Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions

Pei Yu; Xiangyu Wang; Pei Yu; Xiangyu Wang

doi:10.3934/mbe.2019263

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5263-5286. doi: 10.3934/mbe.2019263

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Research article

Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions

Pei Yu ^,,
Xiangyu Wang

Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

Received: 10 April 2019 Accepted: 14 April 2019 Published: 10 June 2019

In this paper, we present a new method based on dynamical system theory to study certain type of slow-fast motions in dynamical systems, for which geometric singular perturbation theory may not be applicable. The method is then applied to consider recurrence behavior in an oscillating network model which is biologically related to organic reactions. We analyze the stability and bifurcation of the equilibrium of the system, and find the conditions for the existence of recurrence, i.e., there exists a "window" in bifurcation diagram between a saddle-node bifurcation point and a Hopf bifurcation point, where the equilibrium is unstable. Simulations are given to show a very good agreement with analytical predictions.
- Slow-fast motion,
- oscillating network,
- organic reaction,
- sustained oscillation,
- recurrence,
- stability and bifurcation,
- saddle-node bifurcation,
- Hopf bifurcation,
- limit cycle
Citation: Pei Yu, Xiangyu Wang. Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5263-5286. doi: 10.3934/mbe.2019263

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Abstract

In this paper, we present a new method based on dynamical system theory to study certain type of slow-fast motions in dynamical systems, for which geometric singular perturbation theory may not be applicable. The method is then applied to consider recurrence behavior in an oscillating network model which is biologically related to organic reactions. We analyze the stability and bifurcation of the equilibrium of the system, and find the conditions for the existence of recurrence, i.e., there exists a "window" in bifurcation diagram between a saddle-node bifurcation point and a Hopf bifurcation point, where the equilibrium is unstable. Simulations are given to show a very good agreement with analytical predictions.

References

[1]	W. Zhang, L. M. Wahl and P. Yu, Modelling and analysis of recurrent autoimmune disease, SIAM J. Appl. Math., 74 (2014), 1998–2025.
[2]	H. J. Girschick, C. Zimmer, G. Klaus, et al., Chronic recuurent multifocal osteomyelitis: What is it and how should it be treated? Nat. Clin. Practice Rheumatol., 3 (2007), 733–738.
[3]	R. S. Iyer, M. M. Thapa and F. S. Chew, Chronic recurrent multifocal osteomyelitis: Review, Amer. J. Roentgenol., 196 (2011), S87–S91.
[4]	D. M. Fergusson, L. J. Horwood and F. T. Shannon, Early solid feeding and recurrent childhood eczema: A 10-year longitudinal study, Pediatrics, 86 (1990), 541–546.
[5]	D. D. Munro, Recurrent subacute discoid lupus erythematosus, Proc. Roy. Soc. Med., 56 (1963), 78–79.
[6]	T. Vollmer, The natural history of relapses in multiple sclerosis, J. Neurolog. Sci., 256 (2007), s5–s13.
[7]	N. Fenichel, Geometri singular perturbation theory, J. Diff. Eqns., 31 (1979) 879–910.
[8]	X. F. Chen, P. Yu, M. Han, et al., Canard solutions of 2-D singularly perturbed systems, Chaos Soliton. Fract., 23 (2005), 915–927.
[9]	W. Zhang, L. M. Wahl and P. Yu, Conditions for transient viremia in deterministic in-host models: viral blips need no exogenous trigger, SIAM J. Appl. Math., 73 (2013), 853–881.
[10]	W. Zhang, L. M. Wahl and P. Yu, Viral blips may not need a trigger: how transient viremia can arise in deterministic in-host models, SIAM Review, 56 (2014), 127–155.
[11]	P. Yu, W. Zhang and L.M. Wahl, Dynamical analysis and simulation of a 2-dimensional disease model with convex incidence, Commun. Nonlinear Sci. Numer. Simulat., 37 (2016), 163–192.
[12]	N. S. Semenov, J. K. Lewis, A. Alar, et al., Autocatalytic, bistable, oscillatory networks of biolog-ically relevant organic reactions, Nature, 573 (2016), 656–660.
[13]	P. Nghe, W. Hordijk, S. A. Kauffman, et al., Prebiotic network evolution:six key parameters, Mol. Biosyst., 11 (2015), 3206–3217.
[14]	F. J. Dyson, A model for the origin of life, J. Mol. Evol., 18 (1982), 344–350.
[15]	B. H. Patel, C. Percivalle, D. J. Ritson, et al., Common origins ofRNA, protein and lipid precursors in a cyanosulfidic protometabolism, Nat. Chem., 7 (2015), 301–307.
[16]	J. J. Tyson, Modeling the cell -division cycle: cdc2 and cyclin interactions, Proc. Natl. Acad. Sci. USA, 88 (1991), 7328–7332.
[17]	J. J. Tyson, K. C. Chen, B. Novak, et al., Toggles and blinkers:dynamics of regulatory and signaling pathways in the cell, Curr. Opin. Cell Biol., 15 (2003), 221–231.
[18]	J. E. Ferrell, T. Y. C. Tasi and Q. O. Yang, Modeling the cell cycle:why do certain circuits oscillate, Cell, 244 (2011), 874–885.
[19]	A. Goldbeter, A model for circadian oscillations in the Drosophila period protein(PER), Proc. R. Soc. Lond., B261 (1995), 319–324.
[20]	R. FitzHugh, D. M. Fergusson, L. J. Horwood, et al., Impulses and physiological states in theoret-ical models of nerve membrane, Biophys, J1 (1961), 445–466.
[21]	M. T. Laub and W. F. A. Loomis, A molecular network that produces spontaneous oscillations in excitable cells of dictyostelium, Mol. Biol. Cell, 9 (1998), 3521–3532.
[22]	A. D. Lander, Pattern, growth, and control, Cell, 144 (2011), 955–969.
[23]	B. P. Belousov, Periodicheski deistvuyushchaya reaktsia i ee mechanism [in Russian]. In Sbornik Referatov po Radiatsionni Meditsine, (1958), 145–147.
[24]	L. Gyorgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162–7170.
[25]	I. R. Epstein and J. A. Pojman, An introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos, in Chps.2 and 4, Oxford Univ. Press, (1998), 17–47 and 62–83.
[26]	I. U. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3^rd edition, Ch.3: 77-115, Springer, New York, 2004.
[27]	D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory Ⅰ: Modelling, State Space Anal-ysis, Stability and Robustnes, 2^nd edition, Springer-Verlag, New York, 2005.
[28]	P. Yu, Closed-form conditions of bifurcation points for general differential equations, Int. J. Bifur-cat. Chaos, 15 (2005), 1467–1483.
[29]	M. Han and P. Yu, Normal Forms Melnikov Functions and Bifurcations of Limit Cycles, Springer, London, 2012.
[30]	P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), 19–38.
[31]	W. Zhang and P. Yu, Hopf and generalized Hopf bifurcations in a recurrent autoimmune disease model, Int. J. Bifurcat. Chaos, 26 (2016), 1650079–1650102.
[32]	C. Li, J. Li, Z. Ma, et al., Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987–1004.

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