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Global attractivity of multiple species competition Lotka-Volterra system with toxicants effect and feedback controls

  • 1.

    Minnan Science and Technology Institute, Fujian Normal University, Quanzhou 362332, China

  • 2.

    Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2016.08.003
  • Received Date: 15 October 2015
  • Accepted Date: 28 May 2016
  • Rev Recd Date: 28 May 2016
  • Publish Date: 30 August 2016
    Abstract

  • Abstract

    By means of species dynamic theory, a nonautonomous Lotka-Volterra multiple species competition system with toxicants effect and feedback controls was established, and the high order nonlinear function was used in the construction of the feedback control variables. By means of Continuation Theorem based on Gaines and Mawhin coincidence degree theory, Barbalat Lemma and constructing an appropriate Lyapunov function, the sufficient conditions for the uniqueness and global attractivity of positive periodic solutions of the system were obtained.

    Abstract

    By means of species dynamic theory, a nonautonomous Lotka-Volterra multiple species competition system with toxicants effect and feedback controls was established, and the high order nonlinear function was used in the construction of the feedback control variables. By means of Continuation Theorem based on Gaines and Mawhin coincidence degree theory, Barbalat Lemma and constructing an appropriate Lyapunov function, the sufficient conditions for the uniqueness and global attractivity of positive periodic solutions of the system were obtained.
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    • References(16)
  • [1]
    CUI Jingan. Permanence and periodic solution of Lotka-Volterra system with time delay[J]. Acta Mathmatica Sinia, 2000, 47(1):511-519.
    崔景安.时滞Lotka-Volterra系统的持久性和周期解[J]. 数学学报, 2000, 47(1):511-519.
    [2]
    HUI Jing, CHEN Lansun. Periodity and stability in impulsive equation with delay[J]. Acta Mathmatica Sinia, 2005, 48(6):1 137-1 144.
    惠静，陈兰荪. 脉冲时滞微分方程的周期性和稳定性[J]. 数学学报，2005，48(6): 1 137-1 144.
    [3]
    CHEN Fengde, SHI Jinlin, CHEN Xiaoxing. Periodicity in a Lotka-Volterra facultative mutualism system with several delays[J]. Chinese Journal of Engineering Mathematics, 2004, 21(3): 403-409.
    [4]
    CHENG Yongkuan, DOU Dou.On almost automorphic solutions of the competing species problems[J].Journal of University of Science and Technology of China, 2006，36（9）：956-959.
    程永宽，窦斗.竞争系统的几乎自守解[J].中国科学技术大学学报，2006，36（9）：956-959.
    [5]
    ZHAO Ming, CHENG Rongfu. Existence of periodic solution of a food chain system with biocontrol and ratio functional response[J]. Journal of Jilin University (Science Edition), 2009, 47(4): 730-736.
    赵明,程荣福. 一类具生物控制和比率型功能反应的食物链系统周期解的存在性[J]. 吉林大学学报(理学版), 2009, 47(4): 730-736.
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    SHAO Yuanfu, DAI Binxiang. Multiple positive periodic solutions of a delayed ratio-dependent predator-prey model with functional response and impulse[J]. Mathem Atica Applicata, 2011, 24(1): 30-39.
    邵远夫,戴斌祥. 一类含功能反应与脉冲的时滞比率依赖捕食者捕食者-食饵模型的多重正周期解[J]. 应用数学,2011, 24(1): 30-39.
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    LIU Kaili, DOU Jiawei. The periodic solutions and globally asymptotic properties of L-V system with impulsive effects[J]. Journal of Xian University of Technology, 2012，28(2): 235-239.
    刘凯丽,窦家维.一类脉冲L-V系统的周期解和全局渐近性质[J]. 西安理工大学学报,2012，28(2): 235-239.
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    WEI F Y, WANG S H. Positive periodic solutions of nonautonomous competitive systems with infinite delay and diffusion [J]. Journal of Biomathematics, 2012, 27(2): 193-202.
    [9]
    GOPALSAMY K, Weng P X. Feedback regulation of logistic growth[J]. Internat Math and Math Sci, 1993, 16: 177-192.
    [10]
    HUANG Zhenkun, CHEN Fengde. Almost periodic solution in two species competitive system with feedback controls [J]. Journal of Biomathematics, 2005, 20(1): 28-32.
    黄振坤,陈凤德. 具有反馈控制的两种群竞争系统的概周期解[J]. 生物数学学报, 2005, 20(1): 28-32.
    [11]
    DING Xiaoquan, CHENG Shuhan. The stability of delayed stage-structured population growth model with feedback controls[J]. Journal of Biomathematics, 2006, 21(2): 225-232.
    丁孝全,程述汉.具反馈控制的时滞阶段结构种群模型[J]. 生物数学学报, 2006, 21(2): 225-232.
    [12]
