2017/05/23 by Juncheng Wei, Matthias Winter, Wei, Juncheng +3
Computer Science · Engineering · Physics and Astronomy · #35B25 #35B35 #35B40 #92C15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Slime Mold and Myxomycetes Research #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1705.08284
openalex publication_date 2017/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.