Existence and stability of nonconstant positive steady states of morphogenesis models

In this paper, we study a one‐dimensional morphogenesis model considered by C. Stinner et al. (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. Then we rigorously study the stability of these nonconstant solutions when the sensitivity functions are chosen to be linear and logarithmic, respectively. Finally, we present numerical solutions to illustrate the formation of stable inhomogeneous spatial patterns. Our numerical simulations show that this model can develop very complicated and interesting structures even over one‐dimensional finite domains. Copyright © 2014 John Wiley & Sons, Ltd.