MA2MOD: MATHEMATICAL MODELLING
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MA2MOD: MATHEMATICAL MODELLING
Assignment 2
1. In a National Park in Central East Africa, lions and hyenas are competing for food. A mathematical model was proposed to describe the populations of these two competing predators
x˙ = x(a - by)
y˙ = y(c - dx).
where x and y are the populations of the lions and hyenas, respectively, and a, b, c, d e R are constants.
(a) What are the signs of the constants a, b, c, d? Give reasons based on how the population of one species changes in the absence and presence of the other species.
[5 marks]
(b) Find the equilibrium points of the system, discuss the stability of these equilibrium points and the behaviour of the trajectories in their vicinities.
[13 marks]
(c) Find an analytical solution V (x, y) of the trajectories of the system in the phase plane and discuss their behaviour by finding their extrema about x and y.
[12 marks]
(d) Take a = 0.55, b = 0.01, c = 0.6 and d = 0.02, plot the typical trajectories (contour lines) of the system in the range of x e [0 100], y e [0 150].
(Note: you may consider to use the Matlab code given in Example 2.2.12 of the lecture notes. There the line ”syms x y”may or may not be needed, depending on the version of Matlab used. )
[5 marks]
Submit both the plot and the programming code used for generating the plot in your answer sheet.
(e) Based on the above calculation results and plot, discuss if the model is realistic. [5 marks]
2. John was roasting a turkey for the Christmas dinner. The total cooking time t of the turkey is related to its weight m, initial temperature Θt and thermal conductivity k (of dimension L2T − 1 ), as well as the interior temperature of the oven Θo . The temperatures Θo and Θt are measured in units of energy per volume and so have the dimension of ML− 1T −2 .
Use dimensional matrix and Buckingham’s theorem to find
(a) How the cooking time t depends on other factors in the system.
[25 marks]
(b) Assuming that the cooking time t was experimentally found to be proportional to Θo(−)1/5. By what factor t is going to change if the initial temperate of the turkey Θt is doubled while all other factors remain unchanged?
(Hint: If the Buckingham’s theorem f(Π1 , Π2 , . . . , Πp ) = 0 is used to solve (a) by assuming Π 1 = h(Π2 , . . . , Πp ), the unknownfunction h(Π2 , . . . , Πp ) can befurther rewritten as h(Π2 , . . . , Πp ) = Π2(α)g(Π3 , . . . , Πp ) where the exponent α is a rational number to be determined. Depending on the number of dimensionless products p in the complete set, g can be either a constant if p = 2 or some unknownfunction if p > 2.)
[10 marks]
3. A stone of mass m was dropped from rest at time t = 0 from a helicopter suspending at height H in a uniform gravitational field. The air resistance force the stone experienced is proportional to the square of its speed v as mkv2 where k > 0 is a constant. Find out
(a) The speed V at which the stone hit the ground and give the terminal velocity in case of H → o.
[12 marks]
(b) The amount of time that the stone flied until hitting the ground.
[13 marks]
(Hints: (i) You may consider to use differentforms of equation of motion to solve Parts (a) and (b); (ii) If needed, you can use the integral 
= tanh− 1 (y)for |y| < 1 without proof.)
[End of Question Paper]
2021-12-07