BiMed-Math Lecture Series

First I review the work of Alan Turing (1952) and related results by Gierer-Meinhardt, James Murray, Shigeru Kondo and other people.  Then I will talk about various results on the stability of stationary solutions of scalar reaction-diffusion equations developed in 70's, 80's and 90's, including my own results on the existence of stable nonconstant solutions in dumbell-shaped domains (1979), and my results with Morton Gurtin on the stability of stationary solutions of Cahn-Hilliard equations (1988).  Various other related results are also mentioned.

We consider curvature-driven motion of plane curves (in other words, motion by curvature with a driving force) in a two-dimensional infinite cylinder with sawtooth-shaped boundary.  This equation can be regarded as a sharp-interface limit of a bistable reaction-diffusion equation.  We derive necessary and sufficient conditions on the existence of traveling waves and show that the speed of the traveling waves depends on the angles of the sawtooth. We also discuss the homogenization limit of the problem as the bumps of the boundary become finer and finer.  This is joint work with Bendong Lou and Ken-Ichi Nakamura (2006, 2013).

We consider KPP type reaction-diffusion equations with spatially periodic coefficients and discuss the speed of the traveling waves. I first recall the pioneering work of the three mathematical ecologists, Teramoto, Shigesada and Kawasaki (1986), and the more recent result by Gregoire Nadin (2010), who, with an ingeneous variational approach, answered the open questions raised by Teramoto et al.  Next we consider KPP type equations in spatially heterogeneous 2D media having stripe patterns and discuss how the speed of the traveling wave depends on the direction of propagation (joint work with Xing Liang 2014).  I also discuss similar questions for an epidemiological model describing spread of desease in vineyard (joint work with Arnaud Ducrot 2016).