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Mathematics Institute

MA4N4 Transport Processes in Mathematical Biology

Lecturer: Tom Montenegro-Johnson

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 one hour lectures

Assessment: 100% by 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

Useful background:

Synergies:

Content: 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.

Indicative syllabus:

  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.

Module Aims: The aims of this module are

  1. To develop and understand a range of models for transport processes in biology.
  2. To articulate commonality in these models across systems, and elucidate their differences.
  3. Develop the partial differential equations relating to agent-based transport models, understanding when these are valid.
  4. Quantify a range of wave-like and self-similar transport behaviours displayed in various biological systems.
  5. Understand spatial pattern formation and diffusion-driven instability.

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.

Books: TBC

Additional Resources