2017 Volume 7 Issue 2
Article Contents
Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(2): 439-454
Next Article Previous Article

Xiaowei Tang, Xilin Fu. ON PERIOD-K SOLUTION FOR A POPULATION SYSTEM WITH STATE-DEPENDENT IMPULSIVE EFFECT[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 439-454. doi: 10.11948/2017028
Citation: Xiaowei Tang, Xilin Fu. ON PERIOD-K SOLUTION FOR A POPULATION SYSTEM WITH STATE-DEPENDENT IMPULSIVE EFFECT[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 439-454. doi: 10.11948/2017028
shu

ON PERIOD-K SOLUTION FOR A POPULATION SYSTEM WITH STATE-DEPENDENT IMPULSIVE EFFECT

  • 1 School of Mathematical Sciences, Shandong Normal University, Ji'nan, 250014, China;

  • 2 Mathematical School, Qilu Normal University, Jinan, 250013, China

  • Fund Project:
    Abstract
  • The period-k solutions of population differential system with statedependent impulsive effect are investigated by the theory of discontinuous dynamical system. Through G-function theory, the necessary and sufficient conditions are obtained for trajectory direction of a population differential system, and the results are better than the previous work. Also, the local stability of period-k solutions is studied by the mapping structure and the theory of eigenvalue analysis. Furthermore, the existence of period-1 solution is investigated for a special impulsive population differential system, and the analytical condition is established. Finally, the trajectory of period-1 solution and the relationship between different parameters and the module of eigenvalues are illustrated.
    MSC: 34K13;49N25
    • FullText(HTML)
  • 加载中
    • References
  • [1] E. Angelis, Optimal semi-active control and non-linear dynamics response of variable stiffness structures, Journal of Vibration and Control, 2015, 11(10), 1253-1289.

    Google Scholar

    [2] H. Baek, An impulsive two-prey one-predator system with seasonal effects, Discrete Dynamics in Nature and Society, 2009, 57(2), 332-337.

    Google Scholar

    [3] H. Baek, Qualitative analysis of Beddington-DeAngelis type impulsive predatorprey models, Nonlinear Analysis, 2010, 11(1), 1312-1322.

    Google Scholar

    [4] L. Dong, L. Chen and L. Sun, Extinction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively, Mathematical Methods in the Applied Sciences, 2006, 29(4), 415-425.

    Google Scholar

    [5] A. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans., 1964, 42(2), 199-231.

    Google Scholar

    [6] X. Fu, J. Qi and Y. Liu, The existence of periodic orbits for nonlinear impulsive differential systems, Communications in Nonlinear Science and Numerical Simulation, 1999, 4(1), 50-53.

    Google Scholar

    [7] X. Fu, B. Yan and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005.

    Google Scholar

    [8] X. Fu, B. Yan and Y. Liu, Nonlinear Impulsive Differential Systems, Science Press, Beijing, 2008.

    Google Scholar

    [9] X. Fu and S. Zheng, New approach in dynamics of regenerative chatter research of turning, Communications in Nonlinear Science and Numerical Simulation, 2014, 19(11), 4013-4023.

    Google Scholar

    [10] G. Grzegorz, Viabe periodic solutions in state-dependent impulsive problems, Collect. Math., 2015, 66(3), 1-15.

    Google Scholar

    [11] P. Georgescua and G. Moroanub, Impulsive perturbations of a three-trophic prey-dependent food chain system, Mathematical and Computer Modelling, 2008, 48(7), 975-997.

    Google Scholar

    [12] M. He, F. Chen and Z. Li, Almost periodic solution of an impulsive differential equation model of plankton allelopathy, Nonlinear Analysis, 2010, 11(4), 2296-2301.

    Google Scholar

    [13] M. Huang and X. Song, Study on species cooperative systems with impulsive state feedback control, Journal of System Science and Mathematical Science, 2012, 32(3), 265-276.

    Google Scholar

    [14] G. Jiang, Complex dynamics of a Holling type Ⅱ prey-predator system with state feedback control, Chaos, Solutions and Fractals, 2007, 31(2), 448-461.

    Google Scholar

    [15] G. Jiang and Q. Lu, Impulsive state feedback control of predator-prey model, Journal of Computational and Applied Mathematics, 2007, 200(1), 193-207.

