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Home » Issues » Archive of Issues » 2021 - 3 issue » pp. 424-442

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Russia Rostov-on-Don; Vladikavkaz
Year
2021
Volume
31
Issue
3
Pages
424-442
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Section Mathematics
Title Sufficient Turing instability conditions for the Schnakenberg system
Author(-s) Revina S.V.ab, Lysenko S.A.a
Affiliations Southern Federal Universitya, Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciencesb
Abstract A classical reaction-diffusion system, the Schnakenberg system, is under consideration in a bounded domain $\Omega\subset\mathbb{R}^m$ with Neumann boundary conditions. We study diffusion-driven instability of a stationary spatially homogeneous solution of this system, also called the Turing instability, which arises when the diffusion coefficient $d$ changes. An analytical description of the region of necessary and sufficient conditions for the Turing instability in the parameter plane is obtained by analyzing the linearized system in diffusionless and diffusion approximations. It is shown that one of the boundaries of the region of necessary conditions is an envelope of the family of curves that bound the region of sufficient conditions. Moreover, the intersection points of two consecutive curves of this family lie on a straight line whose slope depends on the eigenvalues of the Laplace operator and does not depend on the diffusion coefficient. We find an analytical expression for the critical diffusion coefficient at which the stability of the equilibrium position of the system is lost. We derive conditions under which the set of wavenumbers corresponding to neutral stability modes is countable, finite, or empty. It is shown that the semiaxis $d>1$ can be represented as a countable union of half-intervals with split points expressed in terms of the eigenvalues of the Laplace operator; each half-interval is characterized by the minimum wavenumber of loss of stability.
Keywords reaction-diffusion systems, Schnakenberg system, Turing space, critical wavenumber
UDC 517.957
MSC 35K57
DOI 10.35634/vm210306
Received 5 May 2021
Language English
Citation Revina S.V., Lysenko S.A. Sufficient Turing instability conditions for the Schnakenberg system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 3, pp. 424-442.
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