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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 ›› 2026, Vol. 43 ›› Issue (2): 245-259.doi: 10.3969/j.issn.1005-3085.2026.02.003cstr: 32411.14.cjem.CN61-1269/O1.2026.02.003

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吸烟与肺癌增长关系双时滞模型的Hopf分支

王晶囡,  刘童星   

  1. 哈尔滨理工大学理学院，哈尔滨  150080
  • 收稿日期:2023-06-27 接受日期:2023-12-15 出版日期:2026-04-15 发布日期:2026-06-15
  • 基金资助:
    国家自然科学基金 (11801122)；黑龙江省自然科学基金 (A2018008).

Hopf Bifurcation for a Smoking and Lung Cancer Growth Interaction Model with Double Time Delays

WANG Jingnan,  LIU Tongxing   

  1. School of Science, Harbin University of Science and Technology, Harbin 150080
  • Received:2023-06-27 Accepted:2023-12-15 Online:2026-04-15 Published:2026-06-15
  • Supported by:
    The National Natural Science Foundation of China (11801122); the Natural Science Foundation of Heilongjiang Province (A2018008).

摘要：

考虑到潜在吸烟者和被动吸烟者转变为主动吸烟者存在的时滞因素，建立了一类吸烟与肺癌增长关系的双时滞动力学模型。通过分析特征方程根的分布与得到的稳定性转换曲线、平衡点渐近稳定性与Hopf分支存在条件，来研究吸烟对肺癌增长的具体影响。根据Hopf分支周期解的稳定性和分支方向的计算公式，利用数值模拟展示了所建模型在环境承载力、被动吸烟率与时滞等参数变化时，肺癌增长的变化规律，如稳态、周期振荡和混沌等现象。得到的理论结果和数值结果有助于理解吸烟对肺癌增长的重要影响，为预防和控制吸烟提供了理论指导策略。

关键词: 双时滞, 稳定性, Hopf分支, 肺癌, 吸烟

Abstract:

In consideration of the time delays required for the transform from potential smokers and passive smokers into active smokers, the paper establishes a smoking and lung cancer growth interaction model with double time delays. By analyzing the distribution of characteristic equation roots and the stability switching curves, as well as the conditions for the equilibrium asymptotic stability and the Hopf bifurcation, the specific effects of smoking on lung growth are studied. On the basis of  the computing formulas for the stability and the direction of Hopf bifurcating periodic solutions, numerical simulations are performed to show the changing laws of lung cancer growth, such as steady states, periodic oscillations and chaos, when the parameters of the environmental carrying capacity, the passive smoking rate and the time delays vary. The theoretical and numerical results are helpful to understand the relationship between smoking and lung cancer growth, which provide theoretical guidelines for preventing and controlling smoking.

Key words: double time delays, stability, Hopf bifurcation, lung cancer, smoke

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