**Mathematical Biology [Fall 2019]**

**Prerequisites**

Basic knowledge about linear algebra (e.g. determinant and trace of matrices, eigenvalues), analysis, ODEs (steady states and their stability, bifurcations) and PDEs (e.g. separation of variables), and stochastic processes. (The key point, however, is the attitude: students should be willing to quickly fill in gaps in background knowledge.)

**Aim of the course**

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. At the end of the course the student is capable of reading a scientific paper on a topic in Mathematical Biology in depth and can summarize and discuss the contents and impact of the paper in a scientific presentation.

**Lecturers**

Sander Hille: (shille@math.leidenuniv.nl)

Assistant Professor at the Department of Mathematics, Leiden University

Bob Planqué (r.planque@vu.nl)

Assistant Professor at the Mathematics Department, Vrije Universiteit, Amsterdam

**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:

Sep 10: Chapter 2: Exploiting time-scale differences

Sep 17: Chapter 4: Movement in space

Sep 24 : Chapter 5: Linear diffusion

Oct 1: Section 6.1, 6.2, 6.3: Linear diffusion, Lyapunov-Schmidt (Appendix)

Oct 8: Section 6.4, 6.5: travelling waves, non-existence of patterns

Oct 15: Section 6.6, up to 6.6.11: Turing patterns (rest of 6.6 is to be handed in, on Nov 3)

Oct 22: Chapter 7: Chemotaxis

Oct 29: Chapter 8: Physiologically Structured Population Models

Nov 5: Chapter 10: Adaptive Dynamics

Nov 12: Chapter 11: Noise in biological systems

Nov 19: Chapter Chemical Reactions

Nov 26: Pitfalls

**Assignments**

Five assignments are planned. Percentages in brackets indicate how much they contribute to the homework part of your grade (see below for more information).

Hand in on October 1 : Chapter 3 (35%)

Hand in on October 8 : Section 4.4 (10%)

Hand in on October 22 : Exercises 6.3.3, 6.3.4, 6.3.5, 6.3.6

(Bonus 6.3.7) (20%)

Hand in on November 5 : Exercise 6.6, numbers 14, 15, 16 and 17 (15%)

Hand in on November 12 : Section 9.9 (excluding 9.9.8!) (20%)

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.

**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.

If you know the paper you'd like to work on, let us know, so we can put a CHOSEN behind it. In principle, each student has a unique paper to work on.

Depending on the total number of final participants, we will determine the length of the presentations. We will most likely hold a one-day final workshop, divided in a morning and afternoon session. All speakers in the morning session are asked to attend the full morning session, and similarly for the afternoon session. We will ask your personal availabilities in due course.

We expect about 15 min per presentation.

The length of the essay should be about 7-10 pages or so. The idea is to show us that you have tried to understand the modelling and calculations in the paper, and present the material in your own words. Try to be critical. Perhaps you didn't like the way things were presented, or you thought the authors were kind of weak in their motivation or analysis or interpretation. Let us see you have thought about the paper and its result, what it taught you about the biological problem, etc.

The planning is as follows :

November 12 : 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)

December 3 : no meeting (work at home on the topic)

December 10 : 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 December 14 by e-mail

December 17 : 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)

- Docent: Sander Hille
- Docent: Bob Planqué