Stability and bifurcation analysis in a mechanochemical model of pattern formation

TL;DR

This paper investigates how mechanochemical feedback mechanisms can lead to stable, single-peaked pattern formation in tissue models, revealing bifurcation structures and stability conditions.

Contribution

It introduces a mechanochemical model coupling morphogen dynamics with tissue mechanics, analyzing stability, bifurcations, and pattern formation without needing a second inhibitor.

Findings

Only unimodal patterns are stable.

Bifurcation analysis shows subcritical and supercritical pitchforks.

Bistable regimes arise from fold bifurcations.

Abstract

We analyze the stability and bifurcation structure of steady states in a mechanochemical model of pattern formation in regenerating tissue spheroids. The model couples morphogen dynamics with tissue mechanics via a positive feedback loop: mechanical stretching enhances morphogen production, while morphogen concentration modulates tissue elasticity. Global strain conservation implements a nonlocal inhibitory effect, realizing a mechanochemical variant of the local activation--long-range inhibition mechanism. For exponential elasticity-morphogen coupling, the system admits a variational formulation. We prove existence of nonconstant steady states for small diffusion and uniqueness of the homogeneous state for large diffusion. Linear stability analysis shows that only unimodal patterns are stable, while multimodal solutions are unstable. Bifurcation analysis reveals subcritical and supercritical pitchforks, with fold bifurcations generating bistable regimes. Our results demonstrate that mechanochemical feedback provides a robust mechanism for single-peaked pattern formation without requiring a second diffusible inhibitor.