Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Part II of Exam 1
3. Consider the linear rst order dierential equation
= + ,
where and are real parameters with = 0.
(a) Find the equilibrium points of the equation.
Solution: Solve the equation + = 0 to get that
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Part I of Exam 1
1. Consider the dierence equation = , where is a nonzero parameter.
(a) Give an interpretation of the equation as a model for population growth.
Answer: This model assumes that the percapita grow
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Review Problems for Exam #1
1. Consider the dierence equation +1 = + , where and are real
parameters, given that 0 is known.
(a) Find a closed form solution, , to the equation and discuss how the
behavior of the
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
Topics for Exam 1
1. Discrete Models of Population Growth: Dierence Equations
1.1 Modeling bacterial growth: A conservation principle
1.2 Malthusian or geometric growth (or decay) models
1.3 Logistic growth
1.4 General discrete
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Exam 2 (Part I)
Monday, April 5, 2010
Name:
Show all signicant work and justify all your answers. This is a closed book exam. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Exam 2 (Part II)
Monday, April 5, 2010
Name:
This is the outofclass portion of Exam 2. There is no time limit for working on the
following problem. This is a closedbook exam; you are only allowed to consult the
distributions
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Part I of Exam 2
1. Suppose that the rate at which a drug leaves the bloodstream and passes into
the urine at a given time is proportional to the quantity of the drug in the blood
at that time.
(a) Write down and
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Review Problems for Exam #2
1. Consider a pond that initially contains 10 million gallons of fresh water.1 Water
containing an undesirable chemical ows into the pond at a rate of 5 million
gallons per year and th
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Review Problems for Exam 2
1. Consider a pond that initially contains 10 million gallons of fresh water.1 Water containing an undesirable chemical ows into the pond at a rate of 5 million gallons per
year and the mixture in t
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Solutions to Part II of Exam 2
3. Luria and Delbrck1 devised the following procedure (known as a uctuation
u
test) to estimate the mutation rate, , for certain bacteria:
Imagine that you start with a single normal bacterium (
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Review Problems for Exam 1
1. Consider the dierence equation +1 = + , where and are real
parameters, given that 0 is known.
(a) Find a closed form solution, , to the equation and discuss how the
behavior of the solution as is
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Exam 1 (Part II)
February 24, 2010
Name:
This is the outofclass portion of Exam 1. There is no time limit for working on
the following two problems. This is a closedbook exam, and you are not allowed to
consult your notes or
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Exam 1 (Part I)
February 24, 2010
Name:
Show all signicant work and justify all your answers. This is a closed book exam. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the foll
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
5
Number of Embryo Cells vs. t
x 10
10
9
Number of Embryo Cells
8
7
6
5
4
3
2
1
0
0
5
10
Time t in units of 30 minutes
15
20
Insect Population Values
9
8
Number of Insects
7
6
5
4
3
2
1
0
0
2
4
6
Time t
8
10
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Plot of N versus t for No = 1
1.8
1.6
N
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Time t
2
2.5
3
N = 2 N 0.01 N2 75
180
160
140
120
N
100
80
60
40
20
0
0
0.5
1
1.5
2
2.5
t
3
3.5
4
4.5
5
N = r N (1 N/K) (N/T 1)
r=1
T=1
K=2
3
2.5
N
2
1.5
1
0.5
0
0
1
2
3
t
4
5
Mathematical and Computational Methods for the Life Sciences
MATH 36

Spring 2010
Math 36. Rumbos
Spring 2010
1
Topics for Exam 2
I. Continuous Models: First Order Dierential Equations
(a) Linear rst order equations
i. General solution
ii. Qualitative analysis: Equilibrium points and stability
(b) Applications: Conservation Principle a