Special Issues

Instability and bifurcation of a cooperative system with periodic coefficients

  • Received: 01 November 2020 Revised: 01 January 2021 Published: 16 March 2021
  • Primary: 34C12, 34D99, 34A26, 34A40

  • In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter $ \lambda $ in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result.

    Citation: Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients[J]. Electronic Research Archive, 2021, 29(5): 3069-3079. doi: 10.3934/era.2021026

    Related Papers:

  • In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter $ \lambda $ in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result.



    加载中


    [1] On the stability of planar randomly switched systems. Ann. Appl. Probab (2014) 24: 292-311.
    [2] The ecology of mutualism. Annual Review of Ecology and Systematics (1982) 13: 315-347.
    [3] Perron-Frobenius Theorem for Nonnegative Tensors. Commun. Math. Sci. (2008) 6: 507-520.
    [4] C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Springer-Verlag, New York, 2006.
    [5] Stability in models of mutualism. Amer. Natur. (1979) 113: 261-275.
    [6] On the stability of switched positive linear systems. IEEE Trans. Automat. Control (2007) 52: 1099-1103.
    [7] M. W. Hirsch and H. L. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, 239–357, Elsevier B. V., Amsterdam, (2005).
    [8] Unstable solutions of nonautonomous linear differential equations. SIAM Review (2008) 50: 570-584.
    [9] Does dormancy increase fitness of bacterial populations in time varying environments?. Bull. Math. Biol. (2008) 70: 1140-1162.
    [10] R. M. May and A. R. McLean, Theoretical Ecology: Principles and Applications, 3nd edition, Oxford, UK, 2007.
    [11] Instability in linear cooperative systems of ordinary differential equations. SIAM Review (2017) 59: 649-670.
    [12] J. D. Murray, Mathematical Biology I: An Introduction, 3nd edition, Springer-Verlag, New York, 2002.
    [13] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995.
    [14] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003.
  • Reader Comments
  • © 2021 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(960) PDF downloads(210) Cited by(0)

Article outline

Figures and Tables

Figures(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog