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Permanence and existence of a positive periodic solution to a periodic stage-structured system with infinite delay. (English) Zbl 1156.34065
The paper introduces a periodic $n$-species nonautonomous stage-structured system which includes competition. For each species the model incorporates a time lag which describes the time from birth to maturity of that species. Also, infinite delay is introduced which represents the effect of the entire past history of the system on the current competition interactions. It is first observed that if the growth rates are sufficiently large then the system is uniformly permanent. By means of Horn’s fixed point theorem, one notes that the system with finite delay has a positive periodic solution. As a consequence of this result, it is shown that even the system with infinite delay admits a positive periodic solution.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K13 Periodic solutions of functional differential equations 34K25 Asymptotic theory of functional-differential equations 92D25 Population dynamics (general)