Stationary multiple spots for reaction–diffusion systems

Abstract

Abstract. In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reactiondiffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activatorsubstrate systems (such as the Gray-Scott system or the Schnakenberg model).The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature.We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator.Existence of multi-spots is proved using tools from nonlinear functional analysis such as LiapunovSchmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots.Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis.The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

Stability of cluster solutions in a cooperative consumer chain model

“…In [12,26] the existence and stability of spiky patterns on bounded intervals is established. In [34] similar results are shown for two-dimensional domains. In [2] it is shown how the degeneracy of the Turing bifurcation can be lifted using spatially varying diffusion coefficients.…”

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

“…, x K are such that e is an eigenvector of G (i.e. G is a circulant matrix as in §5.1), then χ v1 is given by the common value χ v1 = µb 1 e/[χ Next, we show that for D = D 0 /ν the multi-spot stability problem (3.26) and (3.20) reduces to the NLEP problem of [54]. Since V j ∼ ν 1/2 w/χ 0j and U j ∼ ν −1/2 χ 0j from (6.2) and (6.4), then (3.20) with D = D 0 /ν reduces to…”

“…However, this decoupling property for the stability problem does not occur for the case where D is asymptotically large with D = O(ν −1 ). In this limiting regime, we show in §6 that (3.20) and (3.26) reduces to the vectorial nonlocal eigenvalue problem of [54].…”

“…Moreover, in the limit D 1, we show that the stability problem of §3.1 for the m = 0 mode reduces to the vectorial nonlocal eigenvalue problem (NLEP) of [54] that governs the stability of a K-spot quasi-equilibrium pattern to locally radially symmetric perturbations near each spot. The stability requirement of [54] that D 0 < D 0K , for some explicit threshold D 0K , is then recovered. Finally, some open problems suggested by this study are listed in §7.…”

“…With regards to the stability of equilibrium multi-spot patterns for singularly perturbed reaction diffusion systems, an analytical theory based on the rigorous derivation and analysis of certain nonlocal eigenvalue problems (NLEP) has been developed in [48], [49], [50], [51], [52], and [53], for the Gierer-Meinhardt and Gray-Scott models. A survey of this theory, together with a further application of it to the Schnakenburg model, is given in [54].…”