  An important component in studying mathematical models in many biochemical
systems, such as those found in developmental biology, is phase transition. The
purpose of this work is to analyze the phase transition property of a
diffusion-chemotaxis model with proliferation source, as a macroscopic model of
behavior of mobile species. Along the way, we will discuss that the system
exhibits very rich pattern-forming behavior. In particular, a portion of the
present work is devoted to the proof of existence of hexagonal patterns as a
result of instability of two Fourier modes. It is also shown that they are
either saddle points or attracting nodes. Moreover, they belong to an attractor
which consists of finite number of steady-state solutions and their connecting
heteroclinic orbits. The structure of this attractor will be precisely
determined as well.
