Existence and stability of nonconstant positive steady states of morphogenesis models

TL;DR

This paper investigates the existence and stability of nonconstant positive steady states in a one-dimensional morphogenesis model, using bifurcation theory and numerical simulations to reveal complex pattern formations.

Contribution

It provides rigorous proofs of nonconstant steady states and analyzes their stability for specific sensitivity functions, complemented by numerical illustrations of pattern formation.

Findings

Existence of nonconstant positive steady states proven.

Stability analyzed for linear and logarithmic sensitivity functions.

Numerical simulations show complex spatial patterns can develop.

Abstract

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.