doi: 10.3934/dcdsb.2021258
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Robust uniform persistence for structured models of delay differential equations

University of Louisiana at Lafayette, Lafayette, LA 70503, USA

Received  April 2021 Revised  August 2021 Early access November 2021

Fund Project: The author is supported by a Simons Foundation Collaboration Grant for Mathematicians. Award ID: 524761

We consider a general class of delay differential equations systems, typically used to model the dynamics of structured biological populations, and establish necessary conditions for the part of the attractor contained in the boundary of the state space to repel the complementary dynamics contained in the interior of the state space. The conditions are formulated in terms of Lyapunov exponents and invariant probability measures and we use them to prove a robust uniform persistence result.

Citation: Paul L. Salceanu. Robust uniform persistence for structured models of delay differential equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021258
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Sringer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[3]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

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

[5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[6]

J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.  doi: 10.1214/aop/1176996798.  Google Scholar

[7]

J. MierczyńskiW. Shen and X.-Q. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems, J. Differ. Equations, 204 (2004), 471-510.  doi: 10.1016/j.jde.2004.02.014.  Google Scholar

[8]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231.  doi: 10.1007/s10884-013-9337-y.  Google Scholar

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S. NovoR. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.  Google Scholar

[10]

G. RothP. L. Salceanu and S. J. Schreiber, Robust permanence for ecological maps, SIAM J. Math. Anal., 49 (2017), 3527-3549.  doi: 10.1137/16M1066440.  Google Scholar

[11]

G. Roth and S. J. Schreiber, Persistence in fluctuating environments for interacting structured populations, J. Math. Biol., 69 (2014), 1267-1317.  doi: 10.1007/s00285-013-0739-6.  Google Scholar

[12]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Advances in Mathematics, 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[13]

P. L. Salceanu and H. L. Smith, Lyapunov exponents and persistence in discrete dynamical systems, Discrete Continuous Dynam. Systems - B, 12 (2009), 187-203.  doi: 10.3934/dcdsb.2009.12.187.  Google Scholar

[14]

S. J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differ. Equations, 148 (1998), 334-350.  doi: 10.1006/jdeq.1998.3471.  Google Scholar

[15]

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

[16]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, Amer. Math. Soc., 1995.  Google Scholar

[17]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/gsm/118.  Google Scholar

[18] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[19]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Sringer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[3]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

[4]

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

[5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[6]

J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.  doi: 10.1214/aop/1176996798.  Google Scholar

[7]

J. MierczyńskiW. Shen and X.-Q. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems, J. Differ. Equations, 204 (2004), 471-510.  doi: 10.1016/j.jde.2004.02.014.  Google Scholar

[8]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231.  doi: 10.1007/s10884-013-9337-y.  Google Scholar

[9]

S. NovoR. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.  Google Scholar

[10]

G. RothP. L. Salceanu and S. J. Schreiber, Robust permanence for ecological maps, SIAM J. Math. Anal., 49 (2017), 3527-3549.  doi: 10.1137/16M1066440.  Google Scholar

[11]

G. Roth and S. J. Schreiber, Persistence in fluctuating environments for interacting structured populations, J. Math. Biol., 69 (2014), 1267-1317.  doi: 10.1007/s00285-013-0739-6.  Google Scholar

[12]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Advances in Mathematics, 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[13]

P. L. Salceanu and H. L. Smith, Lyapunov exponents and persistence in discrete dynamical systems, Discrete Continuous Dynam. Systems - B, 12 (2009), 187-203.  doi: 10.3934/dcdsb.2009.12.187.  Google Scholar

[14]

S. J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differ. Equations, 148 (1998), 334-350.  doi: 10.1006/jdeq.1998.3471.  Google Scholar

[15]

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

[16]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, Amer. Math. Soc., 1995.  Google Scholar

[17]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/gsm/118.  Google Scholar

[18] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[19]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

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