Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation

Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

“…In Section 5, we apply our theorems to the two specific reaction-diffusion systems ( 6) and (7). We emphasize here that our result can be applied to the Gray-Scott model [18], the Gierer-Meinhardt system with saturation effect [10], and so on, that are more or less related to multi-spot patterns. If the readers are interested in these models, see these papers and references cited therein.…”

“…(A9): λ = µ > 0 and λ = 0 are eigenvalues of (10). For small γ > 0, there is no essential spectrum of (10) in {λ ∈ C | Reλ ≥ −γ}.…”

“…(A9): λ = µ > 0 and λ = 0 are eigenvalues of (10). For small γ > 0, there is no essential spectrum of (10) in {λ ∈ C | Reλ ≥ −γ}. The eigenfunction ψ corresponding to µ is uniquely defined, positive and radially symmetric, while the eigenspace associated with the zero eigenvalue consists of S yj = ∂S/∂y j for j = 1, .…”

“…In particular, it is important to verify the condition (A11). First, we introduce results which show that L 0 given in (10) satisfies (A9).…”

Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

“…Chen et al [5] investigated the stability of the equilibria, the Hopf bifurcation and the steady state bifurcation of a Gierer-Meinhardt model with saturation in the activator production. Kurata et al [10] explored the multi-peak stationary solution of a saturated Gierer-Meinhardt model with a bounded smooth domain in R N . The stripe solution, spotted solution, mixed solution and labyrinthine-like solution in a two-dimensional space of a saturated Gierer-Meinhardt model were discussed by Song et al [26].…”