  This paper investigates the Keller-Segel model with quadratic cellular
diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its
nontrivial patterns. We obtain explicit formulas of radially symmetric
stationary solutions and such configurations give rise to the ring patterns and
concentric airy patterns. These explicit formulas empower us to study the
global bifurcation and asymptotic behaviors of these solutions, within which
the cell population density has $\delta$-type spiky structures when the
chemotaxis rate is large. The explicit formulas are also used to study the
uniqueness and quantitative properties of nontrivial stationary radial patterns
ruled by several threshold phenomena determined by the chemotaxis rate. We find
that all nonconstant radial stationary solutions must have the cellular density
compactly supported unless for a discrete sequence of bifurcation values at
which there exist strictly positive small-amplitude solutions. The hierarchy of
free energy shows that in the radial class the inner ring solution has the
least energy while the constant solution has the largest energy, and all these
theoretical results are illustrated through bifurcation diagrams. A natural
extension of our results to $\mathbb R^2$ yields the existence, uniqueness and
closed-form solution of the problem in this whole space. Our results are
complemented by numerical simulations that demonstrate the existence of
non-radial stationary solutions in the disk.
