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June  2020, 19(6): 2965-3031. doi: 10.3934/cpaa.2020130

Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

Yuta Ishii

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

Received  February 2019 Revised  October 2019 Published  March 2020

In this paper, we consider the following one-dimensional Schnakenberg model with periodic 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*} $
where
$ d,c,D>0 $
 are given constants,
$ \varepsilon >0 $
 is sufficiently small, and
$ g(x) $
 is a given positive function. Let
$ N \ge 1 $
 be an arbitrary natural number. We assume that
$ g(x) $
 is a periodic and symmetric function, namely
$ g(x) = g(-x) $
 and
$ g(x) = g(x+2N^{-1}) $
. We study the stability of
$ N $
-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity
$ g(x) $
 above on their stability. For the standard Schnakenberg model, namely the case of
$ g(x) = 1 $
, with
$ d = 0 $
, the stability of
$ N $
-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of
$ N $
-peak solution. In particular, we investigate how
$ N $
-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case
$ g(x) = 1 $
.
Keywords: Pattern formation, Schnakenberg model, multi-peak solution, stability, reaction-diffusion systems.
Mathematics Subject Classification: Primary: 35K57, 35J66; Secondary: 35B35, 35Q92.
Citation: Yuta Ishii. Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2965-3031. doi: 10.3934/cpaa.2020130
References:
[1]

W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081.  Google Scholar

[2]

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

[3]

A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar

[4]

A. Doelman, A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 155 (2002), xii+64. doi: 10.1090/memo/0737.  Google Scholar

[5]

A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar

[6]

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.  Google Scholar

[7]

Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118.  Google Scholar

[8]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar

[9]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar

[10]

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.  Google Scholar

[11]

T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Physica D, 2019. Available from: https://doi.org/10.1016/j.physd.2019.132247. doi: 10.1016/j.physd.2019.132247.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[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.  Google Scholar

[15]

J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, Discrete Contin. Dyn. Syst., 25 (2009), 363-398.  doi: 10.3934/dcds.2009.25.363.  Google Scholar

[16]

J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetic, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar

[17]

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

[18]

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.  Google Scholar

show all references

References:
[1]

W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081.  Google Scholar

[2]

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

[3]

A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar

[4]

A. Doelman, A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 155 (2002), xii+64. doi: 10.1090/memo/0737.  Google Scholar

[5]

A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar

[6]

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.  Google Scholar

[7]

Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118.  Google Scholar

[8]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar

[9]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar

[10]

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.  Google Scholar

[11]

T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Physica D, 2019. Available from: https://doi.org/10.1016/j.physd.2019.132247. doi: 10.1016/j.physd.2019.132247.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[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.  Google Scholar

[15]

J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, Discrete Contin. Dyn. Syst., 25 (2009), 363-398.  doi: 10.3934/dcds.2009.25.363.  Google Scholar

[16]

J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetic, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar

[17]

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

[18]

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.  Google Scholar

Figure 1.  For $ D_2^{2,+}(\xi_2) > D = (3+\sqrt{17})/16-0.1 $, two-peak solution is stable. For $ D_2^{2,+}(\xi_2) < D = (3+\sqrt{17})/16+0.1 $, two-peak solution is unstable
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