Chemotaxis is the directed movement of cells in response to the concentration gradient of chemical substances in their environment. Since the seminal work of Keller and Segel in 1970, where they first wrote down what now are called Keller-Segel models, there has been an intensive interest and large amount of work on these models, especially in the last 20 years. The most important phenomenon concerning chemotaxis is cell aggregation. To model this phenomenon mathematically, three methods have been developed. The first one, first proposed by Nanjundia and advanced by Childress and Percus formally, is to show the solution of the model blows up in finite time to form delta-singularity at several spots(pores) in the cell habitat; the second one, as pioneered by Lin, Ni and Takagi, is to use variational method (and Lyapunov-Schmidt method) to show the model has spikey steady states; the third one is to use the speaker’s method of combining Global Bifurcation Theorem and Helly’s compactness theorem to show the existence of spikey and transition layer solutions. Until now, the third method has been successful in dealing with some chemotaxis systems only in 1D or radially symmetric domains. The difficulty stems from proving the monotonicity of the second Neumann-Laplace eig
