UBC Math Biology Seminar: Vincent Calvez

Concentration waves of swimming bacteria Escherichia coli were described in his seminal paper by Adler (Science 1966). These experiments gave rise to intensive PDE modelling and analysis, after the original model by Keller and Segel (J. Theor. Biol. 1971), and the work of Alt (J. Math. Biol. 1980) and his co-authors. Together with Bournaveas, Perthame, Raoul and Schmeiser, we have revisited this old problem from the point of view of kinetic transport equations. This framework is very much adapted to the so-called run-and-tumble motion, in which bacteria modulate the frequency of reorientation (tumble) -- and thus the duration of free runs -- depending on chemical variations in the environment.

I will present some recent analytical and numerical results about the existence of traveling wave solutions for a coupled kinetic-parabolic system describing concentration waves of bacteria in a micro-channel. The parabolic-parabolic problem obtained in the diffusive limit admits unique traveling wave solutions without any restriction on the parameters. This is in opposition to the kinetic-parabolic system for which solutions may be not unique, or may not exist for some extreme range of parameters.