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Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

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  • Rao, Ruofeng
  • Yang, Xinsong
  • Tang, Rongqiang
  • Zhang, Yulin
  • Li, Xinggui
  • Shi, Lei

Abstract

Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

Suggested Citation

  • Rao, Ruofeng & Yang, Xinsong & Tang, Rongqiang & Zhang, Yulin & Li, Xinggui & Shi, Lei, 2021. "Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 571-590.
    • Handle: RePEc:eee:matcom:v:188:y:2021:i:c:p:571-590
    DOI: 10.1016/j.matcom.2021.04.025
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    References listed on IDEAS

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    1. Zhang, Lan & Yang, Xinsong & Xu, Chen & Feng, Jianwen, 2017. "Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 22-30.
    2. Moore, Christopher M. & Catella, Samantha A. & Abbott, Karen C., 2018. "Population dynamics of mutualism and intraspecific density dependence: How θ-logistic density dependence affects mutualistic positive feedback," Ecological Modelling, Elsevier, vol. 368(C), pages 191-197.
    3. Ruofeng Rao & Quanxin Zhu & Jialin Huang & Eulalia Mart nez, 2021. "Existence, Uniqueness, and Input-to-State Stability of Ground State Stationary Strong Solution of a Single-Species Model via Mountain Pass Lemma," Complexity, Hindawi, vol. 2021, pages 1-11, April.
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    Cited by:

    1. Ruofeng Rao & Jialin Huang & Xinsong Yang, 2021. "Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup," Mathematics, MDPI, vol. 9(19), pages 1-11, October.
    2. Ruofeng Rao, 2021. "Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle," Mathematics, MDPI, vol. 9(16), pages 1-11, August.

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