Fall 2002

Course Head: Thomas W. Judson
Office: SC 435
Tel: 495-4744

Office Hours: 3-4 pm on Monday, 3:30-4:30 Thursday or by appointment.

Course Assistants: Blythe Adler (bmadler@fas.harvard.edu) and Nick Ma (nma@fas.harvard.edu)

Textbook: Clifford Taubes. Modeling Differential Equations in Biology, Prentice-Hall, Upper Saddle River, NJ, 2001.

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: To understand how mathematics and other subjects, in this instance biology, can enrich and enliven each other. After taking this course one should be able to read and interpret articles in Biology or related subjects which employ the use of differential equations. Moreover, one 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.

It is hoped that after taking this course a student will be comfortable formulating and analyzing elementary differential equations. If such a student should become a researcher, they should have the ability to employ mathematical techniques or seek collaboration with a mathematician in order to model problems of interest.

Grading: Your course grade will be determined as follows:

  • Homework 25%
  • Two Midterms 25% each
  • Final Project 25%


  • First midterm Wednesday, October 23, in regular classroom and during regular class time.
  • Second midterm Monday, November 25, in regular classroom and during regular class time.
Make-up exams will be administered only if a documented serious illness or personal tragedy prevents of person from taking an exam at the scheduled time.

Homework: Homework problems are an integral part of this course. It is difficult to understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Please think about the problems posed, your strategy, the meaning of the computations you perform, and the answers you get. Although discussion of the homework with your peers is encouraged, copying any part of another person's assignment is at odds with Harvard's Code of Student Conduct. No late homework will be accepted. The lowest three homework scores will be dropped.

Course Outline: We will cover all 28 chapters of the textbook in order.

  • Chapter 1. Introduction
  • Chapter 2. Exponential growth
  • Chapter 3. Introduction to Differential Equations
  • Chapter 4. Stability in a one component system
  • Chapter 5. Systems of First Order Differential Equations
  • Chapter 6. Phase plane analysis
  • Chapter 7. Introduction to vectors
  • Chapter 8. Equilibrium in 2-Component, linear systems
  • Chapter 9. Stability in non-linear systems
  • Chapter 10. Non-linear stability again
  • Chapter 11. Matrix notation
  • Chapter 12. Remarks about Australian Predators
  • Chapter 13. Introduction to Advection
  • Chapter 14. Diffusion equations
  • Chapter 15. Two key properties of the advection and diffusion equations
  • Chapter 16. The no trawling zone
  • Chapter 17. Separation of variables
  • Chapter 18. The diffusion equation and pattern formation
  • Chapter 19. Stability criteria
  • Chapter 20. Summary of advection/diffusion
  • Chapter 21. Traveling waves
  • Chapter 22. Traveling wave velocities
  • Chapter 23. Periodic solutions
  • Chapter 24. Fast and slow
  • Chapter 25. Estimating elapsed time
  • Chapter 26. Switches
  • Chapter 27. Testing for Periodicity
  • Chapter 28. Causes of Chaos
Web contact: Mathematics 19
URL: http://www.courses.fas.harvard.edu/~math19/syllabus/
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