Spring 2005
Math 312 Final Project
The final project for Math 312 can be either an extension of a model
that we have discussed in class, or a problem of your choice that you
analyze using the tools discussed in the class.
The three main types of models that we have discussed are
- Differential equations.
- Discrete maps.
- Markov chains.
Here are some of the models that we have discussed in class.
- Population growth: the logistic equation.
- The Solow growth model of macroeconomics.
- The pendulum.
- Spread of a disease (SIR and SIQR models).
Even a brief search of the web will find many
variations and extensions of these models.
- Romance, including the simple "Romeo and Juliet" model, and the
more complicated "Laura and Petrarch" model.
- The Richardson arms race model.
- Lanchester's model of combat. Many interesting modifications
of this model are easily developed.
- Competing Species.
- Predator-Prey models.
- Periodic drug doses.
- Probabilities of the occupation categories of succeeding generations.
- Analysis of a simple game using Markov chains.
- and more...
The possibilities for modifying or extending these models
are endless. For example:
- Consider a higher dimensional problem--perhaps an arms
race between three countries, or a love triangle.
- Consider using different (maybe more realistic) assumptions about the behavior of the system.
- If in class we used differential equations to develop a model, consider how the problem
would be modeled using a discrete map instead. How would the analysis
change?
You may also have seen problems in other classes that could be
analyzed with the tools we have discussed in Math 312.
These could be problems in economics, political science,
sociology, biology, etc. For example, a model from another
class in which the mathematical details were glossed over
because the instructor wanted to avoid differential equations
would make a great project for Math 312.
Numerical Computations
Several of the models that we discussed in class were simulated
on the computer using programs that I wrote and made available
on the web. If you have a problem that requires numerical
computations (for example, a set of nonlinear differential equations
with three or more dimensions), talk to me about setting up some
tools for you to use. (It would not take long to create web pages
similar to the "Laura and Petrarch" or "Linear 3D Map" pages.)
