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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (10): 32-42, 53.doi: 10.6040/j.issn.1671-9352.0.2023.279

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一类具有时滞的植被-水反应扩散模型的Hopf分支

郭改慧(),王晶晶,李旺瑞   

  1. 陕西科技大学数学与数据科学学院, 陕西 西安 710021
  • 收稿日期:2023-06-27 出版日期:2023-10-20 发布日期:2023-10-17
  • 作者简介:郭改慧(1980—),女,教授,博士生导师,博士,研究方向为反应扩散方程及其应用. E-mail: guogaihui@sust.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(61872227);国家自然科学基金资助项目(12126420)

Hopf bifurcation of a vegetation-water reaction-diffusion model with time delay

Gaihui GUO(),Jingjing WANG,Wangrui LI   

  1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, Shaanxi, China
  • Received:2023-06-27 Online:2023-10-20 Published:2023-10-17

摘要:

以时滞τ为分支参数, 通过分析特征方程, 给出时滞对正平衡点稳定性的影响以及Hopf分支产生的条件。利用规范型理论和中心流形定理, 得到Hopf分支方向和周期解稳定性的判定条件。最后借助数值模拟验证理论结果。

关键词: 植被-水模型, 时滞, Hopf分支, 稳定性

Abstract:

Taking the time delay τ as the bifurcation parameter, the effect of time delay on the stability of the positive steady state point and the existence of Hopf bifurcation are given by analyzing the characteristic equation. The criteria for the direction of Hopf bifurcation and the stability of periodic solutions are obtained by the normal form theory and the center manifold theorem. Finally, the theoretical results are verified by numerical simulations.

Key words: vegetation-water model, time delay, Hopf bifurcation, stability

中图分类号: 

  • O175.26

图1

参数a=14.74>a0, 系统(3)在没有时滞影响和扩散的情况下正平衡点(u*, v*)局部渐近稳定"

图2

参数a=14.240, 系统(3)在没有时滞影响和扩散的情况下产生稳定的周期闭轨"

图3

参数τ=0.46 < τ00, 系统(3)的正平衡点(u*, v*)局部渐近稳定"

图4

参数τ=0.57>τ00, 系统(3)产生稳定的分支周期解"

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