• Previous Article
    Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise
  • DCDS-B Home
  • This Issue
  • Next Article
    Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions
October  2021, 26(10): 5407-5419. doi: 10.3934/dcdsb.2020349

Effective reduction of a three-dimensional circadian oscillator model

1. 

School of Mathematics and Statistics & Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

2. 

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

3. 

School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Shuang Chen

Received  July 2020 Published  October 2021 Early access  November 2020

Fund Project: This work was partly supported by the NSFC grants 11531006, 11771449, 11771161, and the Hubei provincial postdoctoral science and technology activity project

We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.

Citation: Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5407-5419. doi: 10.3934/dcdsb.2020349
References:
[1]

S. BoieV. KirkJ. Sneyd and M. Wechselberger, Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.  doi: 10.1016/j.jtbi.2015.12.011.

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[3]

B. Deng, The Šil'nikov problem, exponential expansion, strong $\lambda$-lemma, $C^{1}$ linearization, and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.  doi: 10.1016/0022-0396(89)90100-9.

[4]

B. Deng and G. Hines, Food chain chaos due to transcritical point, Chaos, 13 (2003), 578-585.  doi: 10.1063/1.1576531.

[5]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[6]

F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., Vol. 577, Providence, 1996. doi: 10.1090/memo/0577.

[7]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[8]

J. Dunlap, Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.  doi: 10.1016/S0092-8674(00)80566-8.

[9]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[10]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.

[11]

N. Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), 81-93.  doi: 10.1512/iumj.1977.26.26006.

[12]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[13] D. B. Forger, Biological Clocks, Rhythms, and Oscillations, MIT Press, Cambridge, MA, 2017. 
[14]

A. GoekeS. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.

[15]

D. Gonze, Modeling circadian clocks: From equations to oscillations, Cent. Eur. J. Bio., 6 (2011), 699-711.  doi: 10.2478/s11535-011-0061-5.

[16]

D. A. Goussis and H. N. Najm, Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.  doi: 10.1137/060649768.

[17]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.

[18]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems (eds. R. Johnson), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.

[19]

C. K. R. T. JonesT. J. Kaper and N. Kopell, Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558-577.  doi: 10.1137/S003614109325966X.

[20]

H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66-93.  doi: 10.1016/S0167-2789(02)00386-X.

[21]

J. Keener and J. Sneyd, Mathematical Physiology, Int. Appl. Math., 8, Springer-Verlag, New York, 1998. doi: 10.1007/978-0-387-79388-7.

[22]

I. Kosiuk and P. Szmolyan, Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.  doi: 10.1007/s00285-015-0905-0.

[23]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.

[24]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[25]

C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015. doi: 10.1007/978-3-319-12316-5.

[26]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.

[27]

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.  doi: 10.1006/jdeq.2000.3778.

[28]

U. Maas and S. Pope, Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.  doi: 10.1016/0010-2180(92)90034-M.

[29]

D. McMillenN. KopellJ. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684. 

[30]

J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146. doi: 10.1016/S1874-575X(02)80024-8.

[31]

S. Schecter, Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.  doi: 10.1016/j.jde.2007.08.011.

[32]

S. Schecter, Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.  doi: 10.1016/j.jde.2007.10.021.

[33]

J. TysonC. HongC. Thron and B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.  doi: 10.1016/S0006-3495(99)77078-5.

[34]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4312-0.

[35]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992. doi: 10.1007/978-1-4757-4969-4_4.

show all references

References:
[1]

S. BoieV. KirkJ. Sneyd and M. Wechselberger, Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.  doi: 10.1016/j.jtbi.2015.12.011.

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[3]

B. Deng, The Šil'nikov problem, exponential expansion, strong $\lambda$-lemma, $C^{1}$ linearization, and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.  doi: 10.1016/0022-0396(89)90100-9.

[4]

B. Deng and G. Hines, Food chain chaos due to transcritical point, Chaos, 13 (2003), 578-585.  doi: 10.1063/1.1576531.

[5]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[6]

F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., Vol. 577, Providence, 1996. doi: 10.1090/memo/0577.

[7]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[8]

J. Dunlap, Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.  doi: 10.1016/S0092-8674(00)80566-8.

[9]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[10]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.

