September  2019, 24(9): 5203-5224. doi: 10.3934/dcdsb.2019129

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

1. 

School of Mathematics and Statistics, Yulin University, Yulin 719000, China

2. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

* Corresponding author: Meihua Wei

Received  August 2017 Revised  January 2019 Published  September 2019 Early access  July 2019

In this paper, a glycolysis model subject to no-flux boundary condition is considered. First, by discussing the corresponding characteristic equation, the stability of constant steady state solution is discussed, and the Turing's instability is shown. Next, based on Lyapunov-Schmidt reduction method and singularity theory, the multiple stationary bifurcations with singularity are analyzed. In particular, under no-flux boundary condition we show the existence of nonconstant steady state solution bifurcating from a double zero eigenvalue, which is always excluded in most existing works. Also, the stability, bifurcation direction and multiplicity of the bifurcation steady state solutions are investigated by the singularity theory. Finally, the theoretical results are confirmed by numerical simulations. It is also shown that there is no Hopf bifurcation on basis of the condition $ (C) $.

Citation: Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
References:
[1]

M. Archetti, Evolutionary dynamics of the Warburg effect: Glycolysis as a collective action problem among cancer cells, J. Theor. Biol., 341 (2014), 1-8.  doi: 10.1016/j.jtbi.2013.09.017.

[2]

D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in non-flux boundary value problems, Math. Proc. Cambridge Philos. Soc., 101 (1987), 167-192.  doi: 10.1017/S0305004100066500.

[3]

M. Ashkenazi and H. G. Othmer, Spatial patterns in coupled biochemical oscillators, J. Math. Biol., 5 (1978), 305-350.  doi: 10.1007/BF00276105.

[4]

J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of reaction-diffusion equations. Ⅲ. Chemical oscillations, Bull. Math. Biol., 38 (1976), 325-350.  doi: 10.1007/bf02462209.

[5]

S. C. Bhargava, On the higgins model of glycolysis, Bull. Math. Biol., 42 (1980), 829-836.  doi: 10.1007/BF02461061.

[6]

L. L. Bonilla and M. G. Velarde, Singular perturbations approach to the limit cycle and global patterns in a nonlinear diffusion-reaction problem with autocatalysis and saturation law, J. Math. Phys., 20 (1979), 2692-2703.  doi: 10.1063/1.524034.

[7]

M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[8]

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.

[9]

Y. H. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 777-809.  doi: 10.1017/S0308210500023064.

[10]

L. K. Forbes and C. A. Holmes, Limit-cycle behaviour in a model chemical reaction: the cubic autocatalator, J. Engrg. Math., 24 (1990), 179-189.  doi: 10.1007/BF00129873.

[11]

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.

[12]

M. A. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer, New York, 1985. doi: 10.1007/978-1-4612-5034-0.

[13]

W. Han and Z. H. 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.

[14]

M. Herschkowitz-Kaufman, Bifurcation analysis of nonlinear reaction-diffusion equations. Ⅱ. Steady state solutions and comparison with numerical simulations, Bull. Math. Biol., 37 (1975), 589-636.  doi: 10.1007/bf02459527.

[15]

J. Higgins, A chemical mechanism for oscillations of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. USA, 51 (1964), 989-994.  doi: 10.1073/pnas.51.6.989.

[16]

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.

[17]

H. X. Li, Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response, Comput. Math. Appl., 68 (2014), 693-705.  doi: 10.1016/j.camwa.2014.07.018.

[18]

H. X. LiY. L. Li and W. B. Yang, Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion, Nonlinear Anal. Real World Appl., 27 (2016), 261-282.  doi: 10.1016/j.nonrwa.2015.07.010.

[19]

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.

[20]

J. LuM. Tan and Q. Cai, The Warburg effect in tumor progression: Mitochondrial oxidative metabolism as an anti-metastasis mechanism, Cancer Lett., 356 (2015), 156-164.  doi: 10.1016/j.canlet.2014.04.001.

