# 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 and 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, Springer, New York, 2011. [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507. 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), 65-104 (electronic). 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), 181-226. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 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), 25-62. 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), 313-363 (electronic). doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. [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), 3601-3610. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. [13] J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 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), 8429-8434. 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), 9137-9142. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, in "Dynamics Reported (New Series)", 3 (1994), Springer, Berlin, 25-103. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. Translated in J. Soviet Math., 45 (1989), 1205-1218. 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), 339-350. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. 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), 29-54. [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), 117-145. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems, SIAM J. Math. Anal., 25 (1994), 99-134. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. 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), 283-320. 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), 433-476. 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), 415-458. doi: 10.1007/s00332-001-0380-1.

show all references

##### References:
 [1] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011. [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507. 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), 65-104 (electronic). 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), 181-226. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 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), 25-62. 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), 313-363 (electronic). doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. [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), 3601-3610. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. [13] J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 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), 8429-8434. 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), 9137-9142. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, in "Dynamics Reported (New Series)", 3 (1994), Springer, Berlin, 25-103. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. Translated in J. Soviet Math., 45 (1989), 1205-1218. 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), 339-350. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. 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), 29-54. [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), 117-145. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems, SIAM J. Math. Anal., 25 (1994), 99-134. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. 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), 283-320. 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), 433-476. 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), 415-458. doi: 10.1007/s00332-001-0380-1.
 [1] Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363 [2] Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651 [3] Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437 [4] Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583 [5] Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115 [6] Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885 [7] Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192 [8] Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443 [9] Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033 [10] Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158 [11] Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2275-2312. doi: 10.3934/dcdsb.2021132 [12] Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013 [13] Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008 [14] Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529 [15] Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025 [16] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [17] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [18] Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems and Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043 [19] Qilong Zhai, Ran Zhang. Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 403-413. doi: 10.3934/dcdsb.2018091 [20] Hao Li, Hai Bi, Yidu Yang. The two-grid and multigrid discretizations of the $C^0$IPG method for biharmonic eigenvalue problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1775-1789. doi: 10.3934/dcdsb.2020002

2020 Impact Factor: 1.213