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  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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
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