David Tong: Mathematical Biology

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.


PDF


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