Localized Patterns in the Gierer Meinhardt Model on a Cycle Graph

Abstract

Abstract. In this study, we provide a detailed analysis of the spike solutions and their stability for the Gierer–Meinhardt model on discrete lattices. We explore several phenomena that have no analogs 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 [Formula: see text]-spike solution. The stability of this solution is determined by a [Formula: see text] by [Formula: see text] matrix. We further delve into the scenarios where [Formula: see text] and [Formula: see text], 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 that arise when the inhibitor’s diffusion is not significantly large, a condition under which the continuum limit approximation of the solution no longer holds. In this regime, 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 localized patterns for the Gierer–Meinhardt model.

The authors' abstract, as published at the source. SIAM Journal on Applied Mathematics, 2026 · DOI ↗