Mathematical Biology

Credits 8 credit points
Instructors Planqué, R. (Vrije Universiteit), Diekmann, O. (Universiteit Utrecht)
E-mail R.Planque@few.vu.nlO.Diekmann@uu.nl
Aim This is a master course for math students about mathematical methods to gain insight in the mechanisms
underlying biological phenomena. 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.
Description

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 (case study: antibiotic resistant bacteria in the ICU)

8. Adaptive Dynamics

and additional topics, as time permits.

Organization -- 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 both pen and paper and computer tools (MatLab).
Examination Grades are to a large extent based on the handed in written texts and on oral presentations.
Literature Lecture notes will be prepared by the instructors. See also the course website for the latest details:
http://www.few.vu.nl/~rplanque/Onderwijs/MathBio/
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.)
  Last changed: 09-04-2014 16:35