The University of Southampton
Courses

# MATH3052 Mathematical Biology

## Module Overview

Biology is undergoing a quantitative revolution, generating vast quantities of data that are analysed using bioinformatics techniques and modelled using mathematics to give insight into the underlying biological processes. This module aims to give a flavour of how mathematical modelling can be used in different areas of biology. Typically the models that are used in biology cannot be solved analytically. Nonetheless they give very useful information about the behaviour of the system. We will start by studying what we can say about differential equations that we cannot solve. For example, we cannot solve the equation of a simple pendulum analytically, but we can still say under what conditions it has periodic solutions. For biological oscillators this is usually what matters: it is important that your heart beats regularly, but whether your pulse rate is 68 or 71 beats per minute is less critical. Having introduced the mathematical tools needed to study ordinary differential equations, we will apply them to simple models of population dynamics and biochemical reaction networks. We will finish by considering pattern formation: how animals get their spots can be modelled using partial differential equations. We will first see how the ordinary differential equation tools studied earlier in the module can be extended to partial differential equations and then apply them to simple pattern formation models.

### Aims and Objectives

#### Module Aims

The principal aim of this module is to indicate how techniques used to study ordinary differential equations are used in different areas of biology.

#### Learning Outcomes

##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Develop the ability to explain mathematical results in language understandable by biologists.
##### Disciplinary Specific Learning Outcomes

Having successfully completed this module you will be able to:

• Understand and apply the concept of stability of a fixed point solution of a system of ordinary and partial differential equations.
• Solve mathematically and interpret biologically simple problems involving one- and two-species ecosystems and biochemical reactions.
• Understand and apply the basis of pattern formation in PDE-based models of biological systems.

### Syllabus

A. Introduction to dynamical systems theory 1. Phase space of an ordinary differential equation; 2. Linear stability analysis of fixed points. B. Biology and ODEs 1. Population dynamics: how species spread, reproduce and die; 2. Gene regulatory networks: how many genes work together in a series of reactions to help us live. C. Patterns in biology 1. Linear stability of partial differential equations; 2. Simple pattern forming models.

### Learning and Teaching

#### Teaching and learning methods

Lectures, lecture notes, web support materials, private study.

TypeHours
Teaching60
Independent Study90
Total study time150

Glendinning, P. (1995). Stability, Instability and Chaos.

Jones D.S. & Sleeman B.D. (2010). Differential Equations and Mathematical Biology.

Alon, U. (2007). An introduction to systems biology: Design principles of biological circuits.

Leah Edelstein-Keshet (2005). Mathematical Models in Biology.

Murray J.D.. Mathematical Biology.

De Vries G., Hillen G., Lewis M., Müller J. and Schonfisch B. (2006). A Course in Mathematical Biology:Quantitative Modeling with Mathematical & Computational Methods.

Strogatz, S.H,. Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering.

Nicholas Britton. Essential Mathematical Biology.

### Assessment

#### Summative

MethodPercentage contribution
Exam  (2 hours) 80%
In-class Test 20%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH2038Partial Differential Equations

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.