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
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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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