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MA4N4-15 Transport Processes in Mathematical Biology

Department
Warwick Mathematics Institute
Level
Undergraduate Level 4
Module leader
Tom Montenegro-Johnson
Credit value
15
Module duration
10 weeks
Assessment
100% exam
Study location
University of Warwick main campus, Coventry

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

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.

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.

Study time

Type Required
Lectures 30 sessions of 1 hour (29%)
Seminars 9 sessions of 1 hour (9%)
Private study 66 hours (63%)
Total 105 hours

Private study description

Homework problems.

Costs

No further costs have been identified for this module.

You must pass all assessment components to pass the module.

Assessment group B1
Weighting Study time Eligible for self-certification
Assessment component
In-person Examination 100% 45 hours No

Standard 3 hour written exam.


  • Answerbook Pink (12 page)
Reassessment component is the same
Feedback on assessment

Written feedback on the outcome of the exam.

Past exam papers for MA4N4

Courses

This module is Optional for:

  • Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
  • TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
    • Year 1 of G1PC Mathematics (Diploma plus MSc)
    • Year 2 of G1PC Mathematics (Diploma plus MSc)

This module is Option list A for:

  • Year 4 of UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)

This module is Option list B for:

  • Year 1 of TPXA-F345 Postgraduate Taught Modelling of Heterogeneous Systems (PGDip)
  • Year 4 of UCSA-G4G3 Undergraduate Discrete Mathematics
  • Year 5 of UCSA-G4G4 Undergraduate Discrete Mathematics (with Intercalated Year)

This module is Option list C for:

  • UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
    • Year 4 of G105 Mathematics (MMath) with Intercalated Year
    • Year 5 of G105 Mathematics (MMath) with Intercalated Year
  • UMAA-G103 Undergraduate Mathematics (MMath)
    • Year 3 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
  • Year 4 of UMAA-G107 Undergraduate Mathematics (MMath) with Study Abroad