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PAPER CODE NO.
MATH 122
THE UNIVERSITY
of LIVERPOOL
SUMMER 2004 EXAMINATIONS
Bachelor of Science : Year 1 Bachelor of Science : Year 2 Master of Mathematics : Year 1 Master of Mathematics : Year 2 Master of Physics : Year 1
DYNAMIC MODELLING
TIME ALLOWED : Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
Candidates should answer the WHOLE of Section A and THREE questions from Section B. Section A carries 55% of the available marks. Take g = 9. 81 ms−^2. Give numerical answers to 3 significant figures. You may use
dx
dv v dt
dx dx
dv dt
dv = =.
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SECTION A
1. Unmolested the concentration of a species of phytoplankton would grow at a rate of 5% per day, but grazing losses due to a predatory species of zooplankton impose a constraint, decreasing the concentration by a rate equivalent to 3% per day. Write down a differential equation for the phytoplankton concentration at a time t , where the unit of t is one day. If the phytoplankton concentration at
C^ ( ) t
t = 0 was , how many days will it take for the concentration to treble?
C 0
[6 marks]
2. A dead body has been discovered in a house, and the police, suspecting foul play, wish to determine the time of death. The temperature θ ( t )of the body at time t , is assumed to decrease at a rate proportional to the difference between
θ and the ambient room temperature , with constant of proportionality k.
When the body was first found, at
T R
t = 0 , its temperature was and 2 hours later this had fallen to. Write down a differential equation for
300 C
280 C
θ ( ) t , and integrate it to show that
R
R T
T
k 28
ln 2
If and assuming the body’s temperature was the normal at the time of death, how long had it lain undiscovered? [9 marks]
TR = 200 C 370 C
3. A house buyer takes out a mortgage of £80000 to buy a new house. The building society lends him the money at an interest rate of 0.48% per month and the customer makes monthly repayments of £500 to pay off the loan. Write down the equation for , the balance owed at the end of month , in terms of. If
um + 1 ( m + 1 ) u (^) m x (^) m = um − N , where N is the equilibrium solution of this equation, write down a second equation for in terms of. How long does it take the customer to pay off his mortgage and how much does he pay in total?
x m x 0
[8 marks]
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8. Let and represent the levels of two populations governed by the
following coupled differential equations
x^ ( ) t y ( ) t
= ( 15 − ) = 5 ( x − 5 )
dt
dy y dt
dx .
Initially x = 7 and. Obtain and solve the differential equation for y in terms of x. From your results draw a phase diagram for this situation, indicating which way around the curve the point
y = 11
( x , y )moves. [5 marks]
SECTION B
9. Consider a two-state stochastic system, with states A and B. In the usual notation,
P ( At ) P ( B t ) W ( B A ) P ( At ) W ( A B
dt
d
) )
Write down what each term in this equation represents. [5 marks] A businessman commutes to work by train. If his train is late on a particular day, the probability that it is late the following day is 0.5. However if his train is on time on a particular day the probability that it is on time the next day is 0.95. If and are the probabilities of the businessman being on time or late for work at time t , show that
P ( OT , t P ( L , t
+. POT t =. dt
dP OT t .
Solve this equation if P ( OT , 0 ) = 0. 8. The businessman cannot afford to be late more than 5% of the time. Is his probability of being late ever likely to fall below this figure, or should he seek alternative transport? [10 marks]
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10. Let be the population of lions in a game reserve at time t in years and
be the corresponding population of buffalo upon which they prey. Suppose these populations satisfy the differential equations
x ( ) t y ( ) t
x dt
dy
y dt
dx
The game warden carries out an initial survey in which he estimates the population of lions to be 248 and buffalo to be 768. Obtain and solve the differential equation for y in terms of x. From your results draw a phase diagram for this situation, indicating which way around the curve the point ( x , y )moves. [8 marks] By using the above equations derive the second order equation
2
d x .
Solve this equation to find x ( t )and hence y ( t ). Deduce that it takes about 15 months after the survey for the buffalo population to reach its maximum. [7 marks]
11. A sheep farmer in the Australian outback maintains a large flock of sheep whose population , where t is the time in years, satisfies the differential equation
n ( ) t
n^2 n dt
dn = −.
Find the equilibrium values of the flock and determine their stability. [5 marks] Integrate the above equation (use partial fractions) to find n ( t )assuming n = 200 when t = 0. [5 marks] The farmer wishes to sell 150 of his sheep at the market each year. The proposed sale would alter the differential equation to
2 = − −
n n dt
dn .
Find the equilibrium population values and their stability in this case. Decide, by means of a diagram, if the farmer can sell 150 animals at market and ensure the long-term size of his flock doesn’t fall below 250 animals. (Again assume n = 200 when t = 0 .) [5 marks]
