Mathematical Modeling

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Course Meeting Time:

  • MWF at 1:00 to 2:00 pm.

Course Meeting Place:

  • Emerson 305 (New Location beginning October 1, 2004).


  • Tuesday at 6:30 to 8:00 PM in Science Center 310.
  • Thursday at 6:30 to 8:00 PM in Science Center 101b.

Course Head: Thomas W. Judson

  • Office: SC 429
  • Tel: 495-5735
  • Email:
  • Office Hours: 2:30-3:30 Monday, 9:00-10:00 Friday, or by appointment.

Course Assistants:


  • Clifford Taubes. Modeling Differential Equations in Biology, Prentice-Hall, Upper Saddle River, NJ, 2001.
  • Computer software such as Matlab or Mathematica is very useful for studying differential equations. Both Matlab and Mathematica can be downloaded from FAS Computer Services.

Course Description

Considers the construction and analysis of mathematical models that arise in the environmental sciences, biology, the ecological sciences, and in earth and atmospheric sciences. Introduces mathematics that includes multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).

Goals of the Course

One of the major goals of this course is to understand how mathematics and other subjects, in this instance biology, can enrich and enliven each other. After taking this course, you should be able to read and interpret articles in biology or related subjects which employ the use of differential equations. Moreover, you should be able to identify whether the mathematics is merely "window dressing" or whether it is used in a substantial fashion to gain insight to an interesting phenomenon.

We hope that students will become comfortable formulating and analyzing elementary differential equations. If you should become a researcher, you should have the ability to employ mathematical techniques or seek collaboration with a mathematician in order to model problems of interest.

Learning Objectives

Upon successfully completing Math 19 you should have acquired a solid foundation of the following topics and be able to move directly into subsequent courses, including Ordinary Differential Equations (Mathematics 106).
  • Introduction to differential equations, exponential growth, and stability in a one component system.
  • Systems of first order differential equations, phase plane analysis, equilibrium in 2-component, linear systems, stability in non-linear systems, and periodic solutions, chaos.
  • Introduction to vectors, matrices, eigenvalues, and eigenvectors.
  • The advection and diffusion equations, separation of variables, pattern formation, stability criteria, and traveling waves.

Grading and Exams

The will be two midterm exams and a final project in lieu of a final exam. Your course grade will be determined as follows:

Component Date Percentage
Homework - 25%
Midterm I Wednesday, November 3, 2004
7:30-9:30 PM in Science Center D.
Midterm II Wednesday, December 15, 2004
7:30-9:30 PM in Science Center D.
Final ProjectTBA 25%

Semester numerical scores will be converted into letter grades according to the following method.

Range of numerical values Corresponding Letter
90-100 A
80-89 B
65-79 C
50-64 D
0-49 E

When we calculate your final grade at the end of the course, we will calculate a score on a 0-100 point scale using the scores that you have obtained during the course, and using the grade breakdown given above. Your course grade will then be obtained using this table. In the event of a fractional score, we will always round up to the nearest integer. We may modify these letter grades with a "+" or a "-" if we believe that your performance in the course warrants this. Make-up exams will be administered only if a documented serious illness or personal tragedy prevents a person from taking an exam at the scheduled time.


There is no question that the best way to learn math is by doing math, and homework exercises are an essential part of any math course. If you just go to a math class and watch the teacher work problems, but do not actually try doing any problems on your own, then there is very little chance you will really learn the subject. It is also very unlikely that you will do well on exams without working through homework problems ahead of time. While doing homework, do not just write down answers. Think about the problems posed, your strategies, the meaning of your computations, and the answers you get. The main point is not to come up with specific answers to the specific problems you are working on, but to develop an understanding of what you are doing so that you can apply your reasoning to a wide range of similar situations. It is very unlikely that later on in life you will see exactly the same math problems you are working on now, so learn the material in such a way that you are prepared to use your general knowledge of mathematics in the future, not just how to apply particular formulas for very specific problems.

We encourage you to form study groups with other students in the class so that you can discuss your work with each other; however, all work submitted must be written up individually. Make sure that even if you do work in groups, that you come away with the ability to explain everything you end up writing up in your homework.

There will generally be two problem sets each week. Assignments will be graded by your course assistant and will typically be returned to you at the following class meeting. We will then post solutions to the homework on the course website. Check the solutions so that you can learn from your work. In order for us to post solutions as soon as possible, and in light of the fact that getting behind in a math class is one of the most uncomfortable things you can do to yourself, homework must be turned in on time. Since we will drop your 3 lowest homework grades, please do not try to harass your course assistant into accepting a late homework assignment.

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Fall 2004

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Last modified: 9/27/2004 Harvard University FAS Shield
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