David Tong: Lectures on Mathematical Biology
This is a course on Mathematical Biology, given to final year undergraduates. It mostly focusses on population dynamics, with a number of digressions to other biological systems that can be modelled by similar equations. Please do email me if you find any typos or mistakes.
Content
- 1. Population Dynamics and Other Stories: PDF
Malthusian exponential growth, the logistic equation, fixed points; Time delay differential equations, Hutchinson-Wright equation, Nicholson's blowflies, breathing; Age structured populations, von Foerster equation; Predator-Prey systems, Lotka-Volterra equations, competition, dengue fever, May stability criterion; epidemiology, SIR model; Chemical reactions, law of mass action, enzyme reactions, Michaelis-Menten reaction; Excitable systems, FitzHugh-Nagumo model. - 2. Discrete Time:
PDF
The logistic map, fixed points, bifurcation, chaos; Universality, renormalisation, Feigenbaum constants. - 3. Spatial Variations:
PDF
Reaction-Diffusion equations, cooking a turkey, diffusion with growth, non-linear diffusion; Travelling waves, Fisher equation, front propagation; Turing instability, pattern formation; Chemotaxis. - 4. Random Variations:
PDF
Discrete outcomes, Poisson process, extinction; Fokker-Planck equation, constant drift, fluctuation and dissipation.
Problem Sheets
- Problem Sheet 1: PDF Dynamical systems for populations.
- Problem Sheet 2: PDF More dynamical systems, discrete time, and diffusion.
- Problem Sheet 3: PDF Reaction-diffusion equations
- Problem Sheet 4: PDF Random Processes
Mathematical Biology on the Web
- Mathematical Biology by Julia Gog
- Mathematical Biology by Peter Haynes
- Mathematical Biology by Jeffrey Chasnov
- Lectures on Mathematical Systems Biology by Eduardo Sontag
- Lectures on Game Theory by Marcos Marino
- Introduction to universality and renormalization group techniques by Alessandro Sfondrini
- Lectures on Community Ecology by Wenping Cui, Robert Marsland, and Pankaj Mehta