Entry requirements

120 credit points and including Ordinary Differential Equations I

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 three parts chosen from the four areas listed below. In each part 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.

1, One, two and three dimensional dynamical systems

Applications: Population growth models in biology and chemistry; predator-prey models; disease spread; decision-making by groups; social dynamics and elections. Techniques: Non-dimensionalisation; stability of equilibria; phase plane analysis; perturbation theory; separation of time scales; and chaos.

2, Dynamical game theory

Applications: Natural selection and evolution in biology; designing economic systems that preserve a common good (e.g. the natural environment); and game theory in sociology. Techniques: optimisation; replicator equations; and adaptive dynamics.

3, High dimensional dynamical systems

Applications: thermodynamics; magnetisation; firefly flashing and cricket chirping; and flocking of birds and schooling of fish. Techniques: Hamiltonian systems; the Ising model; coupled oscillators and the Kuramoto model; and self-propelled particle models.

4, Spatial models

Applications: spread of disease and animal populations; spiral waves in chemical reactions; cell movement; developmental biology and pattern formation. Techniques: Introduction to partial differential equations; diffusion equation; reaction-diffusion equations; chemotaxis; and cellular automata.

Instruction

Lectures, problem solving and computer laboratories.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course which will combine analysis with numerical methods.