Abstract
Solutions of a model reaction-diffusion system inspired by a model for hair follicle initiation in mice are constructed and analysed for the case of a one-dimensional domain. It is shown that all regular spatially heterogeneous solutions of the problem are unstable. Numerical tests show that the only asymptotically stable weak solutions are those with large jump discontinuities.
Highlights
A parabolic reaction-diffusion system is proposed in [14] to model the WNT signaling pathway in primary hair follicle initiation in mice
It is demonstrated that heterogeneous solutions arise because of diffusion-driven instability, and due to convergence to far-from-equilibrium solution branches
Of particular interest are the properties of the non-negative stationary solutions of (1), i.e. those pairs (u, v) such that ut = vt = 0
Summary
A parabolic reaction-diffusion system is proposed in [14] to model the WNT signaling pathway in primary hair follicle initiation in mice. The authors in [14] use a modified version of the well-known activator-inhibitor (Gierer-Meinardt) model [3], [4] with saturation and without source terms. It is demonstrated that heterogeneous solutions arise because of diffusion-driven instability, and due to convergence to far-from-equilibrium solution branches. This short note compares stationary solutions in the singularly perturbed problem (letting the inhibitor’s diffusion rate tend to 0) and the reduced problem (setting the inhibitor’s diffusion rate equal to 0) based on the modified equations from [12] for the case of a one-dimensional domain. We show that all strictly positive, spatially heterogeneous, regular solutions of the reduced problem are unstable.
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