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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 and 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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

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

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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

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

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