EXISTENCE AND STABILITY ANALYSIS OF SPIKY SOLUTIONS FOR THE GIERER-MEINHARDT SYSTEM WITH LARGE REACTION RATES THEODORE KOLOKOLNIKOV, JUNCHENG WEI, AND MATTHIAS WINTER Abstract. We study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either stable or unstable, depending on the parameters. We fully classify the stability in this case. We characterize the destabilizing eigenfunctions in all cases. Numerical simulations are shown which are in full agreement with the analytical results. 1. Introduction In this paper, we study the Gierer-Meinhardt system in the limit of large reaction rates. Let us first put it in the context of Turing’s diffusion-driven instability. Since the work of Turing [16] in 1952, many models have been established and investigated to explore the so-called Turing instability [10]. One of the most famous models in biological pattern formation is the Gierer-Meinhardt system [4], [8], [9], which in one dimension can be stated as follows: A t = D A ΔA − A + A p H q , x ∈ (−1, 1),t> 0, τH t = D H ΔH − H + A m H s x ∈ (−1, 1),t> 0, (1) A x (±1,t)= H x (±1,t)=0,t> 0, where all of the parameters are positive and (p, q, m, s) satisfy 1 < qm (s + 1)(p − 1) < +∞, 1 <p< +∞. In all of the recent mathematical investigations it was assumed that the activator diffuses much slower than the inhibitor, that is (2) D H ≫ D A , 1991 Mathematics Subject Classification. Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40. Key words and phrases. Stability, Multiple-peaked solutions, Singular perturbations, Turing instability. 1