    LI Hui, WANG Yifei. Stability of a delayed stage structured population growth model with feedback controls [J]. Journal of Beihua University (Natural Science), 2008, 9(5): 391-395.
    李辉,王艺霏.一个具反馈控制的时滞阶段结构种群模型的稳定性[J]. 北华大学学报(自然科学版), 2008，9(5): 391-395.
    [13]
    CHEN Fengde, YUAN Yuqing, WU Yumen. Research progress of single population model with feedback control [J]. Journal of Fuzhou University (Natural Science), 2011, 39(5): 617-621.
    陈凤德,阮育清,吴玉敏.具反馈控制的单种群模型研究进展[J].福州大学学报(自然科学版), 2011,39(5): 617-621.
    [14]
    GUI Zhanji. Biodynamics Model and Computer Simulation[M]. Beijing: Science Press,2005.
    桂占吉.生物动力学模型与计算机仿真[M]. 北京：科学出版社，2005.
    [15]
    GAINES R E, MAWHIN J L. Coincidence Degree and Nonlinear Differential Equations[M]. Berlin: Springer-Verlag, 1977: 40-45.
    [16]
    BARBALAT I. System dequation differentilles doscilltion nonlinears [J].Rev Math Pure and Appl, 1959, 4:267-270.
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    [1]
    CUI Jingan. Permanence and periodic solution of Lotka-Volterra system with time delay[J]. Acta Mathmatica Sinia, 2000, 47(1):511-519.
    崔景安.时滞Lotka-Volterra系统的持久性和周期解[J]. 数学学报, 2000, 47(1):511-519.
    [2]
    HUI Jing, CHEN Lansun. Periodity and stability in impulsive equation with delay[J]. Acta Mathmatica Sinia, 2005, 48(6):1 137-1 144.
    惠静，陈兰荪. 脉冲时滞微分方程的周期性和稳定性[J]. 数学学报，2005，48(6): 1 137-1 144.
    [3]
    CHEN Fengde, SHI Jinlin, CHEN Xiaoxing. Periodicity in a Lotka-Volterra facultative mutualism system with several delays[J]. Chinese Journal of Engineering Mathematics, 2004, 21(3): 403-409.
    [4]
    CHENG Yongkuan, DOU Dou.On almost automorphic solutions of the competing species problems[J].Journal of University of Science and Technology of China, 2006，36（9）：956-959.
    程永宽，窦斗.竞争系统的几乎自守解[J].中国科学技术大学学报，2006，36（9）：956-959.
    [5]
    ZHAO Ming, CHENG Rongfu. Existence of periodic solution of a food chain system with biocontrol and ratio functional response[J]. Journal of Jilin University (Science Edition), 2009, 47(4): 730-736.
    赵明,程荣福. 一类具生物控制和比率型功能反应的食物链系统周期解的存在性[J]. 吉林大学学报(理学版), 2009, 47(4): 730-736.
    [6]
    SHAO Yuanfu, DAI Binxiang. Multiple positive periodic solutions of a delayed ratio-dependent predator-prey model with functional response and impulse[J]. Mathem Atica Applicata, 2011, 24(1): 30-39.
    邵远夫,戴斌祥. 一类含功能反应与脉冲的时滞比率依赖捕食者捕食者-食饵模型的多重正周期解[J]. 应用数学,2011, 24(1): 30-39.
    [7]
    LIU Kaili, DOU Jiawei. The periodic solutions and globally asymptotic properties of L-V system with impulsive effects[J]. Journal of Xian University of Technology, 2012，28(2): 235-239.
    刘凯丽,窦家维.一类脉冲L-V系统的周期解和全局渐近性质[J]. 西安理工大学学报,2012，28(2): 235-239.
    [8]
    WEI F Y, WANG S H. Positive periodic solutions of nonautonomous competitive systems with infinite delay and diffusion [J]. Journal of Biomathematics, 2012, 27(2): 193-202.
    [9]
    GOPALSAMY K, Weng P X. Feedback regulation of logistic growth[J]. Internat Math and Math Sci, 1993, 16: 177-192.
    [10]
    HUANG Zhenkun, CHEN Fengde. Almost periodic solution in two species competitive system with feedback controls [J]. Journal of Biomathematics, 2005, 20(1): 28-32.
    黄振坤,陈凤德. 具有反馈控制的两种群竞争系统的概周期解[J]. 生物数学学报, 2005, 20(1): 28-32.
    [11]
    DING Xiaoquan, CHENG Shuhan. The stability of delayed stage-structured population growth model with feedback controls[J]. Journal of Biomathematics, 2006, 21(2): 225-232.
    丁孝全,程述汉.具反馈控制的时滞阶段结构种群模型[J]. 生物数学学报, 2006, 21(2): 225-232.
    [12]
    LI Hui, WANG Yifei. Stability of a delayed stage structured population growth model with feedback controls [J]. Journal of Beihua University (Natural Science), 2008, 9(5): 391-395.
    李辉,王艺霏.一个具反馈控制的时滞阶段结构种群模型的稳定性[J]. 北华大学学报(自然科学版), 2008，9(5): 391-395.
    [13]
    CHEN Fengde, YUAN Yuqing, WU Yumen. Research progress of single population model with feedback control [J]. Journal of Fuzhou University (Natural Science), 2011, 39(5): 617-621.
    陈凤德,阮育清,吴玉敏.具反馈控制的单种群模型研究进展[J].福州大学学报(自然科学版), 2011,39(5): 617-621.
    [14]
    GUI Zhanji. Biodynamics Model and Computer Simulation[M]. Beijing: Science Press,2005.
    桂占吉.生物动力学模型与计算机仿真[M]. 北京：科学出版社，2005.
    [15]
    GAINES R E, MAWHIN J L. Coincidence Degree and Nonlinear Differential Equations[M]. Berlin: Springer-Verlag, 1977: 40-45.
    [16]
    BARBALAT I. System dequation differentilles doscilltion nonlinears [J].Rev Math Pure and Appl, 1959, 4:267-270.
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