March  2021, 26(3): 1305-1335. doi: 10.3934/dcdsb.2020164

Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria

1. 

Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison

2. 

Department of Mathematics, University of Wisconsin-Madison

3. 

Department of Mathematics, West Virginia University

Received  June 2019 Revised  February 2020 Published  March 2021 Early access  May 2020

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.

Citation: Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164
References:
[1]

D. F. Anderson, Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math., 68 (2008), 1464-1476.  doi: 10.1137/070698282.

[2]

D. F. Anderson and A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858.  doi: 10.1137/090764098.

[3]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.  doi: 10.1137/11082631X.

[4]

A. ArnoldP. MarkowichG. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.

[5]

W. X. ChenC. M. Li and E. S. Wright, On a nonlinear parabolic system-modeling chemical reactions in rivers, Communications On Pure And Applied Analysis, 4 (2005), 889-899.  doi: 10.3934/cpaa.2005.4.889.

[6]

M. ChoulliL. Kayser and E. M. Ouhabaz, Observations on Gaussian upper bounds for Neumann heat kernels, Bulletin of the Australian Mathematical Society, 92 (2015), 429-439.  doi: 10.1017/S0004972715000611.

[7]

G. CraciunA. DickensteinA. Shiu and B. Sturmfels, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.

[8]

G. CraciunF. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.  doi: 10.1137/100812355.

[9]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2016), arXiv: 1501.02860.

[10]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.

[11]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.  doi: 10.4171/RMI/541.

[12]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. 

[13]

L. DesvillettesK. FellnerM. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Adv. Nonlinear Stud., 7 (2007), 491-511.  doi: 10.1515/ans-2007-0309.

[14]

L. DesvillettesK. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion system arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.  doi: 10.1137/16M1073935.

[15]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972/73), 187-194.  doi: 10.1007/BF00255665.

[16]

K. Fellner, W. Prager and B. Q. Tang, The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks, Kinet. Relat. Models, 10 (2017), 1055–1087, arXiv: 1504.08221. doi: 10.3934/krm.2017042.

[17]

K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., 69 (2018), Paper No. 54, 30 pp. doi: 10.1007/s00033-018-0948-3.

[18]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x.

[19]

W. E. FitzgibbonJ. Morgan and R. Sanders, Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems, Nonlin. Anal., 19 (1992), 885-899.  doi: 10.1016/0362-546X(92)90057-L.

[20]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.  doi: 10.1137/130928170.

[21]

M. Gopalkrishnan, E. Miller and A. Shiu, A projection argument for differential inclusions, with applications to persistence of mass-action kinetics, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 025, 25 pp. doi: 10.3842/SIGMA.2013.025.

[22]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.

[23]

F. Horn, The dynamics of open reaction systems, Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, SIAM-AMS Proceedings, Amer. Math. Soc., Providence, R.I., 8 (1974), 125-137. 

[24]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs, AMS, 23 (1995).

[25]

A. MielkeJ. Haskovec and P. A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.  doi: 10.1007/s10884-014-9394-x.

[26]

M. Minchevaand and D. Siegel, Stability of mass action reaction-diffusion systems, Nonlinear Anal., 56 (2004), 1105-1131.  doi: 10.1016/j.na.2003.10.025.

[27]

F. MohamedC. Pantea and A. Tudorascu, Chemical reaction-diffusion networks: Convergence of the method of lines, J. Math. Chem., 56 (2018), 30-68.  doi: 10.1007/s10910-017-0779-z.

[28]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.  doi: 10.1137/110840509.

[29]

M. PierreT. Suzuki and H. Umakoshi, Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria, J. Appl. Anal. Comp., 8 (2018), 836-858.  doi: 10.11948/2018.836.

[30]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[31]

F. Rothe, Global Solutions of Reaction-Diffusion System, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[32]

D. Siegel and D. MacLean, Global stability of complex balanced mechanisms, J. Math. Chem., 27 (2000), 89-110.  doi: 10.1023/A:1019183206064.

[33]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[34]

E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028-1047.  doi: 10.1109/9.935056.

[35]

M. E. Taylor, Partial Differential Equation Ⅲ. Nonlinear Equations, Springer Series Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[36]

P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L^p$-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51.  doi: 10.1090/S1079-6762-02-00104-X.

show all references

References:
[1]

D. F. Anderson, Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math., 68 (2008), 1464-1476.  doi: 10.1137/070698282.

