| Title:
|
Analysis of pattern formation using numerical continuation (English) |
| Author:
|
Janovský, Vladimír |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
67 |
| Issue:
|
6 |
| Year:
|
2022 |
| Pages:
|
705-726 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with the issue of self-organization in applied sciences. It is particularly related to the emergence of Turing patterns. The goal is to analyze the domain size driven instability: We introduce the parameter $L$, which scales the size of the domain. We investigate a particular reaction-diffusion model in 1-D for two species. We consider and analyze the steady-state solution. We want to compute the solution branches by numerical continuation. The model in question has certain symmetries. We define and classify them. Our goal is to calculate a global bifurcation diagram. (English) |
| Keyword:
|
pattern formation |
| Keyword:
|
reaction-diffusion model |
| Keyword:
|
Turing instability |
| Keyword:
|
diffusion-driven instability |
| Keyword:
|
bifurcation |
| MSC:
|
34B24 |
| MSC:
|
35B36 |
| MSC:
|
35K57 |
| MSC:
|
92C15 |
| idZBL:
|
Zbl 07613020 |
| idMR:
|
MR4505701 |
| DOI:
|
10.21136/AM.2022.0126-21 |
| . |
| Date available:
|
2022-10-31T13:25:31Z |
| Last updated:
|
2025-01-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151053 |
| . |
| Reference:
|
[1] Allgower, E. L., Georg, K.: Introduction to Numerical Continuation Methods.Classics in Applied Mathematics 45. SIAM, Philadelphia (2003). Zbl 1036.65047, MR 2001018, 10.1137/1.9780898719154 |
| Reference:
|
[2] Baker, R. E., Gaffney, E. A., Maini, P. K.: Partial differential equations for self-organization in cellular and developmental biology.Nonlinearity 21 (2008), R251--R290. Zbl 1159.35003, MR 2448226, 10.1088/0951-7715/21/11/R05 |
| Reference:
|
[3] Böhmer, K., Govaerts, W., Janovský, V.: Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies.Math. Comput. 68 (1999), 1097-1108. Zbl 0918.65039, MR 1627846, 10.1090/S0025-5718-99-01052-2 |
| Reference:
|
[4] Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems.Advanced Series in Nonlinear Dynamics 15. World Scientific, Singapore (2000). Zbl 0968.37001, MR 1831950, 10.1142/4062 |
| Reference:
|
[5] Dhooge, A., Govaerts, W., Kuznetsov, Y. A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs.ACM Trans. Math. Softw. 29 (2003), 141-164. Zbl 1070.65574, MR 2000880, 10.1145/779359.779362 |
| Reference:
|
[6] Dhooge, A., Govaerts, W., Kuznetsov, Y. A.: MATCONT and CL$_-$MATCONT: Continuation Toolboxes in MATLAB.University Gent, Gent (2011). |
| Reference:
|
[7] Gierer, A., Meinhardt, H.: A theory of biological pattern formation.Kybernetik 12 (1972), 30-39. Zbl 1434.92013, 10.1007/BF00289234 |
| Reference:
|
[8] Golubitsky, M., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory. I.Applied Mathematical Sciences 51. Springer, New York (1985). Zbl 0607.35004, MR 0771477, 10.1007/978-1-4612-5034-0 |
| Reference:
|
[9] Golubitsky, M., Stewart, I., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory. II.Applied Mathematical Sciences 69. Springer, New York (1988). Zbl 0691.58003, MR 0950168, 10.1007/978-1-4612-4574-2 |
| Reference:
|
[10] Govaerts, W. J. F.: Numerical Methods for Bifurcations of Dynamical Equilibria.SIAM, Philadelphia (2000). Zbl 0935.37054, MR 1736704, 10.1137/1.9780898719543 |
| Reference:
|
[11] Janovský, V., Plecháč, P.: Numerical applications of equivariant reduction techniques.Bifurcation and Symmetry: Cross Influence Between Mathematics and Applications International Series of Numerical Mathematics 104. Birkhäuser, Basel (1992), 203-213. Zbl 0764.65031, MR 1248618 |
| Reference:
|
[12] Klika, V.: Significance of non-normality-induced patterns: Transient growth versus asymptotic stability.Chaos 27 (2017), Article ID 073120, 9 pages. Zbl 1390.92021, MR 3679679, 10.1063/1.4985256 |
| Reference:
|
[13] Klika, V., Baker, R. E., Headon, D., Gaffney, E. A.: The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation.Bull. Math. Biol. 74 (2012), 935-957. Zbl 1235.92010, MR 2903016, 10.1007/s11538-011-9699-4 |
| Reference:
|
[14] Klika, V., Kozák, M., Gaffney, E. A.: Domain size driven instability: Self-organization in systems with advection.SIAM J. Appl. Math. 78 (2018), 2298-2322. Zbl 1396.35035, MR 3850305, 10.1137/17M1138571 |
| Reference:
|
[15] Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory.Applied Mathematical Sciences 112. Springer, New York (1998). Zbl 0914.58025, MR 1711790, 10.1007/b98848 |
| Reference:
|
[16] Marciniak-Czochra, A., Karch, G., Suzuki, K.: Instability of Turing patterns in reaction-diffusion-ODE systems.J. Math. Biol. 74 (2017), 583-618. Zbl 1356.35043, MR 3600397, 10.1007/s00285-016-1035-z |
| Reference:
|
[17] Murray, J. D.: Mathematical Biology. II. Spatial Models and Biomedical Applications.Interdisciplinary Applied Mathematics 18. Springer, New York (2003). Zbl 1006.92002, MR 1952568, 10.1007/b98869 |
| Reference:
|
[18] Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour.J. Theoret. Biol. 81 (1979), 389-400. MR 0558661, 10.1016/0022-5193(79)90042-0 |
| Reference:
|
[19] Trefethen, L. N., Embree, M.: Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators.Princeton University Press, Princeton (2005). Zbl 1085.15009, MR 2155029 |
| Reference:
|
[20] Turing, A. M.: The chemical basis of morphogenesis.Philos. Trans. R. Soc. Lond., Ser. B, Biol. Sci. 237 (1952), 37-72. Zbl 1403.92034, MR 3363444, 10.1098/rstb.1952.0012 |
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