Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 911–935 | Cite as

Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

  • Teresa FariaEmail author
  • Rafael Obaya
  • Ana M. Sanz


The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.


Delay differential equation Non-autonomous Nicholson system Quasi-monotone condition Persistence Permanence Global asymptotic stability 

Mathematics Subject Classification

34K25 34K12 34K27 34K20 92D25 



This work was partially supported by Fundação para a Ciência e a Tecnologia under project UID/MAT/04561/2013 (T. Faria) and by Ministerio de Economía y Competitividad under project MTM2015-66330, and the European Commission under project H2020-MSCA-ITN-2014 (R. Obaya and A. M. Sanz). The authors are very grateful to the referee, whose careful reading and valuable comments led to significant improvements of the manuscript.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAF-CIO, Faculdade de CiênciasUniversidade de LisboaCampo Grande, LisboaPortugal
  2. 2.Departamento de Matemática Aplicada, E. Ingenierías IndustrialesInstituto de Matemáticas, Universidad de Valladolid (IMUVA)ValladolidSpain
  3. 3.Departamento de Didáctica de las Ciencias Experimentales, Sociales y de la Matemática, Facultad de EducaciónInstituto de Matemáticas, Universidad de Valladolid (IMUVA)PalenciaSpain

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