Synopsis for B8a: Mathematical Ecology and Biology

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A47)

Recommended Prerequisites:

Part A core material (especially differential equations). The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

Mathematical Ecology and Biology introduces the applied mathematician to practical applications in an area that is growing very rapidly. The course mainly focusses on situations where continuous models are appropriate and where these may be modelled by deterministic ordinary and partial differential equations. By using particular modelling examples in ecology, chemistry, biology, physiology and epidemiology, the course demonstrates how various applied mathematical techniques, such as those describing 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 ordinary and partial differential equations.

Synopsis

Continuous and discrete population models for a single species, including Ludwig's 1978 insect outbreak models hysterisis and harvesting.

Modelling interacting populations, including predator-prey and the Principle of competitive exclusion.

Epidemic models.

Michaelis–Menten model for enzyme-substrate kinetics.

Travelling wave propagation with biological examples.

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

Excitable systems. Threshold phenomena (nerve pulses) and nerve signal propagation.

J.D. Murray, Mathematical Biology, Volume I: An Introduction (2002); Volume II: Spatial Models and Biomedical Applications (2003) (3rd edition, Springer–Verlag).

1. Volume I: 1.1, 1.2, 1.6, 2.1–2.4, 3.1, 3.3–3.6, 3.8, 6.1–6.3, 6.5, 6.6, 8.1, 8.2, 8.4, 8.5, 10.1, 10.2, 11.1–11.5, 13.1–13.5, Appendix A.
2. Volume II: 1.6, 2, 3.1, 3.2, 5.1, 5.2, 13.1–13.4.

Further Reading

1. J. Keener and J. Sneyd, Mathematical Physiology (First Edition Springer, Berlin, 1998) 1.1, 1.2, 9.1, 9.2.
2. N. F . Britton, Essential Mathematical Biology (Springer, London, 2003). 1.1, 1.2, 1.3, 1.5, 2.1, 2.3, 2.4, 2.5, 2.7, 3.1, 3.2, 3.3, 5.1, 5.2, 5.3, 5.6, 7.1, 7.2, 7.3, 7.4, 7.5.