MA 137 - Calculus I for the Life Sciences SPRING 2014
Possible Final Projects Due Date: 04/25/2014
This document provides some suggestions for possible final projects for MA 137. You can select any
other project of your choice provided it has both a substantial mathematical component (related to what
we learned in MA 137) and an adequate biological and/or medical interest. You can freely draw your
projects from the World Wide Web. The following website maintained by the author of our textbook is
also a valuable source of information:
http://bioquest.org/numberscount/
Here are some guidelines with which your paper must comply:
You have to turn in a typewritten paper, at least four (4) pages long. It should be double spaced,
it should use 12pt fonts (Times New Roman, Helvetica, or Arial);
Your paper should not be in an itemized form but it should be written in a narrative/expository
form. You can use sections and subsections. There must be an abstract, an introductory preamble
and a final conclusion;
You must quote, at the end of your paper, all the references that you used for your work;
Your paper must provide historical and contextual facts and/or preliminary background material
on the topic of your paper;
Your paper must contain all the steps of your calculations (say, derivatives and/or limits). You
must also indicate which properties you are using while doing your calculations (say, product
rule, chain rule, etc...);
Your paper must contain illustrative graphs and tables (Maple or Mathematica outputs, Excel
spreadsheets, etc...);
Your paper can be written by yourself or by a group of at most two students.
Your project paper will be worth at most 20 points.
Please seek help from your instructor if you are uncertain on what to do. You can also seek a preliminary
opinion from your instructor to determine whether you are doing satisfactory work. DO NOT
WAIT until the last minute to complete your project and/or seek help.
Projects for MA 137
Project Ideas/Suggestions
1. (U.S. Population Growth Analysis)
a) Describe the properties of the discrete logistic equation and the stability of its equilibrium
points.
b) The following table provides population data for the United States, 1790-2000, measured
in millions.
Year 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890
Pop. 3.93 5.31 7.24 9.64 12.87 17.07 23.19 31.44 39.82 50.16 62.95
Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Pop. 75.99 91.97 105.71 122.78 131.67 151.33 179.32 203.21 226.50 249.63 281.42
Which values of the parameters N0, R and K of the discrete logistic equation give a
reasonable fit of the U.S. population data given above? Make a graph with both the
actual data and the discrete logistic model fit.
2. (State/county population Analysis) Do a similar analysis for the state of Kentucky (or your
home state) or your home county. Indicate where you obtained your population data and how
reliable the data is, or might be. The population data should go back at least 150 years.
3. (Model of selection and mutation) Bacterial growth is usually exponential. Let the population
of a bacterium at time t be denoted by bt. Suppose that a mutant wild type with population mt
appears and begins competing. Assume the original type (or wild type) satisfies the difference
equation bt+1 = rbt and the mutant type satisfies the difference equation mt+1 = smt.
If s > r then the mutant type will grow faster than the wild type, while if r > s then the wild
type will outperform the mutants. The establishment of this mutant is an example of selection.
Selection occurs when the frequency of a genetic type changes over time.
If we observe this population over time, counting all of the bacteria each hour would be impossible.
Nonetheless, we could sample the population and measure the fraction of the mutant
type by counting or using a specific stain. If the fraction grows larger and larger, then we know
that the mutant type was taking over.
We are interested in the growth of this fraction of the population. Let pt be the fraction of
mutants at time t. What is this fraction?
pt =
number of mutants
total number
=
number of mutants
number of mutants + number of wild type
=
mt
mt + bt
2
Projects for MA 137
The fraction of the wild type is
fraction of wild type =
number of wild type
total number
=
number of wild type
number of mutants + number of wild type
=
bt
mt + bt
Now, note that these two fractions must add to 1.
bt
mt + bt
+
mt
mt + bt
=
mt + bt
mt + bt
= 1:
Therefore, the fraction of wild type is 1
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