Introductory description

Biological systems are seldom well-mixed, but rather have spatial variations. In such cases, it is important to consider transport processes within the system, for instance in the spread of an invasive species, the swimming of bacteria towards nutrients, or the morphogenesis of a tiger's stripes. This module will cover the main mathematical techniques for modelling biological systems with transport, and will be focused around systems of coupled advection-diffusion-reaction partial differential equations, as well as agent-based equations.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Reynold's Transport Theorem, flux, and dimensionless quantities.
2. From agent-based models to transport PDEs.
3. Biological waves in single and two-species models.
4. Invasive species.
5. Spatial pattern formation.

Learning outcomes

By the end of the module, students should be able to:

• Derive transport PDEs from agent-based models.
• Find travelling wave solutions to single- and multi-species population models.
• Find solutions to invasive species problems in different domains.
• Quantify the relative importance/speed of transport processes in a given biological system.
• Find the conditions for a Turing Instability to occur in a system.

Interdisciplinary

This is a mathematics focused module on the life science interface. Students will be taught the necessary biology.

Subject specific skills

Ability to translate biological mechanisms into mathematical framework, eg growth in algal bioreactors
Ability to analyse wave-like solutions of coupled PDEs representing biological processes.
Ability to derive conditions for pattern formation in coupled PDE systems.

Transferable skills

Ability to translate scientific ideas into mathematical language and back. Ability to think creatively. Ability to discern the validity of a proposed explanation.