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Modelling in Mathematical Biology

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Poster

Prof Graeme Pettet
Queensland University of Technology

Synopsis

Familiarity with instances of the use of mathematical models to describe and explain phenomena observed in biological contexts are now part and parcel of the background of applied mathematicians. The methods employed to develop and to solve such models are varied, and much of the art of modelling is associated with adopting appropriate techniques given the available data and the form of the analysis being sought. In this topic, we will explore a series of key mathematical models that have had an impact on the biological sciences, considering their formulation, solution and impact. Model formulations will involve ordinary and partial differential equations. Analysis will involve elementary phase plane techniques and simulation using NetLogo.

Topics:

(i) A brief overview of phase plane methods.
(ii) Modelling with ordinary differential equations for a series of case studies, including the Keller-Segel model of bacterial chemotaxis, TB in possums and the Fitzhugh-Nagumo model for neuron signalling.
(iii) Modelling with partial differential equations with a focus on models of tumour growth as a paradigm for tissue growth and repair, and Turing models as a basis for pattern formation.
(iv) Simulation and interpretation of model output using NetLogo.

Contact hours

28 hours of hours spread over four weeks, plus consultation as required.

Prerequisites

The course is largely self-contained but previous study of qualitative analysis of ODEs and PDEs would be recommended. No previous experience with NetLogo is required.

Resources

There are no specific texts for this topic, and it is expected that students enrolled in the topic will read widely. Some specific readings will be provided as appropriate.

A set of lecture notes or slides will be provided at the start of the Summer School. These will define the assessable content of the topic. Additional readings will be provided especially in those cases where textbook sources are unavailable. As some of the analysis and simulation activities undertaken in this topic will be performed using the freely downloadable multi-platform package NetLogo, it is assumed that each student will access to this package, typically installed on their own laptop computer.

Reference Reading

There are a number of useful textbooks in the area of mathematical modelling in biology. The following is a short list that would suit most students, but you are strongly encouraged to read around the content specified above. NetLogo has excellent tutorial resources available.

  1. Murray. Mathematical Biology, Springer, 3rd Ed. 2002.
  2. Edelstein-Keshet. Mathematical Models in Biology, McGraw Hill, 1988.
  3. Britton. Essential Mathematical Biology, Springer, 2003.
  4. Netlogo. http://ccl.northwestern.edu/netlogo/

About Graeme Pettet

I am presently a member of the Discipline of Mathematical Sciences at the Queensland University of Technology, where I function as the leader of the Applied Mathematics theme in the Collaborative Centre for Data Analysis, Modelling and Computation (DAMC) and as a member of the Institute of Health and Biomedical Innovation (IHBI).

My research interests lie at the interface of mathematics, life sciences and engineering where there are a wealth of problems for which novel and inventive solutions are sought, with a particular interest in biomechanics and regenerative medicine. Current collaborative projects cover such topics as modelling the lumbar spine, growth and function of the human epidermis, soft-tissue wound healing, tumour growth and fracture repair in bone. From a mathematical perspective, elements from many of these projects have led to models of chemotactically driven travelling waves and other issues of emergent behaviour wherein population or tissue level patterns evolve from small scale or cell level interactions.