Dynamics of Interfaces in the Two-Dimensional Wave-Pinning Model

TL;DR

This paper analyzes the interface dynamics in the wave-pinning reaction-diffusion model, revealing a hierarchy of three distinct timescales governing wave propagation, curvature-driven evolution, and boundary motion.

Contribution

It provides a detailed asymptotic analysis and higher-order approximations of interface motion, uncovering the multi-timescale hierarchy in the wave-pinning model.

Findings

Interface propagates as a fast front

Interface evolves via area-preserving mean curvature flow

Boundary interface drifts toward higher curvature regions

Abstract

We study the mass-conserved reaction-diffusion system known as the wave-pinning model, which serves as a minimal framework for describing cell polarity. In this model, the interplay between reaction kinetics and slow diffusion forms a sharp interface that partitions the domain into high- and low-concentration regions. We perform a detailed asymptotic analysis and derive higher-order approximation equations governing the motion of this interface. Our results show that on a fast timescale, the interface evolves via propagating front dynamics, whereas on a slow timescale, it evolves as an area-preserving mean curvature flow. Furthermore, using the derived free boundary problem, we demonstrate that on a significantly slower timescale, an interface whose endpoints lie on the domain boundary drifts along the boundary toward regions of higher curvature. In summary, our analysis reveals that the interface dynamics in the wave-pinning model exhibit a hierarchy of three distinct timescales: wave propagation on a fast timescale, curvature-driven area-preserving evolution on a slow timescale, and motion along the boundary on a significantly slower timescale.