = 5.7\) and \(S_0 = 2\). She uses a step length of 0.1 years.
(i) Write down expressions for \(M_{n+1}\) and \(S_{n+1}\) in terms of \(M_n\) and \(S_n\):
(ii) Use Euler’s method to find an estimate for the mackerel population density after one year:
Show Solution
(e) (i) an attempt to set up one recursive equation
(ii) EITHER 6.12 (6.11609…)
OR 6120 (6116.09…) (mackerel per km3)
Detailed Solution:
(i) Euler’s method:
\(M_{n+1} = M_n + h \frac{dM}{dt} = M_n + 0.1 (0.549 M_n – 0.236 M_n S_n)\),
\(S_{n+1} = S_n + 0.1 (0.244 M_n S_n – 1.39 S_n)\).
(ii) From \(M_0 = 5.7\), \(S_0 = 2\),
10 steps (1 year): e.g., \(M_1 = 5.7 + 0.1 (0.549 \cdot 5.7 – 0.236 \cdot 5.7 \cdot 2) \approx 5.818\),
Iterating yields \(M_{10} \approx 6.11609 \approx 6.12\).
f Question f [7 marks] – Phase Portrait and Sustainability
(f) 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 \(\text{km}^3\), during the first nine years.
(i) Use Euler’s method to sketch the trajectory of the phase portrait, for \(4 \leq M \leq 7\) and \(1.5 \leq S \leq 3\), over the first nine years:
(ii) Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years:
(iii) State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid:
Show Solution
(f) (i) 
Spiral or closed loop shape approximately 1.25 rotations (can only be awarded if a spiral) correct shape, in approximately correct position (centred at approx. (5.5, 2.5)).
(ii) EITHER approximate minimum is (5.07223…) 5.07 (which is greater than 5)
OR the line M = 5 clearly labelled on their phase portrait
THEN (the density will not fall below 5000) hence sufficient for sustainable fishing.
(iii) Any two from:
• 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
Detailed Solution:
(i)
Using Euler’s method from (5.7, 2), 90 steps plot a spiral around \((\frac{\delta}{\gamma}, \frac{\alpha}{\beta}) \approx (5.7, 2.3)\).
(ii) Minimum \(M \approx 5.07 > 5\),
So above 5000 mackerel/km³,
Sufficient for fishing.
(iii) Parameter estimates may be inaccurate, and Euler’s approximation introduces error.
Syllabus Reference
Syllabus: Mathematics: Applications and Interpretation
Topic 5: Calculus
- Differential equations
- Euler’s method
- Phase portraits
Assessment Criteria: D (Applying mathematics in real-life contexts)