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Persistence, Permanence and Global Stability for an \(n\)-Dimensional Nicholson System

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Abstract

For a Nicholson’s blowflies system with patch structure and multiple discrete delays, we analyze several features of the global asymptotic behavior of its solutions. It is shown that if the spectral bound of the community matrix is non-positive, then the population becomes extinct on each patch, whereas the total population uniformly persists if the spectral bound is positive. Explicit uniform lower and upper bounds for the asymptotic behavior of solutions are also given. When the population uniformly persists, the existence of a unique positive equilibrium is established, as well as a sharp criterion for its absolute global asymptotic stability, improving results in the recent literature. While our system is not cooperative, several sharp threshold-type results about its dynamics are proven, even when the community matrix is reducible, a case usually not treated in the literature.

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References

  1. Berezansky, L., Idels, L., Troib, L.: Global dynamics of Nicholson-type delay systems with applications. Nonlinear Anal. Real World Appl. 12, 436–445 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Faria, T.: Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal. 74, 7033–7046 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Faria, T., Oliveira, J.J.: Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous feedbacks. J. Differential Equations 244, 1049–1079 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ., Kluwer, Dordrecht (1986)

    Book  MATH  Google Scholar 

  6. Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  7. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc, Providence, Rhode Island (1988)

  8. Hofbauer, J.: An index theorem for dissipative systems. Rocky Mountain J. Math. 20, 1017–1031 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, London (1993)

    MATH  Google Scholar 

  10. Liu, B.: Global stability of a class of delay differential systems. J. Comput. Appl. Math. 233, 217–223 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, B.: Global stability of a class of Nicholson’s blowflies model with patch structure and multiple time-varying delays. Nonlinear Anal. Real World Appl. 11, 2557–2562 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu, X., Meng, J.: The positive almost periodic solution for Nicholson-type delay systems with linear harvesting term. Appl. Math. Model. 36, 3289–3298 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)

    Article  Google Scholar 

  14. Röst, G., Wu, J.: Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463, 2655–2669 (2007)

    Article  MATH  Google Scholar 

  15. Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. Am. Math. Soc., Providence, RI (1995)

  16. Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Am. Math. Soc, Providence, RI (2011)

  17. Smith, H.L., Waltman, P.: The Theory of the Chemostat. University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  18. Wang, L.: Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms. Appl. Math. Model. 37, 2153–2165 (2013)

    Article  MathSciNet  Google Scholar 

  19. Zhao, X.-Q., Jing, Z.-J.: Global asymptotic behavior in some cooperative systems of functional differential equations. Can. Appl. Math. Quart. 4, 421–444 (1996)

Download references

Acknowledgments

Work supported by Fundação para a Ciência e a Tecnologia, PEst-OE/MAT/UI0209/2011 (T. Faria) and by ERC Starting Grant Nr. 259559, OTKA K109782 and ESF project FuturICT.hu (TÁMOP-4.2.2.C-11/1/KONV-2012-0013) (G. Röst).

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Authors and Affiliations

  1. Departamento de Matemática and CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 , Lisboa, Portugal

    Teresa Faria

  2. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720 , Szeged, Hungary

    Gergely Röst

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  1. Teresa Faria
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  2. Gergely Röst
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Correspondence to Teresa Faria.

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Faria, T., Röst, G. Persistence, Permanence and Global Stability for an \(n\)-Dimensional Nicholson System. J Dyn Diff Equat 26, 723–744 (2014). https://doi.org/10.1007/s10884-014-9381-2

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  • DOI: https://doi.org/10.1007/s10884-014-9381-2

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