Journal of Dynamics and Differential Equations

, Volume 26, Issue 3, pp 723–744 | Cite as

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

Article

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.

Keywords

Nicholson’s blowflies equation Delays Persistence Permanence Global asymptotic stability 

Mathematics Subject Classification (2010 )

34K20 34K25 34K12 92D25 

Notes

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).

References

  1. 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)CrossRefMATHMathSciNetGoogle Scholar
  2. 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)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Faria, T.: Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal. 74, 7033–7046 (2011)CrossRefMATHMathSciNetGoogle Scholar
  4. 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)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ., Kluwer, Dordrecht (1986)CrossRefMATHGoogle Scholar
  6. 6.
    Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)CrossRefGoogle Scholar
  7. 7.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc, Providence, Rhode Island (1988)Google Scholar
  8. 8.
    Hofbauer, J.: An index theorem for dissipative systems. Rocky Mountain J. Math. 20, 1017–1031 (1990)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, London (1993)MATHGoogle Scholar
  10. 10.
    Liu, B.: Global stability of a class of delay differential systems. J. Comput. Appl. Math. 233, 217–223 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 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)CrossRefMATHMathSciNetGoogle Scholar
  12. 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)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)CrossRefGoogle Scholar
  14. 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)CrossRefMATHGoogle Scholar
  15. 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)Google Scholar
  16. 16.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Am. Math. Soc, Providence, RI (2011)Google Scholar
  17. 17.
    Smith, H.L., Waltman, P.: The Theory of the Chemostat. University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  18. 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)CrossRefMathSciNetGoogle Scholar
  19. 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)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAF, Faculdade de CiênciasUniversidade de Lisboa, Campo GrandeLisboaPortugal
  2. 2.Bolyai InstituteUniversity of SzegedSzegedHungary

Personalised recommendations