Course Information

Lecturer:  

Professor Mary Myerscough, The University of Sydney

Synopsis:

Mathematics has a plethora of applications to biological systems. This subject covers some of the models and techniques of classical mathematical biology, including population biology, epidemiology, oscillating systems and neural action potentials, and associated mathematical techniques.

Course Overview:

Mathematical Biology

  The Mathematical Biology course at AMSI Summer School 2017 is sponsored by ANZIAM

  • Basic techniques of nonlinear ODEs: phase planes and linear analysis of steady states. Example: predator-prey models and models for two competing species. (2 lectures—mainly revision to make sure everyone has the basics.)
  • Limit cycles, the Hopf bifurcation theorem, the Poincare-Bendixson theorem, limit cycle stability. Example: Schnackenberg kinetics, the Brusselator model. (4 lectures, includes finding limit cycle stability using focal values and reconciling different results for particular problems)
  • Slow-fast systems, excitable kinetics and relaxation oscillators. Examples: the Hodgkin-Huxley model for neural conductance, the FitzHugh-Nagumo model, the Belusov-Zhabotinski reaction. (4 lectures)
  • Stationary bifurcations. Classifying bifurcations using singularity theory.  Examples: Spruce budworm population dynamics, the cubic autocatalator. (4 lectures)
  • Travelling wave analysis. Examples: spruce budworm (again); rabies in foxes. (4 lectures)
  • Travelling waves in excitable media. Examples: travelling pulses in the FitzHugh-Nagumo system. (2 lectures)
  • Epidemiological models. SIR models and extensions.  Endemic disease and R0. Examples: SIR, SIS diseases.  Sexually transmitted diseases. (5 lectures. I will probably  include the next-generation approach to finding R0 in structured models.)
  • Continuous age/size population models and McKendrick-von Foerster equations. Discrete age- or stage- structured models (if time permits). Leslie matrices. Coates graphs. The Perron-Frobenius Theorem.  Examples: noxious weeds, Teasel and Spartina.  (3 lectures)

Contact Hours:

28 lectures incorporating student presentations and class discussion. After theory is presented, students will be given examples to do for homework which they will present on the board in a later class.

Timetable:

Please see the Timetable for scheduled class times during the AMSI Summer School 2017.
Please note: Mathematical Biology is held concurrently with Category Theory and Computer Science and students cannot attend both courses in 2017.

Prerequisites:

Assumed knowledge:

Some basic experience:

  • Mathematical modelling.
  • Basic linear algebra.
  • Mathematical maturity.

Note: Exposure to MATLAB may be useful for the last part of the course.

Assessment:

  • Completion of Set Exercises & Participation in Class Discussion: 10%
  • Short Assignments: 10%
  • Reading Assignment: 10%
  • Quiz at Conclusion of Course (closed book): 10%
  • Final Examination (open book): 60%

Resources:

Coming soon.

Lecturer Biography

Mathematical Biology

Professor Mary Myerscough, The University of Sydney

Mary received her first degrees in Applied Mathematics from the University of Sydney and then completed her D.Phil (the equivalent of a PhD) in Mathematical Biology at Oxford University in the United Kingdom.

She returned to Australia to take up a research position in the School of Chemistry at Macquarie University where she studied the mathematics of exothermic chemical reaction kinetics, of ecological models and of honeybee behaviour.

In 1990, Mary started work at Sydney University and since then, she has been working on problems in social insect behaviour in collaboration with biological scientists at the University of Sydney, Macquarie University and CSIRO.

In 2015, she was a Eureka Prize finalist for work on honeybee demography and has recently diversified into modelling the formation of atherosclerotic plaque.

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