In this paper, We study an one--dimensional morphogenesis model considered byC. Stinner et al. in (Math. Meth. Appl. Sci. 2012,35 (445-465). Underhomogeneous boundary conditions, we prove the existence of nonconstant positivesteady states through local bifurcation theories. We also rigorously study thestability of the nonconstant solutions when the sensitivity function are chosento be linear and logarithmic function respectively. Finally, we presentnumerical solutions to illustrate the formation of stable spatiallyinhomogeneous patterns. Our numerical simulations suggests that this model candevelop very complicated and interesting structures even over one--dimensionalfinite domains.
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