American Institute of Mathematical Sciences

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July  2013, 33(7): 2885-2900. doi: 10.3934/dcds.2013.33.2885

Global-in-time behavior of the solution to a Gierer-Meinhardt system

Georgia Karali 1, , Takashi Suzuki 2, and  Yoshio Yamada 3,

1. 

Department of Applied Mathematics, University Crete, P.O. Box 2208, 71409, Heraklion, Crete
2. 

Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531
3. 

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku 169-8555, Tokyo

Received  April 2012 Revised  September 2012 Published  January 2013

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
Keywords: Hamilton structure, Turing pattern, asymptotic behavior of the solution., Reaction-diffusion equation, Gierer-Meinhardt system.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35Q9.
Citation: Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885
References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. Google Scholar

[3]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.  Google Scholar

[4]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., 840, Sprinver-Verlag, Berlin, 1981.  Google Scholar

[5]

H. Hoshino and Y. Yamada, Sovability and smoothing effect for semilinear parabolic equations, Funkcialaj Ekvacioj, 34 (1991), 475-494.  Google Scholar

[6]

D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar

[7]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete and Continuous Dynamical Systems, 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737.  Google Scholar

[8]

H. Jiang and W.-M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732. doi: 10.1512/iumj.2007.56.2982.  Google Scholar

[9]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanism to complex structure, Rev. Modern Physiscs, 66 (1994), 1481-1510. Google Scholar

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E. Latos, T. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion,, to appear in; Math. Meth. Appl. Sci., ().  doi: 10.1002/mma.2524.  Google Scholar

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K. Masuda and T. Takahashi, Reaction-diffusion systems in Gierer-Meinhardt theory in biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58. doi: 10.1007/BF03167754.  Google Scholar

[13]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," third edition, Springer, New York, 2003.  Google Scholar

[14]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465. doi: 10.1016/j.jde.2006.03.011.  Google Scholar

[15]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368. doi: 10.2307/2000473.  Google Scholar

[16]

F. Rothe, "Global Solutions of Reaction-Diffusion Equations," Lecture Notes in Math., 1072, Springer-Verlag, 1984.  Google Scholar

[17]

A. Turing, The chemical basis of morphogenesis, Philos. Transl. Roy. Soc. London, B237 (1952), 37-72. Google Scholar

[18]

J. Wei, Existence and stability of spikes for the Gierer-Meinhardt sytem, Handbook of Differential Equations, Stationary Partial Differential Equations, 5 (ed. M. Chipot), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

[19]

E. Yanagida, Reaction-diffusion systems with skew-gradient structure, Meth. Appl. Anal., 8 (2001), 209-226.  Google Scholar

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show all references

References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. Google Scholar

[3]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.  Google Scholar

[4]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., 840, Sprinver-Verlag, Berlin, 1981.  Google Scholar

[5]

H. Hoshino and Y. Yamada, Sovability and smoothing effect for semilinear parabolic equations, Funkcialaj Ekvacioj, 34 (1991), 475-494.  Google Scholar

[6]

D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar

[7]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete and Continuous Dynamical Systems, 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737.  Google Scholar

[8]

H. Jiang and W.-M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732. doi: 10.1512/iumj.2007.56.2982.  Google Scholar

[9]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanism to complex structure, Rev. Modern Physiscs, 66 (1994), 1481-1510. Google Scholar

[10]

E. Latos, T. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion,, to appear in; Math. Meth. Appl. Sci., ().  doi: 10.1002/mma.2524.  Google Scholar

[11]

F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776. doi: 10.1016/j.jde.2009.04.009.  Google Scholar

[12]

K. Masuda and T. Takahashi, Reaction-diffusion systems in Gierer-Meinhardt theory in biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58. doi: 10.1007/BF03167754.  Google Scholar

[13]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," third edition, Springer, New York, 2003.  Google Scholar

[14]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465. doi: 10.1016/j.jde.2006.03.011.  Google Scholar

[15]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368. doi: 10.2307/2000473.  Google Scholar

[16]

F. Rothe, "Global Solutions of Reaction-Diffusion Equations," Lecture Notes in Math., 1072, Springer-Verlag, 1984.  Google Scholar

[17]

A. Turing, The chemical basis of morphogenesis, Philos. Transl. Roy. Soc. London, B237 (1952), 37-72. Google Scholar

[18]

J. Wei, Existence and stability of spikes for the Gierer-Meinhardt sytem, Handbook of Differential Equations, Stationary Partial Differential Equations, 5 (ed. M. Chipot), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

[19]

E. Yanagida, Reaction-diffusion systems with skew-gradient structure, Meth. Appl. Anal., 8 (2001), 209-226.  Google Scholar

[20]

E. Yanagida, Mini-maximizers for reaction-diffusion systems with skew-gradient structure, J. Differential Equations, 179 (2002), 311-335. doi: 10.1006/jdeq.2001.4028.  Google Scholar

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