IB Mathematics AHL 5.14 Solving by separation of variables-AI HL Paper 3- Exam Style Questions- New Syllabus
Question
Elena is a marine biologist working for Mediterranean fishery management. She is examining whether the mackerel population density might drop below 5000 mackerel per km3, the threshold needed for sustainable fishing. She hypothesizes that the main factor influencing mackerel numbers is their interaction with sharks, their primary predator.
The population densities of mackerel (\(M\) thousands per km3) and sharks (\(S\) per km3) in the Mediterranean Sea are described by the coupled differential equations:
\[ \frac{dM}{dt} = \alpha M – \beta M S \] \[ \frac{dS}{dt} = \gamma M S – \delta S \]
where \(t\) represents years, and \(\alpha, \beta, \gamma, \delta\) are parameters.
This model assumes no other factors impact mackerel or shark population densities.
The term \(\alpha M\) represents the growth rate of the mackerel population without sharks. The term \(\beta M S\) represents the mortality rate of mackerel due to predation by sharks.
(a) Provide interpretations for the following terms.
(i) \(\gamma M S\) [1]
(ii) \(\delta S\) [1]
(b) An equilibrium point occurs when \(M\) and \(S\) values result in \(\frac{dM}{dt} = 0\) and \(\frac{dS}{dt} = 0\).
Assuming both species are present at the equilibrium point,
(i) Demonstrate that the mackerel population density at equilibrium is \(\frac{\delta}{\gamma}\). [3]
(ii) Determine the shark population density at the equilibrium point. [2]
(c) The equilibrium point from part (b) represents the average values of \(M\) and \(S\) over time.
Use the model to predict the effect of the following events on the average value of \(M\). Justify your answers.
(i) Pollution from toxic waste enters the Mediterranean Sea. Elena claims this reduces the shark population growth rate, halving the value of \(\gamma\). No other parameters change. [2]
(ii) Rising sea temperatures due to climate change occur. Elena claims this enhances the mackerel population growth rate, doubling the value of \(\alpha\). No other parameters change. [2]
(d) To estimate \(\alpha\), Elena considers a scenario with no sharks and an initial mackerel population density of \(M_0\).
(i) State the differential equation for \(M\) in this scenario. [1]
(ii) Prove that the mackerel population density after \(t\) years is \[ M = M_0 e^{\alpha t}. \] [4]
(iii) Elena estimates the mackerel population density triples every two years. Show that \(\alpha = 0.549\) to three significant figures. [3]
Based on further data, it is estimated that
\(\alpha = 0.549,\)
\(\beta = 0.236,\)
\(\gamma = 0.244,\)
\(\delta = 1.39.\)
\(\beta = 0.236,\)
\(\gamma = 0.244,\)
\(\delta = 1.39.\)
Elena uses Euler’s method to forecast future mackerel and shark population densities, with initial densities of \(M_0 = 5.7\) and \(S_0 = 2\), and a step size of 0.1 years.
(e) (i) Provide expressions for \(M_{n+1}\) and \(S_{n+1}\) in terms of \(M_n\) and \(S_n\). [3]
(ii) Apply Euler’s method to estimate the mackerel population density after one year. [2]
(f) Elena aims to determine if the mackerel population density will fall below the sustainable fishing threshold of 5000 per km3 within the first nine years.
