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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 ›› 2015, Vol. 32 ›› Issue (1): 72-84.doi: 10.3969/j.issn.1005-3085.2015.01.008

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一个具有时滞和捕食者、食饵均具有阶段结构的捕食模型

王玲书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数学教研室，石家庄 050003
  • 收稿日期:2013-07-01 接受日期:2013-12-06 出版日期:2015-02-15 发布日期:2015-04-15
  • 基金资助:
    国家自然科学基金 (11101117)；河北省教育厅基金 (QN2014040).

A Predator-prey System with Time Delay and Stage Structure for the Predator and the Prey

WANG Ling-shu1,  FENG Guang-hui2   

  1. 1- School of Mathematics and Statistics, Hebei University of Economics & Business, Shijiazhuang 050061
    2- Department of Basic Courses, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003
  • Received:2013-07-01 Accepted:2013-12-06 Online:2015-02-15 Published:2015-04-15
  • Supported by:
    The National Natural Science Foundation of China (11101117); the Scientific Research Foundation of Hebei Education Department (QN2014040).

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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捕食模型的稳定性．首先，通过分析特征方程，运用Hurwitz判定定理，分别给出了该模型的边界平衡点和正平衡点局部稳定的充分条件，并得到了该模型在正平衡点存在Hopf分支的充分条件；其次，运用无穷维动力系统的一致生存定理，得到了该模型持续生存的充分条件；最后，通过构造适当的Lyapunov泛函，运用LaSall不变集原理，分别给出了该模型边界平衡点和正平衡点全局稳定的充分条件．

关键词: 捕食模型, 时滞, 阶段结构, 全局稳定

Abstract:

The stability of a predator-prey model with time delay and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, sufficient conditions are given respectively for the local stability of each of feasible equilibria of the system and the existence of a Hopf bifurcation at the positive equilibrium. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the positive equilibrium exists. By using the Lyapunov functions and the LaSalle invariant principle, sufficient conditions are derived respectively for the global stability of each of feasible equilibria of the model.

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

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