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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

Kota Ikeda 1,

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.
Keywords: eigenvalue problem, Gierer-Meinhardt system, SLEP method..
Mathematics Subject Classification: 35K57, 35K5.
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, Springer, New York, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[6]

P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0.  Google Scholar

[10]

S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. Google Scholar

[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.  Google Scholar

[12]

H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. Google Scholar

[13]

J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differential Equations, 8 (2003), 29-54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. doi: 10.1098/rstb.1952.0012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[6]

P. Hartman, "Ordinary Differential Equations," Birkhäuser Boston, Mass., second edition, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0.  Google Scholar

[10]

S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620. Google Scholar

[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.  Google Scholar

[12]

H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, 1982. Google Scholar

[13]

J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differential Equations, 8 (2003), 29-54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. doi: 10.1098/rstb.1952.0012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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