\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents

Abstract Related Papers Cited by
  • This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence in a class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, generated by maps. Here a unified approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of $\mathbb{R}^m_+$ to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

    [2]

    P. Ashwin, J. Buescu and I. Stewart, From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity, 9 (1996), 703-737.doi: 10.1088/0951-7715/9/3/006.

    [3]

    C. Conley, "Isolated Invariant Sets and the Morse Index," CBMS Regional Conference Series in Mathematics, 38, Amer. Math. Soc., Providence, RI, 1978.

    [4]

    B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM J. Math. Anal., 34 (2003), 1007-1039.doi: 10.1137/S0036141001392815.

    [5]

    J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.doi: 10.1137/0520025.

    [6]

    M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynamics and Diff. Eqns., 13 (2001), 107-131.doi: 10.1023/A:1009044515567.

    [7]

    J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations, J. Diff. Eqns., 248 (2010), 1955-1971.doi: 10.1016/j.jde.2009.11.010.

    [8]

    E. O. Jones, A. White and M. Boots, Interference and the persistence of vertically transmitted parasites, J. Theor. Biol., 246 (2007), 10-17.doi: 10.1016/j.jtbi.2006.12.007.

    [9]

    A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

    [10]

    J. F. Reineck, Continuation to gradient flows, Duke Math. J., 64 (1991), 261-269.doi: 10.1215/S0012-7094-91-06413-6.

    [11]

    P. L. Salceanu, "Lyapunov Exponents and Persistence in Dynamical Systems with Applications to some Discrete Time Models," Ph.D thesis, Arizona State University, 2009.

    [12]

    P. L. Salceanu and H. L. Smith, Lyapunov exponents and persistence in discrete dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 187-203.doi: 10.3934/dcdsb.2009.12.187.

    [13]

    P. L. Salceanu and H. L. Smith, Lyapunov exponents and uniform weak normally repelling invariant sets, in "Positive Systems," Lecture Notes in Control and Inform. Sci., 389, Springer, Berlin, (2009), 17-27.

    [14]

    S. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations, 162 (2000), 400-426.doi: 10.1006/jdeq.1999.3719.

    [15]

    E. Seneta, "Non-negative Matrices. An Introduction to Theory and Applications," Halsted Press, New York, 1973.

    [16]

    H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995.

    [17]

    H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.

    [18]

    H. L. Smith and H. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, 118, Amer. Math. Soc., Providence, RI, 2011.

    [19]

    H. L. Smith, X.-Q. Zhao, Robust persistence for semi-dynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179.doi: 10.1016/S0362-546X(01)00678-2.

    [20]

    H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.doi: 10.1137/0524026.

    [21]

    X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(63) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return