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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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