Localized patterns in the Gierer Meinhardt model on a cycle graph

The paper derives the K-mode linearized spectrum and the stability threshold for symmetric K-spike states on the n-cycle via a continuum Green’s-function/Floquet reduction, yielding d_c = 1/[K arccosh(2 − cos(2π⌊K/2⌋/K))] (in particular, d_c = 1/(K arccosh 3) for even K). The candidate solution arrives at the same threshold by an exact discrete resolvent approach on the cycle, reducing to a K×K circulant eigenproblem and evaluating the spike–spike interaction via a discrete Poisson kernel. The two methods are consistent and produce identical thresholds; the candidate additionally gives an exact finite-n formula that asymptotically agrees with the paper’s continuum reduction.