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2011, 8(3): 807-825. doi: 10.3934/mbe.2011.8.807

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

1. 

Mathematics Department, University of Louisiana at Lafayette, Lafayette, LA 70504, United States

Received  October 2010 Revised  March 2011 Published  June 2011

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.
Citation: Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807
References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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