B5.5 Further Mathematical Biology - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Helen Byrne
General Prerequisites: 

Part A Differential Equations I and Modelling in Mathematical Biology. Part A Differential Equations II is also preferable but not necessary.

Course Term: 
Michaelmas
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
H

Assessment type:

Course Overview: 

Further Mathematical Biology provides an introduction to more complex models of biological phenomena, including spatial models of pattern formation and free boundary problems modelling invasion. The course focuses on applications where deterministic models formulated using discrete, ordinary and/or partial differential equations are appropriate. By using particular modelling examples in ecology, chemistry, biology and physiology, the course demonstrates how applied mathematical techniques, such as linear stability, phase planes, singular perturbation and travelling waves, can yield important information about the behaviour of complex models.

Learning Outcomes: 

Students will have developed a sound knowledge and appreciation of the ideas and concepts related to modelling biological and ecological systems using discrete, ordinary and partial differential equations.

Course Synopsis: 

Enzyme-substrate kinetics, including quasi-steady state assumption inhibition and cooperativity

Trans-membrane ion transport: Hodgkin-Huxley and Fitzhugh-Nagumo models.

Spatial models for a multiple species, including epidemics and morphogen gradients.

Travelling wave propagation with biological examples, including Fisher’s equation.

Biological pattern formation, including Turing's model for animal coat markings, and chemotaxis models.

Moving boundary problems, with biological examples, including colonisation and wound healing.

Discrete-to-continuum models: coagulation and fragmentation models, biased-random walks and chemotaxis.

Reading List: 
  1. J.D. Murray, Mathematical Biology, Volume I: An Introduction, 3rd Edition (Springer, 2002).
  2. J.D. Murray, Mathematical Biology, Volume II: Spatial Models and Biomedical Applications, 3rd edition (Springer, 2003).
Further Reading: 
  1. J. Keener and J. Sneyd, Mathematical Physiology (First Edition Springer, Berlin, 1998)
  2. N. F . Britton, Essential Mathematical Biology (Springer, London, 2003).