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

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.
Citation: 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
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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[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.

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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[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.

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