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Stationary multiple spots for reaction-diffusion systems. (English) Zbl 1141.92007

Summary: We review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We consider two classes of reaction-diffusion systems: activator-inhibitor systems, such as the Gierer-Meinhardt system, and activator-substrate systems, such as the P. Gray and S. K. Scott system [Chem. Eng. Sci. 38, 29–43 (1983); ibid. 39, 1087–1097 (1984)], or the J. Schnakenberg model [J. Theor. Biol. 81, 389–400 (1979)]. The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constants of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Lyapunov-Schmidt 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 the Lyapunov-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 biological, chemical and ecological contexts. They are confirmed by numerical simulations.

92C15Developmental biology, pattern formation
35K57Reaction-diffusion equations
46N60Applications of functional analysis in biology and other sciences
35K45Systems of second-order parabolic equations, initial value problems
35J55Systems of elliptic equations, boundary value problems (MSC2000)
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