Chemotaxis is the biased motion of cells under the influnce of chemicals that attract the cells.
The most important phenomenon about chemotaxis is cell aggregation, for which we use non- constant, especially spiky or transition-layer (step function-like) steady states to model. In the case of 1D spatial domains, we present two methods to establish the existence of such steady states: (i) global bifurcation theory combined with Helly's compactness theorem and Sturm oscillation theorem; (ii) singular perturbation method. We also prove local asymptotic stability and uniqueness of these steady states