[2]

D. F. Anderson and A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858.  doi: 10.1137/090764098.

[3]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.  doi: 10.1137/11082631X.

[4]

A. ArnoldP. MarkowichG. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.

[5]

W. X. ChenC. M. Li and E. S. Wright, On a nonlinear parabolic system-modeling chemical reactions in rivers, Communications On Pure And Applied Analysis, 4 (2005), 889-899.  doi: 10.3934/cpaa.2005.4.889.

[6]

M. ChoulliL. Kayser and E. M. Ouhabaz, Observations on Gaussian upper bounds for Neumann heat kernels, Bulletin of the Australian Mathematical Society, 92 (2015), 429-439.  doi: 10.1017/S0004972715000611.

[7]

G. CraciunA. DickensteinA. Shiu and B. Sturmfels, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.

[8]

G. CraciunF. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.  doi: 10.1137/100812355.

[9]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2016), arXiv: 1501.02860.

[10]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.

[11]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.  doi: 10.4171/RMI/541.

[12]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. 

[13]

L. DesvillettesK. FellnerM. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Adv. Nonlinear Stud., 7 (2007), 491-511.  doi: 10.1515/ans-2007-0309.

[14]

L. DesvillettesK. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion system arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.  doi: 10.1137/16M1073935.

[15]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972/73), 187-194.  doi: 10.1007/BF00255665.

[16]

K. Fellner, W. Prager and B. Q. Tang, The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks, Kinet. Relat. Models, 10 (2017), 1055–1087, arXiv: 1504.08221. doi: 10.3934/krm.2017042.

[17]

K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., 69 (2018), Paper No. 54, 30 pp. doi: 10.1007/s00033-018-0948-3.

[18]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x.

[19]

W. E. FitzgibbonJ. Morgan and R. Sanders, Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems, Nonlin. Anal., 19 (1992), 885-899.  doi: 10.1016/0362-546X(92)90057-L.

[20]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.  doi: 10.1137/130928170.

[21]

M. Gopalkrishnan, E. Miller and A. Shiu, A projection argument for differential inclusions, with applications to persistence of mass-action kinetics, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 025, 25 pp. doi: 10.3842/SIGMA.2013.025.

[22]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.

[23]

F. Horn, The dynamics of open reaction systems, Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, SIAM-AMS Proceedings, Amer. Math. Soc., Providence, R.I., 8 (1974), 125-137. 

[24]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs, AMS, 23 (1995).

[25]

A. MielkeJ. Haskovec and P. A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.  doi: 10.1007/s10884-014-9394-x.

[26]

M. Minchevaand and D. Siegel, Stability of mass action reaction-diffusion systems, Nonlinear Anal., 56 (2004), 1105-1131.  doi: 10.1016/j.na.2003.10.025.

[27]

F. MohamedC. Pantea and A. Tudorascu, Chemical reaction-diffusion networks: Convergence of the method of lines, J. Math. Chem., 56 (2018), 30-68.  doi: 10.1007/s10910-017-0779-z.

[28]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.  doi: 10.1137/110840509.

[29]

M. PierreT. Suzuki and H. Umakoshi, Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria, J. Appl. Anal. Comp., 8 (2018), 836-858.  doi: 10.11948/2018.836.

[30]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[31]

F. Rothe, Global Solutions of Reaction-Diffusion System, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[32]

D. Siegel and D. MacLean, Global stability of complex balanced mechanisms, J. Math. Chem., 27 (2000), 89-110.  doi: 10.1023/A:1019183206064.

[33]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[34]

E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028-1047.  doi: 10.1109/9.935056.

[35]

M. E. Taylor, Partial Differential Equation Ⅲ. Nonlinear Equations, Springer Series Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[36]

P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L^p$-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51.  doi: 10.1090/S1079-6762-02-00104-X.

Figure 1.  Construction of a rectangular invariant region for the reversible reaction $ m_1A+n_1B\rightleftharpoons m_2A+n_2B $ for the cases a. $ \bar m = m_1-m_2, \ \bar n = n_2-n_1 $ nonzero and of the same sign; b. $ \bar m, \bar n $ nonzero and of different signs
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