This talk will report the nonlinear stability of large-amplitude traveling waves of a system of conservation laws derived from the Keller-Segel model with singular logarithmic sensitivity describing the chemotactic wave propagation by the bacterial. The results partially solve a long-standing question of the stability of traveling waves to the Keller-Segel model. The results are proved via the phase portrait analysis and the (weighted) energy estimates. In the talk, the background of the model will be introduced and the main challenges and primary ideas of the proved results will be sketched. The numerical simulations will be shown and the remaining open problems will also be presented.