Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem

The most important phenomenon in chemotaxis is cell aggregation. To model this phenomenon we use spiky or transition layer (step-function-like) steady states. In the case of one spatial dimension, we carry out global bifurcation analysis on the Keller-Segel model and several variants of it, showing that positive steady states exist if the chemotactic coefficient χ is larger than a bifurcation value χ1 which can be explicitly expressed in terms of the parameters in the models; then we use Helly's compactness theorem to obtain the profiles of these steady states when the ratio of the chemotactic coefficient and the cell diffusion rate is large, showing that they are either spiky or have the transition layer structure. Our results provide insights on how the biological parameters affect pattern formation, and reveal the similarities and differences of some popular chemotaxis models.