On the continuity of the function describing the times of meeting impulsive set and its application

  • Received: 25 April 2016 Accepted: 19 September 2016 Published: 01 October 2017
  • MSC : Primary: 34A37; Secondary: 47N60

  • The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

    Citation: Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1399-1406. doi: 10.3934/mbe.2017072

    Related Papers:

  • The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.


    加载中
    [1] [ E. M. Bonotto,M. Federson, Limit sets and the Poincare Bendixson theorem in impulsive semidynamical systems, J. Differ. Equ., 244 (2008): 2334-2349.
    [2] [ K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bulletin of The Polish Academy of Sciences Mathematics, 52 (2004): 71-80.
    [3] [ G. B. Ermentrout,N. Kopell, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991): 195-217.
    [4] [ R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961): 445-466.
    [5] [ G. Gabor, The existence of viable trajectories in the state-dependent impusive systems, Nonlinear Anal. TMA, 72 (2010): 3828-3836.
    [6] [ G. Gabor, Viable periodic solutions in state-dependent impulsive problems, Collect. Math., 66 (2015): 351-365.
    [7] [ P. Goel,B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002): 191-216.
    [8] [ M. Z. Huang,J. X. Li,X. Y. Song,H. J. Guo, Modeling impulsive injections of insulin: Towards aritificial pancreas, SIAM J. Appl. Math., 72 (2012): 1524-1548.
    [9] [ S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990): 120-128.
    [10] [ J. H. Liang,S. Y. Tang,J. J. Nieto,R. A. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Math. Biosci., 245 (2013): 249-257.
    [11] [ B. Liu, Y. Tian and B. L. Kang, Dynamics on a Holling Ⅱ predator-prey model with state-dependent impulsive control, International J. Biomath. , 5 (2012), 1260006, 18 pp.
    [12] [ L. F. Nie,Z. D. Teng,L. Hu, The dynamics of a chemostat model with state dependent impulsive effects, Int. J. Bifurcat. Chaos, 21 (2011): 1311-1322.
    [13] [ J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment, Bull. Math. Biol., 58 (1996): 425-447.
    [14] [ B. Shulgin,L. Stone,Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60 (1998): 1123-1148.
    [15] [ L. Stone,B. Shulgin,Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000): 207-215.
    [16] [ K. B. Sun,Y. Tian,L. S. Chen,A. Kasperski, Nonlinear modelling of a synchronized chemostat with impulsive state feedback control, Math. Comput. Modelling, 52 (2010): 227-240.
    [17] [ S. Y. Tang,R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005): 257-292.
    [18] [ S. Y. Tang,R. A. Cheke, Models for integrated pest control and their biological implications, Math. Biosci., 215 (2008): 115-125.
    [19] [ S. Y. Tang,L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. B, 4 (2004): 759-768.
    [20] [ S. Y. Tang,J. H. Liang,Y. S. Tan,R. A. Cheke, Threshold conditions for interated pest management models with pesticides that have residual effects, J. Math. Biol., 66 (2013): 1-35.
    [21] [ S. Y. Tang, W. H. Pang, R. A. Cheke and J. H. Wu, Global dynamics of a state-dependent feedback control system, Advances in Difference Equations, 2015 (2015), 70pp.
    [22] [ S. Y. Tang,G. Y. Tang,R. A. Cheke, Optimum timing for integrated pest management: Modeling rates of pesticide application and natural enemy releases, J. Theor. Biol., 264 (2010): 623-638.
    [23] [ S. Y. Tang,B. Tang,A. L. Wang,Y. N. Xiao, Holling Ⅱ predator-prey impulsive semi-dynamic model with complex Poincare map, Nonlinear Dynamics, 81 (2015): 1575-1596.
    [24] [ S. Y. Tang,Y. N. Xiao,L. S. Chen,R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol., 67 (2005): 115-135.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2311) PDF downloads(526) Cited by(13)

Article outline

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog