Mathematical Biology

2019-2020

Course Objective

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

Course Content

This course is part of the joint national master programme in
mathematics.
For schedules, course locations and course descriptions see
http://www.mastermath.nl.
Registration required via http://www.mastermath.nl

The course is taught once every two years.

Content:

1. Exploiting time scale differences : the
quasi-steady-state-approximation
-- Michaelis Menten enzyme kinetics
-- Holling's functional response
-- excitable media: Fitzhugh-Nagumo

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)

-- various derivations of the diffusion equation
-- the fundamental solution, superposition
-- transport by diffusion: what distance in how much time?
-- separation of variables, eigenfunctions/modes
-- the asymptotic speed of propagation

4. Reaction-Diffusion (nonlinearity)
-- travelling waves
-- scalar equations do NOT generate stable patterns (in convex domains)
-- Turing instability
-- bifurcation theory
-- transition layers (excitable systems)?

5. Age/size structured populations, cell cycle models

6. Chemotaxis

7. Branching processes, links to epidemiology

8. Adaptive Dynamics

9. Master equations and additional topics, as time permits.

Teaching Methods

-- lectures (notes are in preparation and should be ready by the time
the course is given) which explain and illustrate the methods while
referring to other sources for detailed accounts of the underlying
mathematical theory
-- assignments which provide training in modelling and in the use of
the methods. Students work on assignments, using pen and paper.

Method of Assessment

Grades are to a large extent based on the handed in written assignments
and on
oral presentations. Grading is based on 5 homework assignments and the
final project. The average grade of the 5 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. Given the nature of this course, there is no resit exam.

Literature

Lecture notes will be provided by the instructors. See also the course
website for the latest details:
http://www.few.vu.nl/~rplanque/Onderwijs/MathBio/

Target Audience

Mmath

Additional Information

This course is part of the joint national master programme in
mathematics. For schedules, course locations and course descriptions see
https://elo.mastermath.nl. Registration required via
https://elo.mastermath.nl/login.

Custom Course Registration

You have to register your participation in each Mastermath course via https://elo.mastermath.nl/login. Registration is mandatory and absolutely necessary for transferring your grades from Mastermath to the administration of your university.

Recommended background knowledge

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.)

General Information

Course Code X_400504
Credits 8 EC
Period P1+2
Course Level 400
Language of Tuition English
Faculty Faculty of Science
Course Coordinator dr. R. Planque
Examiner dr. R. Planque
Teaching Staff dr. R. Planque

Practical Information

You cannot register for this course yourself; your faculty's education office carries out registration

Teaching Methods Lecture