Course description: Reaction-diffusion (RD) equations have found applications in numerous fields, from physics and chemistry, being the most classical, to sociology, being the most recent.  While reaction-diffusion models are vast simplifications of reality, there is much one can learn about many complex systems which we constantly struggle to understand.  This class will focus on the application of such models to numerous fields, focusing on biology, ecology, and criminology.  While exploring how different phenomena are modeled though RD equations (or systems) we will learn some of the theory and techniques necessary to analyze these type of models and this will enable us to draw conclusions about the real world applications.  Such techniques will include singular perturbation theory, maximum principles, linear stability/instability analysis, to name a few.  This class is aimed at students interested in modeling systems in biology, physics, chemistry, ecology, and sociology, with a strong mathematical inclination.  We will assume a solid understanding of ordinary differential equations and linear algebra.

Meeting times:  Tuesday and Thursday from  2:00-3:15 in PH 367.

Office Hours:  T and Th from 3:30-4:30 pm, W from 10 – 11 am in PH 314 or by appointment.

Tentative Course Outline:  I hope to be able to cover these topics (tentatively in this order)

  • Introduction to reaction-diffusion equations
  • ODE theory
  • Maximum/comparison principles
  • Separation of variables
  • Critical patch problem
  • Traveling wave solutions
  • Pattern formation
  • Brain tumor growth models
  • Reaction-diffusion equations in the social sciences
  • Singular perturbation theory
  • Neural-network models

Exams: The final exam will consist of student presentations.

Updated course outline:

  • Feb 14: Finish proof of critical patch problem and excitable systems (reference [BR]).
  • Feb 16: Fisher-KPP and excitable systems in higher dimensions (reference [BR]).
  • Feb 21: Traveling wave solutions: existence of fronts (references [MR1- Ch 13]).
  • Feb 22: Wave speed and wave profile (reference [MR1- Ch 13]).
  • Feb 28: More traveling wave examples (predator-prey system) [MR1- Ch 13]; waves in excitable media [KS- Ch 5,6]
  • March 2: Waves in excitable media [KS- Ch 5,6]
  • March 7:Traveling Pulse (singular perturbation theory); global behavior of nerve propagation [KS- Ch 5,6]
  • March 9:  Pattern Formation [MR2 – Ch 2]
  • March 21:

 

References:

[BR]  N. Britton. Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, 1986.

[MR1]  J. Murray. Mathematical Biology, I: An Introduction. Springer, 2002.

[MR2]  J. Murray. Mathematical Biology II Spatial Models and Biomedical Applications.

[EK]  L. Edelstein-Keshet. Mathematical models in biology. McGraw-Hill, 1997.

[KS] J. Keener and J. Sneyd.  Mathematical physiology. I: Cellular Physiology 2009.