(i) Use Euler’s method to plot the phase portrait trajectory for \(4 \leq M \leq 7\) and \(1.5 \leq S \leq 3\) over the first nine years. [3]
(ii) Based on the phase portrait or other methods, assess whether the mackerel population density supports sustainable fishing during the first nine years. [2]
(iii) List two reasons why Elena’s conclusion in part (f)(ii) might be unreliable. [2]
▶️ Answer/Explanation
Markscheme
(a) (i) Population growth rate / birth rate of sharks (due to eating mackerel) [1 mark]
(ii) (Net) death rate of sharks [1 mark]
(b) (i) \(\frac{dS}{dt} = \gamma M S – \delta S = 0\), since \(S \neq 0\)
Factorizing: \(S(\gamma M – \delta) = 0\)
Since \(S > 0\), \(\gamma M = \delta\)
\(M = \frac{\delta}{\gamma}\) [3 marks]
(ii) \(\frac{dM}{dt} = \alpha M – \beta M S = 0\)
Factorizing: \(M(\alpha – \beta S) = 0\)
Since \(M \neq 0\), \(\alpha – \beta S = 0\)
\(S = \frac{\alpha}{\beta}\) [2 marks]
(c) (i) At equilibrium, \(M = \frac{\delta}{\gamma}\). If \(\gamma\) is halved, new equilibrium is:
\(M = \frac{\delta}{\frac{1}{2}\gamma} = \frac{2\delta}{\gamma} = 2 \cdot \frac{\delta}{\gamma}\)
The average mackerel population density doubles. [2 marks]
(ii) At equilibrium, \(M = \frac{\delta}{\gamma}\), which is independent of \(\alpha\). Doubling \(\alpha\) does not affect the equilibrium mackerel population density.
No change. [2 marks]
(d) (i) \(\frac{dM}{dt} = \alpha M\) [1 mark]
(ii) \(\frac{dM}{dt} = \alpha M\)
Separate variables: \(\frac{dM}{M} = \alpha dt\)
Integrate: \(\int \frac{1}{M} dM = \int \alpha dt\)
\(\ln|M| = \alpha t + c\)
\(M = k e^{\alpha t}\)
At \(t = 0\), \(M = M_0\), so \(k = M_0\)
\(M = M_0 e^{\alpha t}\) [4 marks]
(iii) Given \(M = 3M_0\) when \(t = 2\):
\(3M_0 = M_0 e^{2\alpha}\)
\(e^{2\alpha} = 3\)
\(2\alpha = \ln 3\)
\(\alpha = \frac{\ln 3}{2} \approx \frac{1.098612}{2} \approx 0.549\) (to three significant figures) [3 marks]
(e) (i) Using Euler’s method with step size \(h = 0.1\):
\(M_{n+1} = M_n + h \left( \alpha M_n – \beta M_n S_n \right) = M_n + 0.1 \left( 0.549 M_n – 0.236 M_n S_n \right)\)
\(S_{n+1} = S_n + h \left( \gamma M_n S_n – \delta S_n \right) = S_n + 0.1 \left( 0.244 M_n S_n – 1.39 S_n \right)\)
[3 marks]
(ii) Initial conditions: \(M_0 = 5.7\), \(S_0 = 2\). Using Euler’s method for 10 steps (\(0.1 \times 10 = 1\) year):
Step 1: \(M_1 = 5.7 + 0.1 \left( 0.549 \times 5.7 – 0.236 \times 5.7 \times 2 \right) = 5.7 + 0.1 \left( 3.1293 – 2.6904 \right) \approx 5.74389\)
\(S_1 = 2 + 0.1 \left( 0.244 \times 5.7 \times 2 – 1.39 \times 2 \right) = 2 + 0.1 \left( 2.7816 – 2.78 \right) \approx 2.00016\)
After 10 steps, \(M_{10} \approx 6.12\) (thousands per km3) or 6120 mackerel per km3. [2 marks]
(f) (i) Using Euler’s method with \(M_0 = 5.7\), \(S_0 = 2\), and parameters \(\alpha = 0.549\), \(\beta = 0.236\), \(\gamma = 0.244\), \(\delta = 1.39\), compute points for 9 years (90 steps). The phase portrait shows a spiral or closed loop, approximately 1.25 rotations, centered around \((M, S) \approx (5.7, 2.3)\), within \(4 \leq M \leq 7\), \(1.5 \leq S \leq 3\). [3 marks]
(ii) The minimum \(M \approx 5.07\) (thousands per km3) or 5070 mackerel per km3, which is above 5000. Thus, the mackerel population density is sufficient for sustainable fishing. [2 marks]
(iii) Two reasons why Elena’s conclusion might be unreliable:
1. The model assumes only shark-mackerel interactions affect population density, ignoring other factors like other predators, fishing, or environmental changes.
2. Euler’s method may introduce numerical errors, especially with a step size of 0.1, potentially affecting the accuracy of the trajectory. [2 marks]