中国物理B ›› 2013, Vol. 22 ›› Issue (3): 30204-030204.doi: 10.1088/1674-1056/22/3/030204

• GENERAL • 上一篇    下一篇

Stability and Neimark–Sacker bifurcation analysis of food-limited population model with time delay

姜晓伟a, 关治洪a, 张先鹤b, 张顶学c, 刘峰d   

  1. a Department of Control Sclience and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
    b College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi 435002, China;
    c Petroleum Engineering College, Yangtze University, Jingzhou 434023, China;
    d Department of Electronic Information and Mechanics, China University of Geosciences, Wuhan 430074, China
  • 收稿日期:2012-06-11 修回日期:2012-08-15 出版日期:2013-02-01 发布日期:2013-02-01
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 60973012, 61073025, 61073026, 61170031, and 61100076).

Stability and Neimark-Sacker bifurcation analysis of food-limited population model with time delay

Jiang Xiao-Wei (姜晓伟)a, Guan Zhi-Hong (关治洪)a, Zhang Xian-He (张先鹤)b, Zhang Ding-Xue (张顶学)c, Liu Feng (刘峰)d   

  1. a Department of Control Sclience and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
    b College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi 435002, China;
    c Petroleum Engineering College, Yangtze University, Jingzhou 434023, China;
    d Department of Electronic Information and Mechanics, China University of Geosciences, Wuhan 430074, China
  • Received:2012-06-11 Revised:2012-08-15 Online:2013-02-01 Published:2013-02-01
  • Contact: Guan Zhi-Hong E-mail:zhguan@mail.hust.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 60973012, 61073025, 61073026, 61170031, and 61100076).

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摘要： In this paper, a kind of discrete delay food-limited model obtained by Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, the linear stability of this model is studied. It is shown that Neimark-Sacker bifurcation occurs when τ crosses some critical values. The explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form. Finally, numerical simulations are performed to verify the analytical results.

关键词: food-limited model, time delay, Neimark-Sacker bifurcation, periodic solution

Abstract: In this paper, a kind of discrete delay food-limited model obtained by Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, the linear stability of this model is studied. It is shown that Neimark–Sacker bifurcation occurs when τ crosses some critical values. The explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form. Finally, numerical simulations are performed to verify the analytical results.

Key words: food-limited model, time delay, Neimark-Sacker bifurcation, periodic solution

中图分类号:  (Delay and functional equations)

  • 02.30.Ks
02.30.Oz (Bifurcation theory)