一类自催化可逆生化反应模型的Hopf分支及其稳定性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (1): 166-177.

一类自催化可逆生化反应模型的Hopf分支及其稳定性

郭改慧*(),刘晓慧

陕西科技大学文理学院西安 710021

收稿日期:2020-01-09 出版日期:2021-02-26 发布日期:2021-01-29
通讯作者: 郭改慧 E-mail:guogaihui@sust.edu.cn
基金资助:
国家自然科学基金(61872227);国家自然科学基金(61672021);国家自然科学基金(11671243);国家自然科学基金(11901370);大学生创新创业训练计划(201910708010)

Hopf Bifurcation and Stability for an Autocatalytic Reversible Biochemical Reaction Model

Gaihui Guo*(),Xiaohui Liu

School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021

Received:2020-01-09 Online:2021-02-26 Published:2021-01-29
Contact: Gaihui Guo E-mail:guogaihui@sust.edu.cn
Supported by:
the NSFC(61872227);the NSFC(61672021);the NSFC(11671243);the NSFC(11901370);the National Undergraduate Innovation and Entrepreneurship Training Program(201910708010)

摘要/Abstract

摘要：

在齐次Neumann边界条件下，研究一类自催化可逆三分子生化反应模型.首先对常微分系统，给出Hopf分支的存在性及稳定性.其次对偏微分系统，建立由扩散系数引起的Turing不稳定性，同时给出Hopf分支的存在性，并利用规范型理论和中心流形定理建立Hopf分支的方向和稳定性.最后，借助Matlab软件进行数值模拟，验证补充理论分析结果.

关键词: 可逆生化反应, Turing不稳定性, Hopf分支, 稳定性

Abstract:

An autocatalytic reversible three-molecular biochemical reaction model subject to Neumann boundary conditions is considered. Firstly, the existence and stability of the Hopf bifurcation for the ordinary differential system are given. Secondly, the effect of diffusion coefficients on Turing instability is established and the existence of Hopf bifurcation is obtained for the partial differential system with diffusion. Then applying the normal form theory and center manifold theorem, the direction and stability of Hopf bifurcation are also given. Finally, some numerical simulations are carried out with the help of Matlab software to verify and supplement the theoretical results.

Key words: Reversible biochemical reaction, Turing instability, Hopf bifurcation, Stability

中图分类号:

O175.26

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.

Gaihui Guo,Xiaohui Liu. Hopf Bifurcation and Stability for an Autocatalytic Reversible Biochemical Reaction Model[J]. Acta mathematica scientia,Series A, 2021, 41(1): 166-177.

参考文献

1	Jia Y F , Li Y , Wu J H . Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (9): 4785- 4813
2	Gaspar V , Beck M T . Depressing the bistable behavior of the iodate-arsenous acid reaction in a continuous flow stirred tank reactor by the effect of chloride or bromide ions: A method for determination of rate constants. Journal of Physical Chemistry, 1986, 90 (23): 6303- 6305
3	Finlayson A B , Merkin J H . Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system. Journal of Engineering Mathematics, 2000, 38 (3): 279- 296
4	Wei M H , Wu J H , He Y N . Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3): 1147- 1167
5	Guo G H , Li B F , Lin X L . Hopf bifurcation in spatially homogeneous and inhomogeneous autocatalysis models. Computers and Mathematics with Applications, 2014, 67 (1): 151- 163
6	Zhou J . Spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law. Computers and Mathematics with Applications, 2013, 66 (10): 2003- 2018
7	Shi Q Y , Shi J P , Song Y L . Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition. Journal of Differential Equations, 2017, 263 (10): 6537- 6575
8	李海银. 密度制约且具有时滞捕食-被捕食模型的Hopf分支. 数学物理学报, 2019, 39A (2): 358- 371
8	Li H Y . Hopf bifurcation of delayed density-dependent predator-prey model. Acta Mathematica Scientia, 2019, 39A (2): 358- 371
9	Yi F Q , Liu J X , Wei J J . Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model. Nonlinear Analysis: Real World Applications, 2010, 11 (5): 3770- 3781
10	周军. 一类具有自动催化作用和饱和定律的双分子模型的图灵不稳定性和霍普夫分歧. 数学物理学报, 2017, 37A (2): 366- 373
10	Zhou J . Turing instability and Hopf bifurcation of a bimolecular model with autocatalysis and saturation law. Acta Mathematica Scientia, 2017, 37A (2): 366- 373
11	Wiggins S . Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer, 1991: 1- 280
12	Yi F Q , Wei J J , Shi J P . Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. Journal of Differential Equations, 2009, 246 (5): 1944- 1977
13	Hassard B D , Kazarinoff N D , Wan Y H . Theory and Application of Hopf bifurcation. Cambridge: Cambridge University Press, 1981

[1]	王春, 许天周. 空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子[J]. 数学物理学报, 2025, 45(5): 1381-1391.
[2]	邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.
[3]	吕东霆. 两种群空间非均匀反应扩散竞争模型的全局渐近稳定性[J]. 数学物理学报, 2025, 45(4): 1171-1183.
[4]	梁青. 两类带 Markov 切换的随机比例泛函微分方程全局解的存在唯一性和稳定性[J]. 数学物理学报, 2025, 45(4): 1184-1205.
[5]	刘佳, 包雄雄. 非局部时滞扩散方程棱锥形波前解的渐近稳定性[J]. 数学物理学报, 2025, 45(1): 44-53.
[6]	肖江龙, 宋永利, 夏永辉. 一个扩散模型中恐惧效应与集群行为协同诱导的时空动力学研究[J]. 数学物理学报, 2024, 44(6): 1577-1594.
[7]	张咏雪, 贾文生. 有限理性下广义多目标多主多从博弈受控系统解的稳定性[J]. 数学物理学报, 2024, 44(6): 1689-1702.
[8]	樊天娇, 冯立超, 杨艳梅. 不等式的推广以及在加性时变时滞系统中的应用[J]. 数学物理学报, 2024, 44(5): 1335-1351.
[9]	杨红, 张晓光. 单纯复形上的随机 SIS 传染病模型[J]. 数学物理学报, 2024, 44(5): 1392-1399.
[10]	郭燕, 徐小川. 基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性[J]. 数学物理学报, 2024, 44(4): 859-870.
[11]	潘丽君, 吕顺, 翁莎莎. 耦合Aw-Rascle-Zhang模型的Riemann解及其稳定性[J]. 数学物理学报, 2024, 44(4): 885-895.
[12]	章春国, 孙宝楠, 付煜之, 于欣. 具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定[J]. 数学物理学报, 2024, 44(4): 946-959.
[13]	高彩霞, 赵东霞. 带干扰项的 ARZ 交通流模型的时滞控制与 ISS 稳定性[J]. 数学物理学报, 2024, 44(4): 960-977.
[14]	徐瑞, 周凯娟, 白宁. 一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型[J]. 数学物理学报, 2024, 44(3): 771-782.
[15]	邱仰聪, 王其如. 一阶非线性时标动态方程的 Hyers-Ulam-Rassias 稳定性[J]. 数学物理学报, 2024, 44(2): 465-475.