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May  2019, 39(5): 2807-2875. doi: 10.3934/dcds.2019118

Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity

Yuta Ishii , and  Kazuhiro Kurata

Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

* Corresponding author: Yuta Ishii

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: The second author is supported by JSPS KAKENHI Grant Number 16K05240.

In this paper, we consider stationary solutions of the following one-dimensional Schnakenberg model with heterogeneity:
$ \begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ u_x (\pm 1) = v_x (\pm 1) = 0 . \end{cases} \end{equation*} $
We concentrate on the case that
$ d, c, D>0 $
 are given constants,
$ g(x) $
 is a given symmetric function, namely
$ g(x) = g(-x) $
, and
$ \varepsilon>0 $
 is sufficiently small and are interested in the effect of the heterogeneity
$ g(x) $
 on the stability. For the case
$ g(x) = 1 $
 and
$ d = 0 $
, Iron, Wei, and Winter (2004) studied the existence of
$ N- $
peaks symmetric stationary solutions and their stability. In this paper, first we construct symmetric one-peak stationary solutions
$ (u_{\varepsilon}, v_{\varepsilon}) $
 by using the contraction mapping principle. Furthermore, we give a linear stability analysis of the solutions
$ (u_{\varepsilon}, v_{\varepsilon}) $
 in details and reveal the effect of heterogeneity on the stability, which is a new phenomenon compared with the case
$ g(x) = 1 $
.
Keywords: Pattern formation, Schnakenberg model, stability, singular perturbation.
Mathematics Subject Classification: 35K57, 35J66, 35B25, 35B35, 35Q92.
Citation: Yuta Ishii, Kazuhiro Kurata. Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2807-2875. doi: 10.3934/dcds.2019118
References:
[1] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Orderer Elliptic Equations, Princeton Univ. Press, 1982. 
[2]

D. L. Benson, J. A. Sherrat and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. of Math. Biology, 55 (1993), 365-384. 

[3]

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

[4]

K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325.  doi: 10.3934/nhm.2013.8.291.

[5]

D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.

[6]

T. Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-dependent feed rate, SIAM Rev., 60 (2018), 626-645.  doi: 10.1137/17M1116027.

[7]

E. H. Lieb and M. Loss, Analysis, Vol.14 of Graduate Studies in Mathematics, American Math. Society, 2001. doi: 10.1090/gsm/014.

[8]

P. Liu, J. Shi, Y. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Scnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.  doi: 10.1007/s10910-013-0196-x.

[9]

K. Morimoto, Point-condensation phenomena and saturation effect for the one-dimensional Gierer-Meinhardt system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 973-995.  doi: 10.1016/j.anihpc.2010.01.003.

[10]

J. Schnakenberg, Simple chemical reaction system with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.

[11]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc., B237 (1952), 37-72. 

[12]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.

[13]

J. Wei, On single interior spike layer solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.

[14]

J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion system, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.

[15]

J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Vol. 189 of Applied Mathematical Sciences, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.

[16]

J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, Eur. J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.

show all references

References:
[1] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Orderer Elliptic Equations, Princeton Univ. Press, 1982. 
[2]

D. L. Benson, J. A. Sherrat and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. of Math. Biology, 55 (1993), 365-384. 

[3]

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

[4]

K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325.  doi: 10.3934/nhm.2013.8.291.

[5]

D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.

[6]

T. Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-dependent feed rate, SIAM Rev., 60 (2018), 626-645.  doi: 10.1137/17M1116027.

[7]

E. H. Lieb and M. Loss, Analysis, Vol.14 of Graduate Studies in Mathematics, American Math. Society, 2001. doi: 10.1090/gsm/014.

[8]

P. Liu, J. Shi, Y. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Scnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.  doi: 10.1007/s10910-013-0196-x.

[9]

K. Morimoto, Point-condensation phenomena and saturation effect for the one-dimensional Gierer-Meinhardt system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 973-995.  doi: 10.1016/j.anihpc.2010.01.003.

[10]

J. Schnakenberg, Simple chemical reaction system with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.

[11]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc., B237 (1952), 37-72. 

[12]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.

[13]

J. Wei, On single interior spike layer solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.

[14]

J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion system, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.

[15]

J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Vol. 189 of Applied Mathematical Sciences, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.

[16]

J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, Eur. J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.

Figure 1.  For $ D = 0.6 $, the solution is stable
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Figure 2.  For $ D_1 > D = \frac{1}{24}-0.02 $, the solution is stable. For $ D_1 < D = \frac{1}{24}+0.02 $, the solution is unstable
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