The stochastic Gierer-Meinhardt system

TL;DR

This paper investigates the stochastic Gierer-Meinhardt reaction-diffusion system, establishing existence of solutions and pathwise uniqueness in one dimension, with implications for understanding pattern formation in biological morphogenesis.

Contribution

It introduces a stochastic version of the Gierer-Meinhardt system and proves existence of solutions, including pathwise uniqueness in one dimension, advancing mathematical understanding of morphogenesis models.

Findings

Existence of solutions under certain conditions.

Pathwise uniqueness established in one dimension.

Open question of uniqueness in two dimensions.

Abstract

The Gierer-Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development, is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed in his pioneering work different activator-inhibitor systems with different diffusion rates, which could trigger the emergence of such cell structures. Mathematically, one describes these activator-inhibitor systems as a coupled systems of reaction-diffusion equations with hugely different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer-Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer-Meinhardt system with homogeneous Neumann boundary condition on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer-Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the system, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.