Entry requirements

120 credits including at least 60 credits in Mathematics, or the course Modelling in Biology

Learning outcomes

This course is aimed to be accessible both to Master's students of biology who have a good understanding of the introductory course to mathematical modelling and to Master's students in applied mathematics looking to broaden their application areas. The course extends the range of usage of mathematical models in biology, ecology and evolution. Biologically, the course looks at models in evolution, population genetics and biological invasions. Mathematically the course involves the application of multivariable calculus, ordinary differential equations, stochastic models and partial differential equations. In order to pass the course (grade 3) the student should be able to

• formulate and solve mathematical models of evolution in terms of optimisation and game theory problems;
• use techniques from stochastic processes to describe population genetics;
• use techniques from partial differential equations to describe spread of genes, disease and other biological material;
• explain how these techniques are applied in scientific studies and applied in ecology and epidemiology.

Content

The use of mathematical models in biology, ecology and evolution. Several Variable Calculus, Ordinary Differential Equations, Stochastic Modelling, Partial Differential Equations.

The course will consist of some of the following sections:

• Evolutionary Invasion Analysis: introduction to game theory; concept of evolutionary stability; general technique for invasion analysis.
• Population genetics: Stochastic models of genetics; Genetic structure and selection in subdivided populations; Kin selection and limited dispersal.
• Diffusion in biology: Constructing diffusion models; diffusion as approximation of stochastic systems; biological waves; pattern formation and Turing bifurcations; Chemotaxis.
• Networks in biology: Spread of disease in contact networks; random graphs; moment closure techniques in complex graphs.

Instruction

Lectures, problem solving and computer laboratories.

Assessment

Compulsory assignments combining analysis and numerical solution. Written and, possibly, oral examination at the end of the course.