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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Berezansky, L., Idels, L., Troib, L.: Global dynamics of Nicholson-type delay systems with applications. Nonlinear Anal. Real World Appl. 12, 436–445 (2011)
Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)
Faria, T.: Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal. 74, 7033–7046 (2011)
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)
Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ., Kluwer, Dordrecht (1986)
Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc, Providence, Rhode Island (1988)
Hofbauer, J.: An index theorem for dissipative systems. Rocky Mountain J. Math. 20, 1017–1031 (1990)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, London (1993)
Liu, B.: Global stability of a class of delay differential systems. J. Comput. Appl. Math. 233, 217–223 (2009)
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)
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)
Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)
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)
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)
Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Am. Math. Soc, Providence, RI (2011)
Smith, H.L., Waltman, P.: The Theory of the Chemostat. University Press, Cambridge (1995)
Wang, L.: Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms. Appl. Math. Model. 37, 2153–2165 (2013)
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)
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).
About this article
Cite this article
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
- Nicholson’s blowflies equation
- Global asymptotic stability
Mathematics Subject Classification (2010 )