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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 ›› 2018, Vol. 35 ›› Issue (4): 427-444.doi: 10.3969/j.issn.1005-3085.2018.04.006

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具有时滞和阶段结构的生态-流行病模型的稳定性及Hopf分支

王玲书1,   姚   沛2   

  1. 1- 河北经贸大学数学与统计学学院，石家庄 050061
    2- 石家庄信息工程职业学院国际贸易管理系，石家庄  050035
  • 收稿日期:2016-04-11 接受日期:2016-12-30 出版日期:2018-08-15 发布日期:2018-10-15
  • 基金资助:
    国家自然科学基金(11371368)；河北省教育厅科学技术研究项目(ZD2018052)；河北经贸大学基金(2015KYQ01).

Stability and Hopf Bifurcation of an Eco-epidemiological Predator-prey Model with Stage-structure and Time Delay

WANG Ling-shu1,   YAO Pei2   

  1. 1- School of Mathematics and Statistics, Hebei University of Economics & Business, Shijiazhuang 050061 
    2- Department of International Trade Management, Shijiazhuang  Information Engineering Vocational College, Shijiazhuang 050035
  • Received:2016-04-11 Accepted:2016-12-30 Online:2018-08-15 Published:2018-10-15
  • Supported by:
    The National Natural Science Foundation of China (11371368); the Scientific Research Foundation of Hebei Education Department (ZD2018052); the Foundation of Hebei University of Economics & Business (2015KYQ01).

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摘要： 本文研究一类具有时滞和阶段结构的生态-流行病模型的稳定性及其Hopf分支．给出了边界平衡点和正平衡点存在的充分条件；通过分析特征方程，运用Hurwitz判定定理，讨论了边界平衡点和正平衡点的局部稳定性，并得到了正平衡点附近存在Hopf分支的充分条件；通过构造适当的Lyapunov泛函，运用LaSall不变集原理，讨论了边界平衡点和正平衡点的全局稳定性，从而得到了该生态模型永久持续生存与灭绝的充分条件．

关键词: 生态-流行病模型, 阶段结构, 时滞, 稳定性, Hopf分支

Abstract: In this paper, the stability and Hopf bifurcation of an eco-epidemiological model with a time delay and a stage structure is investigated. By analyzing the characteristic equations and applying Hurwitz criterion, the local stability of the boundary equilibria and the positive equilibrium are discussed, respectively. Moreover, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. By using Lyapunov functions and LaSalle's invariance principle, the global stability of the boundary equilibria and the positive equilibrium is addressed, respectively. Therefore, the sufficient conditions are given for the permanence and extinction of the model.

Key words: eco-epidemiological model, stage structure, time delay, stability, Hopf bifurcation

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