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Travelling waves for chemotaxis systems
We study the existence of travelling wave solutions for Keller-Segel models, a general model of chemotaxis, where the species do not reproduce. In the case of logarithmic sensitivity we show that various functionals modelling the reactive feedback on the chemo-attractant do allow for travelling waves. In particular, a wide range of qualitatively different behaviour is possible. We can find monotone fronts as well as pulse solutions in the densities of the population and the chemical. There exists a new pattern of solutions, where both densities travel as pulses.