    Google Scholar

    [16] G. Jiang, Q. Lu and G. Luo, Impulsive control of a stage-structured pest management system, Journal of Mathematical Study, 2003, 36(4), 331-344.

    Google Scholar

    [17] A. Luo, Discontinuous Dynamical Systems, Higher Education Press, Beijing, 2012.

    Google Scholar

    [18] A. Luo and G. Yu, A semi-analytical periodic motion in Duffing oscillator through mapping structures, Discontinuity, Nonlinear and Complexity, 2015, 4(2), 121-150.

    Google Scholar

    [19] Z. Liu, H. Chen and T. Zhou, Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem, Computers and Mathematics with Applications, 2011, 61(6), 1687-1699.

    Google Scholar

    [20] Z. Luo, B. Dai and Q. Zhang, Existence of positive periodic solutions for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays, Applied Mathematics and Computation, 2010, 215(9), 3390-3398.

    Google Scholar

    [21] X. Liu and K. Rohlf, Impulsive control of a Lotka-Volterra system, Ima Journal of Mathematical Control and Information, 1998, 15(3), 269-284.

    Google Scholar

    [22] B. Liu and L. Zhang, Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input, Applied Mathematics and Computation, 2009, 214(1), 155-162.

    Google Scholar

    [23] J. Liang, S. Tang, J. Nieto and R. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Mathemaical Biosciences, 2013, 245(2), 249-257.

    Google Scholar

    [24] J. Nieto, R. Rodrguez-Lpez and M. Pesqueira, Greens function for the periodic boundary value problem related to a first-order impulsive differential equation and applications to functional Problems, Differential Equations and Dynamical Systems, 2011, 19(3), 199-210.

    Google Scholar

    [25] L. Nie, Z. Teng, J. Nieto and I. Jung, State impulsive control strategies for a two-languages competitive model with bilingualism and interlinguistic similarity, Physica A, 2015, 430, 136-147.

    Google Scholar

    [26] J. Qi and X. Fu, Existence of limit cycles of impulsive differential equations with impulses at variable times, Nonlinear Analysis, 2001, 44(3), 345-353.

    Google Scholar

    [27] B. Shen and X. Wang, Ninlinear state-dependent impulsive system in fed-batch culture and its optimal control, Numerical Algebra, Control and Optimization, 2015, 5(4), 369-380.

    Google Scholar

    [28] A. Terry, Impulsive culling of a structured population on two patches, Journal of Mathematical Biology, 2010, 61(6), 843-875.

    Google Scholar

    [29] S. Tang and R. Cheke, State-dependent impulsive models of integrated pest management strategies and their dynamic consequences, Journal of Mathematical Biology, 2005, 50(3), 257-292.

    Google Scholar

    [30] Utkin, Variable structure systems with sliding modes, IEEE Trans Automat control, 1977, 22(2), 212-222.

    Google Scholar

    [31] L. Wang, L. Chen and J. Nieto, The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Analysis, 2010, 11(3), 1374-1386.

    Google Scholar

    [32] Q. Xiao and B. Dai, Dynamics of an impulsive predator -prey logistic population model with state-dependent, Applied Mathematics and Computation, 2015, 259(5), 220-230.

    Google Scholar

    [33] L. Zhao, Impulsive control of a Lotka-Volterra predator-prey system, Journal of Biomathematics, 2002, 17(1), 38-47.

    Google Scholar

    [34] G. Zeng, L. Chen and L. Sun, Existence of periodic solution of order one of planar impulsive autonomous, Journal and Computational and Applied Mathematics, 2006, 186(2), 466-481.

    Google Scholar

    [35] J. Zhao, X.Guo, Z. Han and Z. Chen, Average conditions for competitive system in a nonautonomous two dimensional Lotka-Volterra system, Mathematical and computer modeling, 2013, 57(5), 1131-1138.

    Google Scholar

    [36] X. Zhang, J. Yan and A. Zhao, Existence of positive periodic solutions for an impulsive differential equation, Nonlinear Analysis, 2008, 68(10), 3209-3216.

    Google Scholar

    • Supplements
    • Cited by
Export Citation
PDF
XML

Article Metrics

Article views(2225) PDF downloads(1175) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Published by Wilmington Scientific Publisher