Entry requirements

120 credit points including Ordinary Differential Equations I or Dynamical Systems and Chaos

Learning outcomes

This course aims to provide insight and practice in how dynamical system (i.e. differential equation and difference equation) models can be used to better understand biology, physics, biochemistry, economics and sociology. The focus will be on formulating models, model analysis, using numerical solution tools to better understand models and drawing conclusions based on model outcomes.

Since the course contents is rather wide a focus will be made on increasing overall confidence in mathematical modelling, rather than deep study of each particular area. In order to pass the course the student should be able to

• formulate important models treated during the course;
• outline the mathematical methods and techniques that are used to analyse these models and understand in what situations these methods can be applied;
• understand how to draw a conclusion from a model;
• use a computer package to investigate models numerically;
• solve standard problems within the areas covered by the course.

Content

The course will consist of first an overview/revision of the theory of ordinary differential and difference equation models. This will be followed by a series of cases studies. In each case study there will be an emphasis both on how to build a model of various systems (i.e. applications), mathematical analysis (i.e. techniques), and on how numerical solutions increase understanding.

Case studies

At least three case studies will be chosen from the following possibilities. Solving the case studies will involve a combination of mathematical analysis and numerical solution in MATLAB.

1, Molecular and cellular biology. Non-dimensionalisation; Michaelis-Menten kinetics; matched asymptotic expansions; models of neural firing; oscillations in biochemical systems (Britton 6.1-6.4; Strogatz 8.3).

2, Coupled oscillators. Flows on the circle; driven and coupled pendulums; global bifurcations; firefly flashing; Kuramoto model (Strogatz 4.5-4.6 and 8.4-8.7 and further material)

3, Biological motion. Introduction to partial differential equations; diffusion equation; Fisher’s equation; travelling wave solutions; reaction-diffusion equations and pattern formation; Turing bifurcations (Britton 5; Britton 7).

4, Chaos. Population dynamics and one-dimensional maps. Cobweb diagrams; periodic windows; Liapunov exponent; renormalisation (Strogatz 10.1 – 10.7).

Instruction

Lectures, problem solving and computer laboratories.

Assessment

Written examination combined with assignments given during the course.