October  2017, 10(5): 1009-1023. doi: 10.3934/dcdss.2017053

Steady states of a Sel'kov-Schnakenberg reaction-diffusion system

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

2. 

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author

Received  September 2016 Revised  February 2017 Published  June 2017

Fund Project: B. Li was partially supported by NSF of China (No. 11671175), the Priority Academic Program Development of Jiangsu Higher Education Institutions, and X.Y. Zhang was partially supported by NSF of China (No. 11571200,11425105).

In this paper, we are concerned with a reaction-diffusion model, known as the Sel'kov-Schnakenberg system, and study the associated steady state problem. We obtain existence and nonexistence results of nonconstant steady states, which in turn imply the criteria for the formation of spatial pattern (especially, Turing pattern). Our results reveal the different roles of the diffusion rates of the two reactants in generating spatial pattern.

Citation: Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053
References:
[1]

W. Y. Chen and R. Peng, Stationary patterns created by cross-diffusion for the competitor-mutualist model, J. Math. Anal. Appl., 291 (2004), 550-564.  doi: 10.1016/j.jmaa.2003.11.015.

[2]

X. F. ChenY. W. Qi and M. X. Wang, Steady states of a strongly coupled prey-predator model, Discrete Contin. Dyn. Syst., suppl (2005), 173-180. 

[3]

X. F. ChenY. W. Qi and M. X. Wang, A strongly coupled predator-prey system with nonmonotonic functional response, Nonlinear Anal., 67 (2007), 1966-1979.  doi: 10.1016/j.na.2006.08.022.

[4]

F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507-516.  doi: 10.1017/S0308210500000275.

[5]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895.

[6]

Y. DuP. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173.

[7]

J. C. Eilbeck, Pattern formation and pattern selection in reaction-diffusion systems, in The oretical Biology: Epigenetic and Evolutionary Order (eds. B. Goodwin and P. T. Saunders), Edinburgh University Press, (1989), 31–41.

[8]

J. E. Furter and J. C. Eilbeck, Analysis of bifurcation in reaction-diffusion systems with no flux boundary conditions: The Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413-438.  doi: 10.1017/S0308210500028109.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[10]

M. Ghergu, Non-constant steady states for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.  doi: 10.1088/0951-7715/21/10/007.

[11]

W. Han and Z. Bao, Hopf bifurcation analysis of a reaction-diffusion Sel'kov system, J. Math. Anal. Appl., 356 (2009), 633-641.  doi: 10.1016/j.jmaa.2009.03.058.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981.

[13]

D. IronJ. C. 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.

[14]

J. JangW. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[15]

R. Kapral and K. Showalter, Chemical Waves and Patterns: Understanding Chemical Reactivity, Springer, 1995. doi: 10.1007/978-94-011-1156-0.

[16]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X.

[17]

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

[18]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[19]

Y. Lou and W. M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.

[20]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. 

[21]

W. M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[22]

L. Nirenberg, Topics in Nonlinear Functional Analysis American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.

[23]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.

[24]

P. Y. H. Pang and M. X. Wang, Nonconstant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[25]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

[26]

R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations, 241 (2007), 386-398.  doi: 10.1016/j.jde.2007.06.005.

[27]

R. PengJ. P. Shi and M. X. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 65 (2007), 1479-1503.  doi: 10.1137/05064624X.

[28]

R. PengJ. P. Shi and M. X. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006.

[29]

R. Peng and M. X. Wang, Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction, Nonlinear Anal., 56 (2004), 451-464.  doi: 10.1016/j.na.2003.09.020.

[30]

R. Peng and M. Yang, On steady-state solutions of the Brusselator-type system, Nonlinear Anal., 71 (2009), 1389-1394.  doi: 10.1016/j.na.2008.12.003.

[31]

P. RichterP. Rehmus and J. Ross, Control and dissipation in oscillatory chemical engines, Progress in Theoretical Phys., 66 (1981), 385-405.  doi: 10.1143/PTP.66.385.

[32]

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

[33]

E. E. Sel'kov, Self-oscillations in glycolysis, European J. Biochem., 4 (1968), 79-86. 

[34]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.  doi: 10.1016/0022-0396(86)90119-1.

[35]

Y. Termonia and J. Ross, Oscillations and control features in glycolysis: Analysis of resonance effects, Proc. Nat. Acad. Sci. U.S.A., 78 (1981), 3563-3566.  doi: 10.1073/pnas.78.6.3563.

[36]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Royal Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[37]

J. C. TzouY. Nec and M. J. Ward, The stability of localized spikes for the 1-D Brusselator reaction-diffusion model, European J. Appl. Math., 24 (2013), 515-564.  doi: 10.1017/S0956792513000089.

[38]

H. Uecker and D. Wetzel, Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg reaction-diffusion systems, SIAM J. Appl. Dyn. Syst., 13 (2014), 94-128.  doi: 10.1137/130918484.

[39]

M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620.  doi: 10.1016/S0022-0396(02)00100-6.

[40]

M. J. Ward and J. C. 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.

[41]

C. Xu and J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.

[42]

Y. You, Asymptotical dynamics of Selkov equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 193-219.  doi: 10.3934/dcdss.2009.2.193.

