Hopf bifurcation and stability for an autocatalytic reversible biochemical reaction model with time delay

Abstract

Abstract: Based on the Hopf bifurcation theory, an autocatalytic three-molecular reversible biochemical reaction model with time delay is considered. First, the stability of the equilibrium and the existence of Hopf bifurcation are obtained by choosing the delay τ as parameter. Secondly, the direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations are presented to verify the theoretical results.

Key words: reversible biochemical reaction, Hopf bifurcation, stability, delay

CLC Number:

O175.13

GUO Fei-yan, GUO Gai-hui. Hopf bifurcation and stability for an autocatalytic reversible biochemical reaction model with time delay[J]. Journal of Lanzhou University of Technology, 2023, 49(3): 147-152.

References

[1] JIA Y F,LI Y,WU J H.Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics[J].Discrete and Continuous Dynamical Systems,2017,37(9):4785-4813.
[2] ZHAO J T,JIA Y F.Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model[J].Applied Mathematics Letters,2019,95:98-103.
[3] 郭改慧,刘晓慧.一类自催化可逆生化反应模型的Hopf分支及其稳定性[J].数学物理学报,2021,41(1):166-177.
[4] LI M M,WANG J R.Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations[J].Applied Mathematics and Computation,2018,324:254-265.
[5] SHU H Y,XU W X,WANG X S,et al.Complex dynamics in a delay differential equation with two delays in tick growth with diapause[J].Journal of Differential Equations,2020,269(12):10937-10963.
[6] MERDAN H,KAYAN Ş.Hopf bifurcations in Lengyel-Epstein reaction-diffusion model with discrete time delay[J].Nonlinear Dynamics,2015,79(3):1757-1770.
[7] 翁爱治. 一阶时滞微分方程正周期解的存在性问题[J].工程数学学报,2016,33(1):106-110.
[8] CAO J Z,YUAN R.Bifurcation analysis in a modified Lesile-Gower model with Holling type II functional response and delay[J].Nonlinear Dynamics,2016,84(3):1341-1352.
[9] LI X.Dynamical analysis of toxin producing model with delay[J].Journal of Mathematical Research with Applications,2021,41(1):33-48.
[10] WANG Z,SUN J,BAI Y Q.Stability analysis of Event-Triggered networked control systems with time-varying delay and packet loss[J].Journal of Systems Science and Complexity,2021,34(1):265-280.
[11] FINLAYSON A B,MERKIN J H.Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system[J].Journal of Engineering Mathematics,2000,38(3):279-296.
[12] GUO G H,LI B F,LIN X L.Hopf bifurcation in spatially homogeneous and inhomogeneous autocatalysis models[J].Computers and Mathematics with Applications,2014,67(1):151-163.
[13] GUO G H,LIU L,LI B F,et al.Qualitative analysis on positive steady-state solutions for an autocatalysis model with high order[J].Nonlinear Analysis Real World Applications,2018,41:665-691.
[14] HASSARD B D,KAZARINOFF N D,WAN Y H.Theory and applications of Hopf bifurcation [M].Cambridge,MA:Cambridge University Press,1981.
[15] 李帅,张志信,蒋威.分数阶时滞微分方程边值问题解的存在性与唯一性[J].工程数学学报,2021,38(5):700-708.
[16] 陈建业,张存华.Degn-Harrison化学模型的稳定性和Hopf分支[J].应用数学,2018,31(3):683-689.