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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (2): 277-291.doi: 10.3969/j.issn.1005-3085.2022.02.008

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一类二重饱和可逆生化反应扩散模型的 Hopf 分支和 Turing 不稳定性

郭改慧,   郭飞燕,   刘晓慧   

  1. 陕西科技大学数学与数据科学学院，西安 710021
  • 出版日期:2022-04-15 发布日期:2022-06-15
  • 基金资助:
    国家自然科学基金 (12126420; 61872227).

Hopf Bifurcation and Turing Instability in a Reversible Biochemical Reaction-diffusion Model Combining Second-order Saturation

GUO Gaihui,   GUO Feiyan,   LIU Xiaohui   

  1. School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi'an 710021
  • Online:2022-04-15 Published:2022-06-15
  • Supported by:
    The National Natural Science Foundation of China (12126420; 61872227).

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摘要：

在齐次 Neumann 边界条件下研究一类具有二重饱和度的四分子可逆生化反应模型，以可逆反应率为参数，利用规范型理论和中心流形定理分别给出常微分系统和扩散系统 Hopf 分支的存在性、方向及其稳定性，并细致研究了扩散系数对系统稳定性的影响。结果表明，当可逆反应率较小时，正平衡点不稳定；当可逆反应率较大时，正平衡点稳定；当可逆反应率介于某一范围内时，扩散系数会对系统的稳定性产生较大影响。此时，当催化剂的扩散系数较小时，系统会产生 Turing 不稳定性。最后，选取满足定理条件的参数，通过数值模拟验证了所得结论。

关键词: 二重饱和, 可逆生化反应, Hopf 分支, Turing 不稳定性

Abstract:

In this paper, a four-molecule reversible biochemical reaction-diffusion model combining second-order saturation subject to homogeneous Neumann boundary conditions is studied. Taking the reversible reaction rate as a parameter, the existence, direction and stability of Hopf bifurcation for ordinary differential system and the diffusive system are respectively given applying the normal form theory and the central manifold theorem. Moreover, the effect of diffusive coefficient on the stability of the system is studied extensively. The results show that the positive equilibrium point is unstable when the reversible reaction rate is small. When the reversible reaction rate is large, the positive equilibrium point is stable. When the reversible reaction rate is in a certain range, the diffusive coefficient has a great effect on the stability of the system. In this case, if the catalyst's diffusive coefficient is small, then Turing instability occurs. Finally, the parameters satisfying the conditions of the theorem are selected and the theoretical results are verified by numerical simulations.

Key words: second-order saturation, reversible biochemical reaction, Hopf bifurcation, Turing instability

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