# American Institute of Mathematical Sciences

March  2013, 8(1): 291-325. doi: 10.3934/nhm.2013.8.291

## The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system

 1 Graduate School of Advanced Mathematical Science, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan

Received  March 2012 Revised  January 2013 Published  April 2013

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with a stripe pattern on a rectangular domain, but numerical results suggest that such stripe pattern is unstable. In [8], Kolokolnikov et al. proved the existence of a positive eigenvalue, which is called an unstable eigenvalue, for a stationary solution with a stripe pattern by the NLEP method, which implies the instability of the stripe pattern. In addition, the uniqueness of the unstable eigenvalue was shown under some technical assumptions in [8]. In this paper, we prove the existence and uniqueness of an unstable eigenvalue by using the SLEP method without any extra conditions. We also prove the existence of a single-spike solution in one-dimension.
Citation: Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
##### References:
 [1] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011). [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. [3] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. [4] S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions,, Japan J. Indust. Appl. Math., 19 (2002), 181. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations,", Birkhäuser Boston, (1982). [7] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25. doi: 10.1016/S0167-2789(00)00206-2. [8] T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation,, SIAM J. Appl. Dyn. Syst., 5 (2006), 313. doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus,, Nature, 376 (1995), 765. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation,, Science, 329 (2010), 1616. [11] P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems,, J. Chem. Soc. Faraday Trans., 93 (1997), 3601. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). [13] J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). doi: 10.1007/978-3-662-08539-4. [14] A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns,, Proceedings of the National Academy of Sciences, 106 (2009), 8429. doi: 10.1073/pnas.0808622106. [15] Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer,, Proceedings of the National Academy of Sciences, 108 (2011), 9137. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, in, 3 (1994), 25. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124. [18] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems,, J. Theoret. Biol., 224 (2003), 339. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. [20] M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems,, Adv. Differential Equations, 8 (2003), 29. [21] M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems,, SIAM J. Math. Anal., 25 (1994), 99. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37. doi: 10.1098/rstb.1952.0012. [24] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability,, European J. Appl. Math., 13 (2002), 283. doi: 10.1017/S0956792501004442. [25] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1.

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##### References:
 [1] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011). [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. [3] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. [4] S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions,, Japan J. Indust. Appl. Math., 19 (2002), 181. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations,", Birkhäuser Boston, (1982). [7] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25. doi: 10.1016/S0167-2789(00)00206-2. [8] T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation,, SIAM J. Appl. Dyn. Syst., 5 (2006), 313. doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus,, Nature, 376 (1995), 765. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation,, Science, 329 (2010), 1616. [11] P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems,, J. Chem. Soc. Faraday Trans., 93 (1997), 3601. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). [13] J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). doi: 10.1007/978-3-662-08539-4. [14] A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns,, Proceedings of the National Academy of Sciences, 106 (2009), 8429. doi: 10.1073/pnas.0808622106. [15] Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer,, Proceedings of the National Academy of Sciences, 108 (2011), 9137. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, in, 3 (1994), 25. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124. [18] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems,, J. Theoret. Biol., 224 (2003), 339. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. [20] M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems,, Adv. Differential Equations, 8 (2003), 29. [21] M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems,, SIAM J. Math. Anal., 25 (1994), 99. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37. doi: 10.1098/rstb.1952.0012. [24] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability,, European J. Appl. Math., 13 (2002), 283. doi: 10.1017/S0956792501004442. [25] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1.
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