Mathematical Biology

This is a master course for mathematics students about mathematical methods to gain insight in the mechanisms underlying biological phenomena.

Teachers

Odo Diekmann, Professor at the Department of Mathematics, Utrecht University
Bob Planqué, Assistant Professor at the Mathematics Department, Vrije Universiteit, Amsterdam

Time and location

This course is given in the Spring Semester of 2012.
The course is held at the VU Amsterdam, room S6.55, between 10:15 and 13:00 and runs from February 7 until May 25. (Note that on March 27 we will convene in room C1.21.) Theoretically inclined biologists with an interest in mathematical methods are more than welcome to attend. Everyone is advised to register at the MasterMath website.

Lecture notes

The lecture notes for this course are still a work in progress. New chapters or enhanced versions of older ones will appear here for download.

Preparation

We advise the students to read the relevant part of the syllabus BEFORE coming to the lecture and to read it again after the lecture. Indeed, by the combination of reading about a subject and listening to an exposition about it, one usually understands things much better than by any of these in isolation.

The programme for the coming weeks is:

Setup of the course

The course consists of

During the third hour and at home students work by themselves or in couples on assignments, using mostly pen and paper.

In the course, a lot of attention is paid to "translation": how do we get from biological information to a mathematical formulation of questions ? And what do the mathematical results tell us about biological phenomena ? In addition, the course aims to introduce general physical ideas about time scales and spatial scales and how these can be used to great advantage when performing a mathematical analysis.

Prerequisites: basic knowledge about linear algebra, analysis, ODE, stochastic processes. (The key point, however, is the attitude: students should be willing to quickly fill in gaps in background knowledge.)

Presently four assignments are planned:
Hand in on February 28 : Chapter 3
Hand in on March 6 : Section 4.4
Hand in on March 20 : Exercises 6.3.3, 6.3.4, 6.3.5, 6.3.6 (Bonus 6.3.7)
Hand in on April 3 : Exercise 6.6, numbers 14, 15, 16 and 17

You are expected to hand in a detailed and readable elaboration. During the third hour of every session you can work on the assignments and ask the lecturers for advice. We recommend that you plan at home the questions you want to ask during the next session.

Concerning the final project, the aim is to write and lecture such that in a coherent (and for the students of the course understandable) way a specific topic is introduced and analysed, with attention for both the biological and the mathematical aspects. Whether or not the material comes from one or several papers is irrelevant.
Here is a list of papers from which you can choose.

The planning is as follows :
April 24 : decide about the topic by making a choice from among the various papers (note that you may suggest a paper yourself, but in this case you'll need our approval)
May 1 : no meeting (work at home on the topic)
May 8 : meeting for feedback, advice. Formulate a plan to us on what you will cover in your essay/presentation. It will help if you send such a plan one or two days before May 8 by e-mail
May 15 : no meeting. Send us a preliminary text for constructive critique. Make sure to do this early enough to allow us some time for reading and formulating feedback
May 22 : presentations and handing in of final project (depending on the number of participants, we may need to continue in the afternoon or to start earlier than usual

List of subjects

  1. Exploiting time scale differences: the quasi-steady-state-approximation
  2. Phase plane analysis
    Essentially an assignment: students work in couples through a series of exercises about prey-predator interaction. In a lecture we explain some key notions, such as linearized stability and Poincare-Bendixon.
  3. Diffusion
    (mainly linear theory; partly in the form of assignments)
  4. Reaction-Diffusion (nonlinearity)
  5. Miscellaneous topics and maybe additional topics, like infectious disease epidemiology, if time permits.
  6. Final project
    The last four weeks the students will work on a final project : writing an essay on a paper of their choice, and preparing a presentation about this paper. A list of suitable papers will be provided, but students may also come up with their own suggestions for a topic and/or a paper (note : these are subject to approval). The aim is to write and lecture such that in a coherent (and for the students of the course understandable) way a specific topic is introduced and analysed, with attention for both the biological and the mathematical aspects.

Learning goals and grading

After completion of the course, the student is able to
  1. read and understand the research literature about (deterministic) models of biological phenomena
  2. participate actively in projects that aim to model biological phenomena
  3. derive mathematical equations from bookkeeping considerations
  4. interpret mathematical results in the biological context that motivated the analysis; more precisely the point is that mathematical statements are translated into a relation between phenomena and the underlying mechanisms
  5. use formal arguments (based on differences in the time- or spatial scale of various mechanisms) to simplify equations in a meaningful way
  6. apply various analytical techniques to study phase portraits of planar ODEs representing ecological systems
  7. derive and analyse linear diffusion equations and their solutions
  8. apply bifurcation theory to study systems of nonlinear reaction- diffusion equations
Grading is based on 4 home assignments and the final project. The average grade of the 4 home assignments will contribute 40% to the final grade. The written work on the paper will contribute another 40% and the remaining 20% will come from the oral presentation.

Here we present the ``toetsmatrijs'' for this course The numbers 1-8 in the first column refer to the learning goals listed above.
H1 = first home assignment
H2 = second home assignment, etc.
essay = written essay
pr = presentation
The question marks indicate that, depending on the subject of the final project, these may or may not contribute substantially.

H1 H2 H3 H4 essay pr
1    - - - - 60 40
2 10 10 10 10 40 20
3 20 20 - - 50 10
4 10 10 10 10 30 30
5 100 - - - ? ?
6 100 - - - ? ?
7 - 100 - - ? ?
8 - - 50 50 ? ?