[11]

N. Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), 81-93.  doi: 10.1512/iumj.1977.26.26006.

[12]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[13] D. B. Forger, Biological Clocks, Rhythms, and Oscillations, MIT Press, Cambridge, MA, 2017. 
[14]

A. GoekeS. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.

[15]

D. Gonze, Modeling circadian clocks: From equations to oscillations, Cent. Eur. J. Bio., 6 (2011), 699-711.  doi: 10.2478/s11535-011-0061-5.

[16]

D. A. Goussis and H. N. Najm, Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.  doi: 10.1137/060649768.

[17]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.

[18]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems (eds. R. Johnson), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.

[19]

C. K. R. T. JonesT. J. Kaper and N. Kopell, Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558-577.  doi: 10.1137/S003614109325966X.

[20]

H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66-93.  doi: 10.1016/S0167-2789(02)00386-X.

[21]

J. Keener and J. Sneyd, Mathematical Physiology, Int. Appl. Math., 8, Springer-Verlag, New York, 1998. doi: 10.1007/978-0-387-79388-7.

[22]

I. Kosiuk and P. Szmolyan, Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.  doi: 10.1007/s00285-015-0905-0.

[23]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.

[24]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[25]

C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015. doi: 10.1007/978-3-319-12316-5.

[26]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.

[27]

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.  doi: 10.1006/jdeq.2000.3778.

[28]

U. Maas and S. Pope, Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.  doi: 10.1016/0010-2180(92)90034-M.

[29]

D. McMillenN. KopellJ. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684. 

[30]

J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146. doi: 10.1016/S1874-575X(02)80024-8.

[31]

S. Schecter, Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.  doi: 10.1016/j.jde.2007.08.011.

[32]

S. Schecter, Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.  doi: 10.1016/j.jde.2007.10.021.

[33]

J. TysonC. HongC. Thron and B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.  doi: 10.1016/S0006-3495(99)77078-5.

[34]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4312-0.

[35]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992. doi: 10.1007/978-1-4757-4969-4_4.

Figure 1.  The mechanism for the circadian oscillator model (1). Adapted from [33]
Figure 2.  The attraction of the slow manifold. The surface is the critical manifolds $ \mathcal{M}_{0} $, which is the zeroth-order approximation of the slow manifold, and the discrete orbits, respectively, start from $ (10,10,2) $, $ (15,15,2) $, $ (20,20,2) $, $ (10,20,2) $ and $ (20,10,2) $. Here $ k_{a} = 20000 $, $ k_{d} = 100 $ and the remaining parameters in (2) are chosen as in [33,Table 1,p.2414], that is, $ v_{m} = 1 $, $ k_{3} = k_{m} = 0.1 $, $ v_{p} = 0.5 $, $ k_{1} = 10 $, $ k_{2} = 0.03 $, $ P_{c} = 0.1 $ and $ J_{p} = 0.05 $. System (16) with $ \widetilde{k}_{2} = 1 $ has small parameter $ \varepsilon = 0.0003 $
[1]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783

[2]

Kai Wang, Hongyong Zhao, Hao Wang. Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022122

[3]

Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1775-1794. doi: 10.3934/dcdsb.2012.17.1775

[4]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[5]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807

[6]

Enrico Bernardi, Alberto Lanconelli. Stochastic perturbation of a cubic anharmonic oscillator. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2563-2585. doi: 10.3934/dcdsb.2021148

[7]

Azniv Kasparian, Ivan Marinov. Duursma's reduced polynomial. Advances in Mathematics of Communications, 2017, 11 (4) : 647-669. doi: 10.3934/amc.2017048

[8]

John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501

[9]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403

[10]

Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517

[11]

Y. Chen, L. Wang. Global attractivity of a circadian pacemaker model in a periodic environment. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 277-288. doi: 10.3934/dcdsb.2005.5.277

[12]

Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631

[13]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019

[14]

Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2065-2075. doi: 10.3934/dcdsb.2021122

[15]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397

[16]

Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431

[17]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784

[18]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

[19]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112

[20]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (219)
  • HTML views (302)
  • Cited by (1)

Other articles
by authors

[Back to Top]