Turing instability in the one-parameter Gierer-Meinhardt system

S. V. Revina ; A. S. Ryabov

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 31 (2023) no. 4, pp. 501-522.

Résumé

The purpose of this work is to find the region of necessary and sufficient conditions for diffusion instability on the parameter plane $(\tau,d) $of the Gierer-Meinhardt system, where $\tau$ is the relaxation parameter, $d$ is the dimensionless diffusion coefficient; to derive analytically the dependence of the critical wave number on the characteristic size of the spatial region; to obtain explicit representations of secondary spatially distributed structures, formed as a result of bifurcation of a spatially homogeneous equilibrium position, in the form of series in degrees of supercriticality. Methods. To find the region of Turing instability, methods of linear stability analysis are applied. To find secondary solutions (Turing structures), the Lyapunov-Schmidt method is used in the form developed by V. I. Yudovich. Results. Expressions for the critical diffusion coefficient in terms of the eigenvalues of the Laplace operator for an arbitrary bounded region are obtained. The dependence of the critical diffusion coefficient on the characteristic size of the region is found explicitly in two cases: when the region is an interval and a rectangle. Explicit expressions for the first terms of the expansions of the secondary stationary solutions with respect to the supercriticality parameter are constructed in the one-dimensional case, as well as for a rectangle, when one of the wave numbers is equal to zero. In these cases, sufficient conditions for a soft loss of stability are found, and examples of secondary solutions are given. Conclusion. A general approach is proposed for finding the region of Turing instability and constructing secondary spatially distributed structures. This approach can be applied to a wide class of mathematical models described by a system of two reaction-diffusion equations.

Détail

Keywords: Turing instability, reaction-diffusion systems, necessary and sufficient conditions for diffusion instability, critical diffusion coefficient.

Bibliographie

[1] Wei J., Winter M., Mathematical Aspects of Pattern Formation in Biological Systems, Springer, London, 2014, 319 pp. | DOI | MR | Zbl

[2] Kostin V. A., Osipov G. V., “Neustoichivost odnorodnogo sostoyaniya i dvukhdomennye prostranstvenno-vremennye struktury v reaktsionno-diffuzionnykh sistemakh s globalnoi svyazyu”, Izvestiya vuzov. PND, 29:1 (2021), 186–207 | DOI | MR

[3] Tsibulin V. G., Kha T. D., Zelenchuk P. A., “Nelineinaya dinamika sistemy khischnik–zhertva na neodnorodnom areale i stsenarii lokalnogo vzaimodeistviya vidov”, Izvestiya vuzov. PND, 29:5 (2021), 751–764 | DOI

[4] Kazarnikov A. V., Revina S. V., “Vozniknovenie avtokolebanii v sisteme Releya s diffuziei”, Vestnik YuUrGU. Ser. «Matematicheskoe modelirovanie i programmirovanie», 9:2 (2016), 16–28 | DOI | Zbl

[5] Kazarnikov A. V., Revina S. V., “Asimptotika statsionarnykh reshenii v sisteme Releya s diffuziei”, Izvestiya vuzov. Severo-Kavkazskii region. Estestvennye nauki, 2016, no. 3 (191), 13–19 | DOI

[6] Kazarnikov A. V., Revina S. V., “Bifurkatsii v sisteme Releya s diffuziei”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 27:4 (2017), 499–514 | DOI | MR | Zbl

[7] Kazarnikov A. V., Revina S. V., “Monotonnaya neustoichivost v sisteme FittsKhyu-Nagumo s diffuziei”, Izvestiya vuzov. Severo-Kavkazskii region. Estestvennye nauki, 2018, no. 4 (200), 18–24 | DOI

[8] Turing A. M., “The chemical basis of morphogenesis”, Phil. Trans. R. Soc. Lond. B, 237:641 (1952), 37–72 | DOI | MR | Zbl

[9] Murray J. D., Mathematical Biology II: Spatial Models and Biomedical Applications, 3, Springer, New York, 2003, 814 pp. | DOI | MR

[10] Revina S. V., Lysenko S. A., “Dostatochnye usloviya neustoichivosti Tyuringa dlya sistemy Shnakenberga”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 31:3 (2021), 424–442 | DOI | MR | Zbl

[11] Revina S. V., “Oblast diffuzionnoi neustoichivosti dlya sistem parabolicheskikh uravnenii”, Vladikavkazskii matematicheskii zhurnal, 24:4 (2022), 117–126 | DOI | MR

[12] Gierer A., Meinhardt H., “A theory of biological pattern formation”, Kybernetik, 12:1 (1972), 30–39 | DOI

[13] Meinhardt H., “Models of biological pattern formation: From elementary steps to the organization of embryonic axes”, Current Topics in Developmental Biology, 81 (2008), 1–63 | DOI

[14] Gomez D., Ward M. J., Wei J., “An asymptotic analysis of localized three-dimensional spot patterns for the Gierer\hspace{1pt}–\hspace{1pt}Meinhardt model: Existence, linear stability, and slow dynamics”, SIAM Journal on Applied Mathematics, 81:2 (2021), 378–406 | DOI | MR | Zbl

[15] Chen M., Wu R., Chen L., “Pattern dynamics in a diffusive Gierer\hspace{1pt}–\hspace{1pt}Meinhardt model”, International Journal of Bifurcation and Chaos, 30:12 (2020), 2030035 | DOI | MR | Zbl

[16] Yudovich V. I., “Primer poteri ustoichivosti i rozhdeniya vtorichnogo techeniya zhidkosti v zamknutom sosude”, Matematicheskii sbornik (novaya seriya), 74(116):4 (1967), 565–579

[17] Yudovich V. I., “Issledovanie avtokolebanii sploshnoi sredy, voznikayuschikh pri potere ustoichivosti statsionarnogo rezhima”, Prikladnaya matematika i mekhanika, 36:3 (1972), 450–459 | Zbl

[18] Revina S. V., “Long wavelength asymptotics of self-oscillations of viscous incompressible fluid”, Operator Theory and Differential Equations. Trends in Mathematics., eds. Kusraev A. G., Totieva Z. D., Birkhäuser, Cham, 2021, 185–203 | DOI | MR | Zbl