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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 
References:
[1] 
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, SpringerVerlag, Berlin, 1998. 
[2] 
P. Ashwin, J. Buescu and I. Stewart, From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity, 9 (1996), 703737. doi: 10.1088/09517715/9/3/006. 
[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), 10071039. doi: 10.1137/S0036141001392815. 
[5] 
J. K. Hale and P. Waltman, Persistence in infinitedimensional systems, SIAM J. Math. Anal., 20 (1989), 388395. 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), 107131. doi: 10.1023/A:1009044515567. 
[7] 
J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations, J. Diff. Eqns., 248 (2010), 19551971. 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), 1017. 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), 261269. doi: 10.1215/S0012709491064136. 
[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), 187203. 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), 1727. 
[14] 
S. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations, 162 (2000), 400426. doi: 10.1006/jdeq.1999.3719. 
[15] 
E. Seneta, "Nonnegative 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 semidynamical systems, Nonlinear Analysis, 47 (2001), 61696179. doi: 10.1016/S0362546X(01)006782. 
[20] 
H. R. Thieme, Persistence under relaxed pointdissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407435. doi: 10.1137/0524026. 
[21] 
X.Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, SpringerVerlag, New York, 2003. 
show all references
References:
[1] 
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, SpringerVerlag, Berlin, 1998. 
[2] 
P. Ashwin, J. Buescu and I. Stewart, From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity, 9 (1996), 703737. doi: 10.1088/09517715/9/3/006. 
[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), 10071039. doi: 10.1137/S0036141001392815. 
[5] 
J. K. Hale and P. Waltman, Persistence in infinitedimensional systems, SIAM J. Math. Anal., 20 (1989), 388395. 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), 107131. doi: 10.1023/A:1009044515567. 
[7] 
J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations, J. Diff. Eqns., 248 (2010), 19551971. 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), 1017. 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), 261269. doi: 10.1215/S0012709491064136. 
[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), 187203. 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), 1727. 
[14] 
S. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations, 162 (2000), 400426. doi: 10.1006/jdeq.1999.3719. 
[15] 
E. Seneta, "Nonnegative 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 semidynamical systems, Nonlinear Analysis, 47 (2001), 61696179. doi: 10.1016/S0362546X(01)006782. 
[20] 
H. R. Thieme, Persistence under relaxed pointdissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407435. doi: 10.1137/0524026. 
[21] 
X.Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, SpringerVerlag, New York, 2003. 
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