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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

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

    Teresa Faria

  2. Departamento de Matemática Aplicada, E. Ingenierías Industriales, Instituto de Matemáticas, Universidad de Valladolid (IMUVA), 47011, Valladolid, Spain

    Rafael Obaya

  3. Departamento de Didáctica de las Ciencias Experimentales, Sociales y de la Matemática, Facultad de Educación, Instituto de Matemáticas, Universidad de Valladolid (IMUVA), 34004, Palencia, Spain

    Ana M. Sanz

Authors
  1. Teresa Faria
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  2. Rafael Obaya
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  3. Ana M. Sanz
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Correspondence to Teresa Faria.

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Faria, T., Obaya, R. & Sanz, A.M. Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications. J Dyn Diff Equat 30, 911–935 (2018). https://doi.org/10.1007/s10884-017-9572-8

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  • DOI: https://doi.org/10.1007/s10884-017-9572-8

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