Computer Assisted Mathematical Modeling
Math 354, Spring 2011
Mid-semester Exam
This exam consists of 15 questions. You may use a calculator.
1) What is a difference equation?
2) Write out the first four terms of the sequence satisfying the following difference
equation:
an  3  an , a0  3
3) Examine the following sequence:  3, 8, 13, 18, 23,  
a) Write a difference equation to represent the change during the nth interval as a
function of the previous term in the sequence.
b) Find a formula for the nth term of the sequence.
4) Examine the following sequence: 1, 4, 9, 16, 25,  
a) Write a difference equation to represent the change during the nth interval as a
function of the previous term in the sequence.
b) Find a formula for the nth term of the sequence.
5) Find the equilibrium value of the system a n 1
 2a n  10 if one exists.
6) Choose a value of b so that the following sequence converges to 100.
an1  0.9an  b, a0  50
7) At the time of her retirement, Cindy has $30,000 in a savings account that earns 6%
interest per year. Each year of her retirement, Cindy withdraws $8,000 to pay her
mortgage. Provide a model of the amount of money in Cindy’s savings account in the
(n+1)th year in terms of the amount of money in the nth year (a model that would be
valid until the account runs out of money).
8) Suppose a species of spotted owls competes for survival in a habitat that also supports
hawks. A model for the owl (W) and hawk (H) populations is given by:
H n1  1.2 H n  0.004 Wn H n
Wn1  1.7 Wn  0.002 Wn H n
What is the largest number of owls and hawks that can co-exist in equilibrium?
9) Monarch butterflies are poisonous due to the concentration of cardenolide aglycones
that comes from eating milkweed. Suppose that another species of butterfly looks like
the Monarch butterfly but is not poisonous. A scientist hypothesizes that if most
butterflies are Monarch butterflies, then both butterfly populations will be protected
from predators. If most butterflies are copycat butterflies, the butterfly populations
will not be protected.
a) Write down a model describing the populations of the two butterfly populations if
the population of monarch butterflies is given by M (t ) and the population of
copycat butterflies is given by C (t ) . Assume that both butterfly populations
double their populations every month due to a fast breeding cycle and the rate of
C (t )
butterfly predation increases linearly with the ratio
(the number of copycat
M (t )
butterflies relative to the number of Monarch butterflies).
b) What are the possible equilibrium populations of Monarch and Copycat
butterflies?
10) Write down the difference equation describing the logistic growth of a population P.
Show how the logistic equation may be derived from simple assumptions.
11) Under what conditions will the logistic growth model approach an exponential
growth model?
12) Let X (t ) be the size of a tumor at time t. The growth of a tumor is often modeled by a
k 
Gomperzian function: X  k1 log  2   X . If the growth of a tumor is given by the
X
equation above, what is the growth rate of the tumor?
13) The autonomous differential equation
dP
 3P(100  P)P  50  represents a model
dt
for population growth.
a) If the population size is currently 90, is the population growing or shrinking?
b) Given your answer above, at what size will the population stop growing or
shrinking?
c) Is this equilibrium stable or unstable?
dx
x y4
dt
dy
x y2
dt
14) Consider the following system of two first-order
differential equations:
a) Sketch the x- and y-nullclines on the coordinate axes below.
x-nullcline
y-nullcline
b) Identify the equilibrium points of this system, if any.
15) Consider the following differential equation:
y  x  y
Given that a solution to this differential equation has a value of 4 when x = 0, estimate the
value of the solution when x = 4 by finding the first two approximations y1, y2 using
Euler’s method and step size x = 2. The slope field for this differential equation for a
region of the first quadrant is shown.