Traveling waves for chemotaxis–systems

In this paper we study the existence of traveling wave solutions to the Keller-Segel model, a general model of chemotaxis, where the species do not reproduce. In the case of logarithmic sensitivity we show that various functionals modeling the reactive feedback on the chemo-attractant do allow for traveling waves and a wide range of qualitatively different behavior is possible. We can find monotone fronts as well as pulse solutions in the densities of the population and the chemical. In particular, a new kind of solution exists, where both densities travel as pulses.