Resources
Intended Learning Outcomes
Once you've successfully completed this module you should be able to:
- Interpret differential equation models for populations, relating the expressions appearing in the model to processes
that affect the population.
- Formulate and analyse ordinary differential equation (ODE) models for the population of a single species, finding
equilibrium populations and determining how their stability depends on parameters.
- Analyse delay-differential equation (DDE) models for the population of a single species and use linear stability
analysis to determine which values of the parameters induce oscillatory instabilities.
- Analyse ODE models for the populations of two interacting species, finding equilibria and using information about their
linear stability to characterise the long-term behaviour of the system.
- Define a conserved quantity for a system of ODEs and, where possible, use such quantities to determine the
long-term behaviour of both two-species ODE models and single-species models population models that include diffusion.
- Construct the ODEs associated with a system of chemical reactions subject to mass-action kinetics and analyse them to
discover conserved quantities.
- Construct the Markov process associated with a system of chemical reactions and, for small numbers of reactions,
analyse it to determine the long-term behaviour of the system.
- Analyse two key models, Wolpert's Frech flag model and Turing's reaction-diffusion model, relating the
solutions of the associated PDEs to the processes of pattern-formation in developing organisms.
Lecture Notes
The first half of the course will cover classical topics in mathematical biology, following sections of Jim Murray's famous text,
Mathematical Biology I: An Introduction. This book is available via Reading Lists Online.
Printed notes are posted below. They outline topics that will be discussed in more depth in lectures.
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Feedback
The main channel for formal, written feedback in this module is the coursework. It will be a problem set similar to the ones provided on this page, but devoted to a specific application that uses the ideas from the course. Written solutions will be marked and scripts returned, providing both written comments and a mark. In addition, the weekly examples classes provide further opportunities for verbal feedback and, for students who bring written solutions to the exercises, on-the-spot marking and written feedback as well.
Problem Sets & Solutions
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Coursework
The coursework will be due at 1:00 PM on Friday, 20 March 2020. Please hand it in at Reception. The coursework contributes 20% to your final mark for the module and will be marked out of 20.
The 2020 Coursework problem concerns the SIR model (and variants) for the spread of a disease in a population. Scripts and marks will be returned at the start of Week 10.
Coursework & Exams
Previous exams include short questions on gene network motifs; these are not part of the 2020 version of the course and can be ignored. However the exam format will remain the same as in previous years.
Exam Solutions:
... will be posted here
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Background materials
- Books for key mathematical ideas and techniques (see also Reading Lists Online via the Library Catalogue):
- Nick Britton, Essential Mathematical Biology, Springer ISBN 978-85233-536-6
- James D. Murray, Mathematical Biology I: An Introduction 3rd edition, (Springer, 2002) ISBN 0-387-95223-3. The first half of the course covers classical topics described in this book.
- James D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications 3rd edition, (Springer, 2002, ISBN 0-387-95228-4
- Lee A. Segel, Modeling dynamic phenomena in molecular and cellular biology (Cambridge University Press, 1984). ISBN 0-521-27477-X An text from the early modern era, using deterministic models and ideas from dynamical systems.
- Darren J. Wilkinson, Stochastic Modelling for Systems Biology (Chapman & Hall/CRC, 2006). ISBN 1-58488-540-8 An up-to-date account of a developing field.
- Books for biological background:
- Bruce Alberts, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts and Peter Walter (2002)
Molecular Biology of the Cell. 4th edition, Garland Science. ISBN 0-8153-4072-9 This is detailed, comprehensive and big!
- Uri Alon (2008) An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC, 2007). ISBN 1-58488-642-0
- Evelyn Fox Keller (2002) Making Sense of Life, Harvard University Press. ISBN 0-674-01250-X A historical and philosophical view of mathematical biology.
- James Keener, James Sneyd (2009) Mathematical Physiology, Springer-Verkag New York, ISBN 978-0-387-75847-3
- Edda Klipp, Wolfram Liebermeister, Christoph Wierling, Axel Kowald, Hans Lehrach, Ralf Herwig (2009) Systems Biology: A Textbook. Wiley-Blackwell. ISBN 978-3-527-31874-2
- Bernhard Ø. Palsson (2006) Systems Biology: Properties of Reconstructed Networks. Cambridge University Press. ISBN 0-521-85903-4
- Ron Milo, Rob Philips (2016) Cell Biology by the Numbers. Taylor & Francis. ISBN 978-0-8153-4537-4. A great resource for practical calculations where quantification is needed.
- A tool (created by Chris Johnson) demonstrating reaction-diffusion patterns can be found here. The equations solved relate to those on page 65 of Turing's 1952 Phil. Trans. paper: α scales diffusion coefficients, setting the lengthscale of the pattern; β sets the ratio of diffusion coefficients, leading to stripes or spots; γ sets the ratio of reaction terms.
- 20 equations that changed biology
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