[43]

Y. You, Upper-semicontinuity of global attractors for reversible Schnackenberg equations, Stud. Appl. Math., 130 (2013), 232-263.  doi: 10.1111/j.1467-9590.2012.00565.x.

[44]

J. Zhou and J. P. Shi, Pattern formation in a general glycolysis reaction-diffusion system, IMA J. Appl. Math., 80 (2015), 1703-1738.  doi: 10.1093/imamat/hxv013.

show all references

References:
[1]

W. Y. Chen and R. Peng, Stationary patterns created by cross-diffusion for the competitor-mutualist model, J. Math. Anal. Appl., 291 (2004), 550-564.  doi: 10.1016/j.jmaa.2003.11.015.

[2]

X. F. ChenY. W. Qi and M. X. Wang, Steady states of a strongly coupled prey-predator model, Discrete Contin. Dyn. Syst., suppl (2005), 173-180. 

[3]

X. F. ChenY. W. Qi and M. X. Wang, A strongly coupled predator-prey system with nonmonotonic functional response, Nonlinear Anal., 67 (2007), 1966-1979.  doi: 10.1016/j.na.2006.08.022.

[4]

F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507-516.  doi: 10.1017/S0308210500000275.

[5]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895.

[6]

Y. DuP. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173.

[7]

J. C. Eilbeck, Pattern formation and pattern selection in reaction-diffusion systems, in The oretical Biology: Epigenetic and Evolutionary Order (eds. B. Goodwin and P. T. Saunders), Edinburgh University Press, (1989), 31–41.

[8]

J. E. Furter and J. C. Eilbeck, Analysis of bifurcation in reaction-diffusion systems with no flux boundary conditions: The Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413-438.  doi: 10.1017/S0308210500028109.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[10]

M. Ghergu, Non-constant steady states for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.  doi: 10.1088/0951-7715/21/10/007.

[11]

W. Han and Z. Bao, Hopf bifurcation analysis of a reaction-diffusion Sel'kov system, J. Math. Anal. Appl., 356 (2009), 633-641.  doi: 10.1016/j.jmaa.2009.03.058.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981.

[13]

D. IronJ. C. 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.

[14]

J. JangW. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[15]

R. Kapral and K. Showalter, Chemical Waves and Patterns: Understanding Chemical Reactivity, Springer, 1995. doi: 10.1007/978-94-011-1156-0.

[16]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X.

[17]

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

[18]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[19]

Y. Lou and W. M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.

[20]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. 

[21]

W. M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[22]

L. Nirenberg, Topics in Nonlinear Functional Analysis American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.

[23]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.

[24]

P. Y. H. Pang and M. X. Wang, Nonconstant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[25]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

[26]

R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations, 241 (2007), 386-398.  doi: 10.1016/j.jde.2007.06.005.

[27]

R. PengJ. P. Shi and M. X. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 65 (2007), 1479-1503.  doi: 10.1137/05064624X.

[28]

R. PengJ. P. Shi and M. X. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006.

[29]

R. Peng and M. X. Wang, Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction, Nonlinear Anal., 56 (2004), 451-464.  doi: 10.1016/j.na.2003.09.020.

[30]

R. Peng and M. Yang, On steady-state solutions of the Brusselator-type system, Nonlinear Anal., 71 (2009), 1389-1394.  doi: 10.1016/j.na.2008.12.003.

[31]

P. RichterP. Rehmus and J. Ross, Control and dissipation in oscillatory chemical engines, Progress in Theoretical Phys., 66 (1981), 385-405.  doi: 10.1143/PTP.66.385.

[32]

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

[33]

E. E. Sel'kov, Self-oscillations in glycolysis, European J. Biochem., 4 (1968), 79-86. 

[34]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.  doi: 10.1016/0022-0396(86)90119-1.

[35]

Y. Termonia and J. Ross, Oscillations and control features in glycolysis: Analysis of resonance effects, Proc. Nat. Acad. Sci. U.S.A., 78 (1981), 3563-3566.  doi: 10.1073/pnas.78.6.3563.

[36]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Royal Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[37]

J. C. TzouY. Nec and M. J. Ward, The stability of localized spikes for the 1-D Brusselator reaction-diffusion model, European J. Appl. Math., 24 (2013), 515-564.  doi: 10.1017/S0956792513000089.

[38]

H. Uecker and D. Wetzel, Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg reaction-diffusion systems, SIAM J. Appl. Dyn. Syst., 13 (2014), 94-128.  doi: 10.1137/130918484.

[39]

M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620.  doi: 10.1016/S0022-0396(02)00100-6.

[40]

M. J. Ward and J. C. 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.

[41]

C. Xu and J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.

[42]

Y. You, Asymptotical dynamics of Selkov equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 193-219.  doi: 10.3934/dcdss.2009.2.193.

[43]

Y. You, Upper-semicontinuity of global attractors for reversible Schnackenberg equations, Stud. Appl. Math., 130 (2013), 232-263.  doi: 10.1111/j.1467-9590.2012.00565.x.

[44]

J. Zhou and J. P. Shi, Pattern formation in a general glycolysis reaction-diffusion system, IMA J. Appl. Math., 80 (2015), 1703-1738.  doi: 10.1093/imamat/hxv013.

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