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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 ›› 2016, Vol. 33 ›› Issue (2): 138-150.doi: 10.3969/j.issn.1005-3085.2016.02.004

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一个具有时滞和阶段结构的比率依赖型捕食系统的稳定性

王玲书1,  张雅南1,  冯光辉2   

  1. 1- 河北经贸大学数学与统计学学院，石家庄  050061
    2- 军械工程学院基础部，石家庄 050003
  • 收稿日期:2014-03-25 接受日期:2016-01-05 出版日期:2016-04-15 发布日期:2016-06-15
  • 基金资助:
    国家自然科学基金 (11101117)；河北省教育厅基金 (QN2014040)；河北经贸大学基金 (2015KYQ01).

Stability Analysis on a Ratio-dependent Predator-prey Model with Time Delay and Stage Structure

WANG Ling-shu1,  ZHANG Ya-nan1,  FENG Guang-hui2   

  1. 1- School of Mathematics and Statistics, Hebei University of Economics & Business, Shijiazhuang 050061
    2- Department of Basic Courses, Mechanical Engineering College, Shijiazhuang 050003
  • Received:2014-03-25 Accepted:2016-01-05 Online:2016-04-15 Published:2016-06-15
  • Supported by:
    The National Natural Science Foundation of China (11101117); the Scientific Research Foundation of Hebei Education Department (QN2014040); the Foundation of Hebei University of Economics & Business (2015KYQ01).

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摘要： 本文研究一个具有时滞和捕食者、食饵均具有阶段结构的比率依赖型捕食系统的稳定性．通过分析特征方程，运用Hurwitz判定定理，讨论了该系统的非负边界平衡点和正平衡点的局部稳定性，并得到了Hopf分支存在的充分条件；通过构造辅助系统，运用单调迭代方法和比较定理，讨论了该系统的非负边界平衡点和正平衡点的全局稳定性，从而得到了该生态系统灭绝与永久持续生存的充分条件．

关键词: 捕食系统, 阶段结构, 时滞, 稳定性, Hopf分支

Abstract:

In this paper, a ratio-dependent predator-prey model with time delay due to the gestation of the predator and stage structure for both the predator and the prey is investigated. By analyzing the characteristic equations and applying Hurwitz criterion, the local stability of a semi-trivial boundary equilibrium and a positive equilibrium are discussed, respectively. Moreover, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. By comparison arguments and iteration technique, the global stability of the semi-trivial boundary equilibrium and the positive equilibrium are addressed, respectively.

Key words: predator-prey model, stage structure, time delay, stability, Hopf bifurcation

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