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

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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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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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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Keywords

  • Nicholson’s blowflies equation
  • Delays
  • Persistence
  • Permanence
  • Global asymptotic stability

Mathematics Subject Classification (2010 )

  • 34K20
  • 34K25
  • 34K12
  • 92D25