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Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control

Received: 15 September 2016     Accepted: 27 September 2016     Published: 19 October 2016
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Abstract

By constructing suitable Lyapunov function and developing some new analysis techniques, a Lotka-Volterra predator-prey system with ratio-dependent functional responses and feedback controls is studied and a sufficient condition which guarantees the globally attractive of positive solution for the predator-prey model is obtained. Moreover, the numerical simulation to the system is given to illustrate our results.

Published in Advances in Bioscience and Bioengineering (Volume 4, Issue 5)
DOI 10.11648/j.abb.20160405.13
Page(s) 59-66
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Predator-Prey System, Feedback Control, Ratio-Dependent, Globally Attractive

References
[1] F. Q. Yin, Y. K. Li, Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Applied Mathematics and Computation 153 (2) (2004) 475-484.
[2] F. Chen, Global stability of a single species model with feedback control and distributed time delay, Applied Mathematics and Computation 178 (2006) 474-479.
[3] L. Nie, Z. Teng, L. Hu, J. Peng, Permanence and stability in non-autonomous predator-prey Lotka-Volterra systems with feedback controls, Computers & Mathematics with Applications 58 (2009) 436-448.
[4] Y. Li, T. Zhang, Permanence of a discrete n-species cooperation system with time-varying delays and feedback controls, Mathematical and Computer Modelling 53 (2011) 1320-1330.
[5] Y. Fan, L. Wang, Global asymptotical stability of a Logistic model with feedback control, Nonlinear Anal. Real World Applications 11 (2010) 2686-2697.
[6] K. Gopalsamy, P. Weng, Global attractivity in a competition system with feedback controls, Computers & Mathematics with Applications 45 (2003) 665-676.
[7] F. Chen, The permanence and global attractivity of Lotka–Volterra competition system with feedback controls, Nonlinear Anal: Real World Applications 7 (2006) 133-143.
[8] J. Li, A. Zhao, J. Yan, The permanence and global attractivity of a Kolmogorov system with feedback controls, Nonlinear Anal. Real World Applications 10 (2009) 506-518.
[9] Z. Yang, Positive periodic solutions of a class of single species neutral models with state dependent delay and feedback control, European Journal of Applied Mathematics 17 (2006) 735-757.
[10] B. Liu, Z. Teng, L. Chen, Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy, Journal of Computational and Applied Mathematics 193 (2006) 347-362.
[11] Y. Lv, R. Yuan, Y. Pei, Two types of predator-prey models with harvesting: non-smooth and non-continuous, J. Comput. Appl. Math. 250 (2013)122-142.
[12] Y. Li, D. Xie, J. Cui, Complex dynamics of a predator-prey model with impulsive state feedback control, Applied Mathematics and Computation 230 (2014) 395-405.
[13] T. Zhang, W. Ma, X. Meng, T. Zhang, Periodic solution of a prey-predator model with nonlinear state feedback control, Applied Mathematics and Computation 266 (2015) 95–107.
[14] Jin Yang, Sanyi Tang, Holling type II predator-prey model with nonlinear pulse as state-dependent feedback control, Journal of Computational and Applied Mathematics 291 (2016)225-241.
[15] C. Y. Wang, X. W. Li, H. Yuan, The permanence of a ratio-dependent Lotka-Volterra predator-prey model with feedback control, Advanced Materials Research 765-767 (2013) 2144-2147.
[16] J. Wang, K. Wang, Dynamics of a ratio-dependent one predator-two competing prey model, Mathematica Applicata 17 (2004) 172-178. (In Chinese).
[17] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs: Prentice-Hall, 2002.
Cite This Article
  • APA Style

    Changyou Wang, Hao Liu, Shuang Pan, Xiaolin Su, Rui Li. (2016). Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control. Advances in Bioscience and Bioengineering, 4(5), 59-66. https://doi.org/10.11648/j.abb.20160405.13

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    ACS Style

    Changyou Wang; Hao Liu; Shuang Pan; Xiaolin Su; Rui Li. Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control. Adv. BioSci. Bioeng. 2016, 4(5), 59-66. doi: 10.11648/j.abb.20160405.13

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    AMA Style

    Changyou Wang, Hao Liu, Shuang Pan, Xiaolin Su, Rui Li. Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control. Adv BioSci Bioeng. 2016;4(5):59-66. doi: 10.11648/j.abb.20160405.13

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  • @article{10.11648/j.abb.20160405.13,
      author = {Changyou Wang and Hao Liu and Shuang Pan and Xiaolin Su and Rui Li},
      title = {Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control},
      journal = {Advances in Bioscience and Bioengineering},
      volume = {4},
      number = {5},
      pages = {59-66},
      doi = {10.11648/j.abb.20160405.13},
      url = {https://doi.org/10.11648/j.abb.20160405.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.abb.20160405.13},
      abstract = {By constructing suitable Lyapunov function and developing some new analysis techniques, a Lotka-Volterra predator-prey system with ratio-dependent functional responses and feedback controls is studied and a sufficient condition which guarantees the globally attractive of positive solution for the predator-prey model is obtained. Moreover, the numerical simulation to the system is given to illustrate our results.},
     year = {2016}
    }
    

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    T1  - Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control
    AU  - Changyou Wang
    AU  - Hao Liu
    AU  - Shuang Pan
    AU  - Xiaolin Su
    AU  - Rui Li
    Y1  - 2016/10/19
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    T2  - Advances in Bioscience and Bioengineering
    JF  - Advances in Bioscience and Bioengineering
    JO  - Advances in Bioscience and Bioengineering
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    PB  - Science Publishing Group
    SN  - 2330-4162
    UR  - https://doi.org/10.11648/j.abb.20160405.13
    AB  - By constructing suitable Lyapunov function and developing some new analysis techniques, a Lotka-Volterra predator-prey system with ratio-dependent functional responses and feedback controls is studied and a sufficient condition which guarantees the globally attractive of positive solution for the predator-prey model is obtained. Moreover, the numerical simulation to the system is given to illustrate our results.
    VL  - 4
    IS  - 5
    ER  - 

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Author Information
  • Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing, P.R. China

  • Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing, P.R. China

  • Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing, P.R. China

  • College of Science, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China

  • Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing, P.R. China

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