Stability analysis of Turing patterns generated by the Schnakenberg model

Journal of Mathematical Biology

Volumn 49, Issue 4, 2004, Pages 358-390

  Iron, David ^a   Wei, Juncheng ^b   Winter, Matthias ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Nonlocal eigenvalue problem; Peaked solutions; Symmetric N; Turing instability

Indexed keywords

ARTICLE; AUTOMATED PATTERN RECOGNITION; BIOLOGICAL MODEL; DIFFUSION; KINETICS; MATHEMATICS; SOLUTION AND SOLUBILITY;

DIFFUSION; KINETICS; MATHEMATICS; MODELS, BIOLOGICAL; PATTERN RECOGNITION, AUTOMATED; SOLUTIONS;

EID: 12944263637     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00285-003-0258-y     Document Type: Article

References (28)

1. Dancer, E.N.: On stability and Hopf bifurcations for chemotaxis systems. Meth. Appl. Anal. 8, 245-256 (2001)
2. Doelman, A., Gardner, R.A., Kaper, T.J.: Stability analysis of singular patterns in the 1D gray-Scott model: a matched asymptotics approach. Phys. D 122, 1-36 (1998)
3. Doelman, A., Gardner, R.A., Kaper, T.J.: Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50, 443-507 (2001)
4. Doelman, A., Kaper, T.J., Zegeling, P.A.: Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 523-563 (1997)
5. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30-39 (1972)
6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. 2nd edition, Springer, Berlin, 1983
7. Gui, C., Wei, J.: Multiple interior peak solutions for some singular perturbation problems. J. Diff. Eqs. 158, 1-27 (1999)
8. Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 47-82 (2000)
9. Iron, D., Ward, M.J., Wei, J.: The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Phys. D 150, 25-62 (2001)
10. Benson, D.L., Maini, P.K., Sherratt, J.A.: Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37, 381-417 (1998)
11. Ni, W.-M.: Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc. 45, 9-18 (1998)
12. Ni, W.-M., Takagi, I.: On the shape of least energy solution to a semilinear Neumann problem. Commun. Pure Appl. Math. 41, 819-851 (1991)
13. Ni, W.-M., Takagi, I.: Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247-281 (1993)
14. Ni, W.-M., Takagi, I., Yanagida, E.: Tohoku Math. J. To appear
15. Nishiura, Y.: Coexistence of infinitely many stable solutions to reaction-diffusion equation in the singular limit. In: Dynamics reported: Expositions in Dynamical Systems, Volume 3, Jones, C. K. R. T., Kirchgraber, U. (eds.), Springer Verlag, New York, 1995
16. Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theoret. Biol. 81, 389-400 (1979)
17. Takagi, I.: Point-condensation for a reaction-diffusion system. J. Diff. Eqs. 61, 208-249 (1986)
18. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237, 37-72 (1952)
19. Ward, M.J., Wei, J.: Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability. Eur. J. Appl. Math. 13, 283-320 (2002)
20. Wei, J.: On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem. J. Diff. Eqs. 134, 104-133 (1997)
21. Wer, J.: On the interior spike layer solutions of singularly perturbed semilinear Neumann problem. Tohoku Math. J. 50, 159-178 (1998)
22. Wei, J.: On the interior spike layer solutions for some singular perturbation problems. Proc. Royal Soc. Edinburgh, Section A (Mathematics) 128, 849-874 (1998)
23. Wei, J.: On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10, 353-378 (1999)
24. Ward, M.J., Wei, J.: The existence and stability of asymmetric spike patterns for the Schnakenberg model. Stud. Appl. Math. 109, 229-264 (2002)
25. Wer, J., Winter, M.: On the two-dimensional Gierer-Meinhardt system with strong coupling. SIAM J. Math. Anal. 30, 1241-1263 (1999)
26. Wer, J., Winter, M.: Spikes for Gierer-Meinhardt system in two dimensions: The strong coupling case. J. Diff. Eqs. 178, 478-518 (2002)
27. Wei, J., Winter, M.: Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case. J. Nonlinear Sci. 11, 415-458 (2001)
28. Wei, J., Winter, M.: Existence and stability analysis of multiple-peaked solutions. Submitted