状态反馈脉冲控制Holling-Tanner系统周期解的存在性*

数学物理学报 ›› 2018, Vol. 38 ›› Issue (1): 174-189.

状态反馈脉冲控制Holling-Tanner系统周期解的存在性^*

梁志清, 曾夏萍, 周泽文, 庞国萍, 黄军华

玉林师范学院数学与统计学院, 广西高校复杂系统优化与大数据处理重点实验室广西玉林 537000

收稿日期:2016-12-13 修回日期:2017-04-16 出版日期:2018-02-26 发布日期:2018-02-26
通讯作者: 梁志清 E-mail:lzqysl@sohu.com
基金资助:
国家自然科学基金（61364020）和玉林师范学院重点科研项目（2015YJZD02）

Existence of Periodic Solution of Holling-Tanner System with State Feedback Impulsive Control

Liang Zhiqing, Zeng Xiaping, Zhou Zewen, Pang Guoping, Huang Junhua

Guangxi Colleges and Universities Key Lab of Complex System Optimization and Large Data Processing, College of Mathematics and Statistics, Yulin Normal University, Guangxi Yulin 537000

Received:2016-12-13 Revised:2017-04-16 Online:2018-02-26 Published:2018-02-26
Supported by:
Supported by the NSFC (61364020) and the Major Research Programmes of Yulin Normal University (2015YJZD02)

摘要/Abstract

摘要： 该文研究具有状态反馈脉冲控制的Holling-Tanner系统.在连续系统有唯一极限环及正平衡点为不稳定的焦点的前提下，利用微分方程几何理论、后继函数及数学分析的方法，获得脉冲系统阶1周期解的存在性、唯一性及轨道稳定性的充分条件.利用数值模拟验证主要结论，并且数值结果显示对某些参数在连续系统的极限环内存在脉冲系统的阶k周期解.

关键词: Holling-Tanner系统, 极限环, 后继函数, 阶1周期解, 状态反馈脉冲控制

Abstract: In this paper, we investigate the Holling-Tanner model with state feedback impulsive control. On the premise that the continuous system has a unique limit cycle and the positive equilibrium point is an unstable focus point, by means of the geometry theory of semi-continuous dynamic system, successor function method and mathematical analysis method, we obtain sufficient conditions for the existence, uniqueness and orbital stability of order 1 periodic solution of the impulsive system. The main conclusions are verified by numerical simulation. Moreover, the numerical results show that the impulsive system has order k periodic solutions within the limit cycle for some parameters.

Key words: Holling-Tanner model, Limit cycle, Successor function, Order 1 periodic solution, State feedback impulsive control

中图分类号:

O175

梁志清, 曾夏萍, 周泽文, 庞国萍, 黄军华. 状态反馈脉冲控制Holling-Tanner系统周期解的存在性^*[J]. 数学物理学报, 2018, 38(1): 174-189.

Liang Zhiqing, Zeng Xiaping, Zhou Zewen, Pang Guoping, Huang Junhua. Existence of Periodic Solution of Holling-Tanner System with State Feedback Impulsive Control[J]. Acta mathematica scientia,Series A, 2018, 38(1): 174-189.

参考文献

[1] Leslie P H. Some further notes on the use of matrices in population mathematics. Biometrika, 1948, 35(3/4):213-245
[2] Leslie P H, Gower J C. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika, 1960, 47(3/4):219-231
[3] Hsu S B, Huang T W. Global stability for a class of predator-prey systems. SIAM J Appl Math, 1995, 55(3):763-783
[4] Hsu S B, Huang T W. Uniqueness of limit cycles for a predator-prey system of holling and leslie type. Canadian Appl Math Quaq, 1998, 6(2):91-117
[5] Feng P, Kang Y. Dynamics of a modified Leslie-Gower model with double Allee effects. Nonlinear Dyn, 2015, 80(1):1051-1062
[6] Yang R Z, Zhang C R. Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting. Nonlinear Dyn, 2017, 87(2):863-878
[7] Liang Z Q, Pan H W. Qualitative analysis of a ratio-dependent Holling-Tanner model. J Math Anal Appl, 2007, 334(2):954-964
[8] 梁志清, 陈兰荪. 离散Leslie捕食与被捕食系统周期解的稳定性. 数学物理学报, 2006, 26A(4):634-640Liang Z Q, Chen L S. Stability of periodic solution for a discrete Leslie predator-prey system. Acta Mathematica Scientia, 2006, 26A(4):634-640
[9] Holling C S. Functional response of predators to prey density and its role in mimicry and population regulation. Mem Ent Soc Can, 1965, 97(45):1-60
[10] Tanner J T. The stability and the intrinsic growth rates of prey and predator populations. Ecology, 1975, 56(4):855-867
[11] Murray J D. Mathematical Biology. Berlin:Springer-Verlag, 1989
[12] May R M. Stability and Complexity in Ecosystems. Princeton:Princeton University Press, 2001
[13] Gasull A, Kooij R E, Torregrosa J. Limit cycles in the Holling-Tanner model. Publ Mat, 1997, 41:149-167
[14] Saez E, Gonzalez-Olivares E. Dynamics of predator-prey model. Siam J Appl Math, 1999, 59(5):1867-1878
[15] Stern V M, Smith R F, Rosch V D R, Hagen K S. The integrated control concept. Hilgardia, 1959, 29:81-101
[16] Bainov D D, Simeonov P S. Impulsive Differential Equations:Periodic Solutions and Applications. England:Longman, 1993
[17] Bainov D D, Simeonov P S. Impulsive Differential Equations:Asymptotic Properties of the Solutions. Singapore:World Scientific, 1995
[18] Simenov P S, Bainov D D. Orbital stability of the periodic solutions of autonomous systems with impulse effect. Int J Systems Sci, 1989, 19(12):2561-2585
[19] Bonotto E M, Federson M. Limit sets and the Poincar\'e-Bendixson theorem in impulsive semi-dynamical systems. J Differential Equ, 2008, 244(9):2334-2349
[20] 陈兰荪. 害虫治理与半连续动力系统几何理论. 北华大学学报(自然科学版), 2011, 12(1):1-9 Chen L S. Pest control and geometric theory of semi-continuous dynamical system. J Beihua Univ (Nat Sci), 2011, 12(1):1-9
[21] 陈兰荪. 半连续动力系统理论及应用. 玉林师范学院学报(自然科学版), 2013, 34(2):1-10Chen L S. Theory and application of semi-continuous dynamical system. J Yulin Normal Univ (Nat Sci), 2013, 34(2):1-10
[22] Liang Z Q, Pang G P, Zen X P, Liang Y H. Qualitative analysis of a predator-prey system with mutual interference and impulsive state feedback control. Nonlinear Dyn, 2017, 87(3):1495-1509
[23] Pang G P, Chen L S. Periodic solution of the system with impulsive state feedback control. Nonlinear Dyn, 2014, 78(1):743-753 ewpage