American Institute of Mathematical Sciences

Advanced
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

Gheorghe Craciun 1, , Jiaxin Jin 2, , Casian Pantea 3, and  Adrian Tudorascu 3,

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.

Keywords: Reaction-diffusion systems, global solutions, entropy estimates, chemical reaction networks, complex-balanced equilibrium.
Mathematics Subject Classification: 35B40, 35K57, 35Q92, 80A30, 80A32, 90E20.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Arnold, P. Markowich, G. 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.  Google Scholar

[5]

W. X. Chen, C. 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.  Google Scholar

[6]

M. Choulli, L. 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.  Google Scholar

[7]

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

[8]

G. Craciun, F. 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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. Desvillettes, K. Fellner, M. 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.  Google Scholar

[14]

L. Desvillettes, K. 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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. E. Fitzgibbon, J. 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.  Google Scholar

[20]

M. Gopalkrishnan, E. 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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.   Google Scholar

[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). Google Scholar

[25]

A. Mielke, J. 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.  Google Scholar

[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.  Google Scholar

[27]

F. Mohamed, C. 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.  Google Scholar

[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.  Google Scholar

[29]

M. Pierre, T. 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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Arnold, P. Markowich, G. 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.  Google Scholar

[5]

W. X. Chen, C. 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.  Google Scholar

[6]

M. Choulli, L. 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.  Google Scholar

[7]

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

[8]

G. Craciun, F. 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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. Desvillettes, K. Fellner, M. 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.  Google Scholar

[14]

L. Desvillettes, K. 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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. E. Fitzgibbon, J. 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.  Google Scholar

[20]

M. Gopalkrishnan, E. 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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.   Google Scholar

[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). Google Scholar

[25]

A. Mielke, J. 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.  Google Scholar

[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.  Google Scholar

[27]

F. Mohamed, C. 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.  Google Scholar

[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.  Google Scholar

[29]

M. Pierre, T. 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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042
[2]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
[3]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304
[4]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283
[5]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[6]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[7]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[8]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[9]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189
[10]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427
[11]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[12]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126
[13]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[14]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[15]

M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395
[16]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69
[17]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001
[18]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[19]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[20]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (182)
  • HTML views (299)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences