  In this paper, We study an one--dimensional morphogenesis model considered by
C. Stinner et al. in (Math. Meth. Appl. Sci. 2012,35 (445-465). Under
homogeneous boundary conditions, we prove the existence of nonconstant positive
steady states through local bifurcation theories. We also rigorously study the
stability of the nonconstant solutions when the sensitivity function are chosen
to be linear and logarithmic function respectively. Finally, we present
numerical solutions to illustrate the formation of stable spatially
inhomogeneous patterns. Our numerical simulations suggests that this model can
develop very complicated and interesting structures even over one--dimensional
finite domains.