[21]

J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal.: Real World Appl., 5 (2004), 105-121.  doi: 10.1016/S1468-1218(03)00020-8.

[22]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200.  doi: 10.1007/BF00275897.

[23]

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.

[24]

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

[25]

R. PengM. X. Wang and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling, 44 (2006), 945-951.  doi: 10.1016/j.mcm.2006.03.001.

[26]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[27]

W. H. Ruan, Asymptotic behavior and positive steady-state solutions of a reaction-diffusion model with autocatalysis and saturation law, Nonlinear Anal., 21 (1993), 439-456.  doi: 10.1016/0362-546X(93)90127-E.

[28] L. A. Segel, Mathematical Models in Molecular and Cellular Biology, Cambridge University Press, Cambridge, 1980. 
[29]

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

[30]

J. P. Shi, Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Front. Math. China, 4 (2009), 407-424.  doi: 10.1007/s11464-009-0026-4.

[31]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.

[32]

J. Tyson and S. Kauffman, Control of mitosis by a continuous biochemical oscillation, J. Math. Biol., 1 (1975), 289-310. 

[33]

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.

[34]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[35]

O. Warburg, On the origin of cancer cells, Science, 123 (1956), 309-314.  doi: 10.1126/science.123.3191.309.

[36]

M. H. WeiJ. H. Wu and G. H. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction, Nonlinear Anal.: Real World Appl., 22 (2015), 155-175.  doi: 10.1016/j.nonrwa.2014.08.003.

[37]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the unstirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.

[38]

F. Q. YiJ. X. Liu and J. J. Wei, Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model, Nonlinear Anal., 11 (2010), 3770-3781.  doi: 10.1016/j.nonrwa.2010.02.007.

show all references

References:
[1]

M. Archetti, Evolutionary dynamics of the Warburg effect: Glycolysis as a collective action problem among cancer cells, J. Theor. Biol., 341 (2014), 1-8.  doi: 10.1016/j.jtbi.2013.09.017.

[2]

D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in non-flux boundary value problems, Math. Proc. Cambridge Philos. Soc., 101 (1987), 167-192.  doi: 10.1017/S0305004100066500.

[3]

M. Ashkenazi and H. G. Othmer, Spatial patterns in coupled biochemical oscillators, J. Math. Biol., 5 (1978), 305-350.  doi: 10.1007/BF00276105.

[4]

J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of reaction-diffusion equations. Ⅲ. Chemical oscillations, Bull. Math. Biol., 38 (1976), 325-350.  doi: 10.1007/bf02462209.

[5]

S. C. Bhargava, On the higgins model of glycolysis, Bull. Math. Biol., 42 (1980), 829-836.  doi: 10.1007/BF02461061.

[6]

L. L. Bonilla and M. G. Velarde, Singular perturbations approach to the limit cycle and global patterns in a nonlinear diffusion-reaction problem with autocatalysis and saturation law, J. Math. Phys., 20 (1979), 2692-2703.  doi: 10.1063/1.524034.

[7]

M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[8]

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.

[9]

Y. H. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 777-809.  doi: 10.1017/S0308210500023064.

[10]

L. K. Forbes and C. A. Holmes, Limit-cycle behaviour in a model chemical reaction: the cubic autocatalator, J. Engrg. Math., 24 (1990), 179-189.  doi: 10.1007/BF00129873.

[11]

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.

[12]

M. A. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer, New York, 1985. doi: 10.1007/978-1-4612-5034-0.

[13]

W. Han and Z. H. 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.

[14]

M. Herschkowitz-Kaufman, Bifurcation analysis of nonlinear reaction-diffusion equations. Ⅱ. Steady state solutions and comparison with numerical simulations, Bull. Math. Biol., 37 (1975), 589-636.  doi: 10.1007/bf02459527.

[15]

J. Higgins, A chemical mechanism for oscillations of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. USA, 51 (1964), 989-994.  doi: 10.1073/pnas.51.6.989.

[16]

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.

[17]

H. X. Li, Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response, Comput. Math. Appl., 68 (2014), 693-705.  doi: 10.1016/j.camwa.2014.07.018.

