Perturbative traveling wave solution for a flux-limited reaction-diffusion morphogenesis equation

TL;DR

This paper derives an approximate analytical traveling wave solution for a flux-limited reaction-diffusion equation modeling morphogenesis, revealing a sharp wave front with saturated speed, supported by numerical and experimental comparisons.

Contribution

It introduces a perturbation method to find analytical solutions for a nonlinear flux-limited PDE in morphogenesis, extending the Fisher-KPP framework.

Findings

Wave propagates as a sharp front with saturated speed

Analytical solutions agree with numerical simulations

Theoretical results align qualitatively with experimental data

Abstract

In this study, we investigate a porous medium-type flux limited reaction--diffusion equation that arises in morphogenesis modeling. This nonlinear partial differential equation is an extension of the generalized Fisher--Kolmogorov--Petrovsky--Piskunov (Fisher-KPP) equation in one-dimensional space. The approximate analytical traveling wave solution is found by using a perturbation method. We show that the morphogen concentration propagates as a sharp wave front where the wave speed has a saturated value. The numerical solutions of this equation are also provided to compare them with the analytical predictions. Finally, we qualitatively compare our theoretical results with those obtained in experimental studies.