Analytical solutions for bacterial energy taxis (chemotaxis): Traveling bacterial bands

Motile bacteria may form bands that travel with a constant speed of propagation through a medium containing a dissolved substrate, to which they respond energy tactically. We generalize the analytical solution by Keller and Segel for such bands by accounting for (1) the presence of a porous medium, (2) substrate consumption described by a Monod kinetics model, and (3) an energy tactic response model derived by Rivero et al. Specifically, we determine the concentration profiles of the bacteria and the substrate. We also derive various expressions for the band velocity. The band velocity is also shown to equal the energy tactic velocity at the bacterial peak divided by tortuosity.