[18]

H. X. LiY. L. Li and W. B. Yang, Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion, Nonlinear Anal. Real World Appl., 27 (2016), 261-282.  doi: 10.1016/j.nonrwa.2015.07.010.

[19]

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.

[20]

J. LuM. Tan and Q. Cai, The Warburg effect in tumor progression: Mitochondrial oxidative metabolism as an anti-metastasis mechanism, Cancer Lett., 356 (2015), 156-164.  doi: 10.1016/j.canlet.2014.04.001.

[21]

J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal.: Real World Appl., 5 (2004), 105-121.  doi: 10.1016/S1468-1218(03)00020-8.

[22]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200.  doi: 10.1007/BF00275897.

[23]

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.

[24]

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

[25]

R. PengM. X. Wang and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling, 44 (2006), 945-951.  doi: 10.1016/j.mcm.2006.03.001.

[26]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[27]

W. H. Ruan, Asymptotic behavior and positive steady-state solutions of a reaction-diffusion model with autocatalysis and saturation law, Nonlinear Anal., 21 (1993), 439-456.  doi: 10.1016/0362-546X(93)90127-E.

[28] L. A. Segel, Mathematical Models in Molecular and Cellular Biology, Cambridge University Press, Cambridge, 1980. 
[29]

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

[30]

J. P. Shi, Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Front. Math. China, 4 (2009), 407-424.  doi: 10.1007/s11464-009-0026-4.

[31]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.

[32]

J. Tyson and S. Kauffman, Control of mitosis by a continuous biochemical oscillation, J. Math. Biol., 1 (1975), 289-310. 

[33]

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.

[34]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[35]

O. Warburg, On the origin of cancer cells, Science, 123 (1956), 309-314.  doi: 10.1126/science.123.3191.309.

[36]

M. H. WeiJ. H. Wu and G. H. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction, Nonlinear Anal.: Real World Appl., 22 (2015), 155-175.  doi: 10.1016/j.nonrwa.2014.08.003.

[37]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the unstirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.

[38]

F. Q. YiJ. X. Liu and J. J. Wei, Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model, Nonlinear Anal., 11 (2010), 3770-3781.  doi: 10.1016/j.nonrwa.2010.02.007.

Figure 1.  Local bifurcation of $ g(s,\lambda) = 0 $ at $ (s,\lambda) = (0,0) $. Here, (a) $ g(s,\lambda) = s^3+\lambda s $; (b) $ g(s,\lambda) = -s^3+\lambda s $
Figure 2.  The zero of $ C = 0 $
Figure 3.  The graph of (2) with $ k = 0.1, \delta = 3.0 $ and $ l = 6.0 $. Here, (a) $ d_2 = 0.105 $; (b) $ d_2 = \frac{\sqrt{(\lambda_3+\lambda_4)^2+4g_1\lambda_3\lambda_4} -(\lambda_3+\lambda_4)}{2\lambda_3\lambda_4} = 0.1200 $
Figure 4.  Numerical simulations of the steady state solution characterized by $ \phi_{4} $ for system (1) with $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.105 $ and $ d_1 = 5.8560 $
Figure 5.  Concentration profiles for $ u $ and $ v $ of (1) for $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.105 $ and $ d_1 = 5.8560 $
Figure 6.  Numerical simulations of the steady state solution involved two models $ \phi_{3} $ and $ \phi_{4} $ for system (1) with $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.1200 $ and $ d_1 = 7.0093 $
Figure 7.  Concentration profiles for $ u $ and $ v $ of (1) for $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.1200 $ and $ d_1 = 7.0093 $
[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[3]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[4]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160

[5]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035

[6]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[7]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945

[8]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[9]

Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621

[10]

Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the MAP/M/s+G queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2505-2532. doi: 10.3934/jimo.2021078

[11]

Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613

[12]

Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941

[13]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[14]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105

[15]

Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042

[16]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849

[17]

Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239

[18]

Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805

[19]

Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055

[20]

Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks and Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (322)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]