Topics in Mathematical Biology (MA390)
Prerequisites
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Any one or more of
MA235 Introduction to Mathematical Biology,
MA240 Modelling Nature's Nonlinearity,
PX253 Partial Differential Equations are recommended.
Content
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Why are some outbreaks of insect pests so much harder to contain than others?
How does the brain store and retrieve information?
How many memories can one person retain?
What makes high blood pressure dangerous,
and why
is the coronary circulation particularly prone to disease?
How come some animal coats have only a few blotches of colour while
others are finely striped or spotted,
and
why do you never see striped animals with spotted tails?
What makes biological clocks so eerily precise?
These are all questions that concern
the central theme of this course:
propagational phenomena in living systems.
Motivated by these questions,
a variety of mathematical techniques will be revised or introduced to study
stationary travelling waves in spatio-temporal population dynamics;
generation and propagation of action potentials in neurons;
the evolution of the pulse wave and its reflections in arteries;
and
pattern (standing wave) formation by reaction/diffusion mechanisms
in the developing organism.
Aims
-
To introduce ideas and techniques of mathematical modelling in biology.
Objectives
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To gain an insight into propagational phenomena in biology,
using a variety of mathematical techniques:
analysis of travelling wave solutions of PDEs
(phase plane methods, characteristics, Riemann invariants)
and of standing wave solutions (Fourier analysis);
establishing the existence of waves; computation of wave speeds;
dynamical systems and phase plane analysis
(stability of fixed points, characterization of special orbits).
Leads on to
-
MA498 Mathematics in Medicine.
Books
Essential:
-
J D Murray, Mathematical Biology, Springer,
or part 1 of the new two-part edition which will appear in 2002.
Highly recommended:
-
J Keener & J Sneyd, Mathematical Physiology, Springer.
-
JD Logan Nonlinear Partial Differential Equations, Wiley-Interscience.
-
F Verhulst Nonlinear Differential Equations and Dynamical
Systems, Springer.
-
L Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill.
Assessment
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3-hour examination (100%)
Lecturer
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Hugo van den Berg