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Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

Author

Listed:
  • Zuxiong Li

    (College of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404120, China)

  • Shengnan Fu

    (Guangzhou Huangpu District (Development Zone) Taxation Bureau, Guangdong Provincial Tax Service, State Taxation Administration, Guangzhou 510760, China
    School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China)

  • Huili Xiang

    (School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China)

  • Hailing Wang

    (School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China)

Abstract

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A 1 of system is globally asymptotically stable when b r − c > 0 and η < 0 . Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when b r − c > 0 and η > 0 . By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Suggested Citation

  • Zuxiong Li & Shengnan Fu & Huili Xiang & Hailing Wang, 2021. "Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest," Mathematics, MDPI, vol. 9(20), pages 1-26, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2560-:d:655074
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    References listed on IDEAS

    as
    1. Xiangrui Li & Shuibo Huang, 2019. "Stability and Bifurcation for a Single-Species Model with Delay Weak Kernel and Constant Rate Harvesting," Complexity, Hindawi, vol. 2019, pages 1-17, December.
    2. Dou, Jiawei & Li, Shundong, 2017. "Optimal impulsive harvesting policies for single-species populations," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 145-155.
    3. Neves, Luis R.T. & Maia, Leonardo Paulo, 2021. "A simple individual-based population growth model with limited resources," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    4. Yao, Zichen & Yang, Zhanwen & Zhang, Yusong, 2021. "A stability criterion for fractional-order complex-valued differential equations with distributed delays," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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