Localized patterns in the Gierer Meinhardt model on a cycle graph

TL;DR

This paper analyzes spike solutions and their stability in the Gierer-Meinhardt model on discrete cycle graphs, revealing unique patterns and stability properties not present in the continuum limit.

Contribution

It provides a detailed stability analysis of spike solutions on discrete lattices, including new phenomena like asymmetric patterns and zigzag configurations.

Findings

Discrete systems retain spike patterns with zero activator diffusion.

Stable asymmetric spike patterns exist for K=2 and K=3.

Symmetric K-spike solutions are generally the most stable.

Abstract

In this study, we provide a detailed analysis of the spike solutions and their stability for theGierer-Meinhardt model on discrete lattices. We explore several phenomena that have no analogues in the continuum limit. For example in the discrete case, the system retains spike patterns even when diffusion of the activator is set to zero. In this limit, we derive a simplified algebraic system to determine the presence of a K-spike solution. The stability of this solution is determined by a K by K matrix. We further delve into the scenarios where K = 2 and K = 3, revealing the existence of stable asymmetric spike patterns. Our stability analysis indicates that the symmetric two-spike solution is the most robust. Furthermore, we demonstrate that symmetric K-spike solutions are locally the most stable configurations. Additionally, we explore spike solutions under conditions where the inhibitor's diffusion rate is not significantly large. In doing so, we uncover zigzag and mesa patterns that do not occur in the continuous system. Our findings reveal that the discrete lattices support a greater variety of stable patterns for the Gierer-Meinhardt model.