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Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. (English) Zbl 1147.92031

Summary: By using the Krasnoselskii cone fixed point theorem, we obtain a sufficient condition as well as a necessary condition for the existence of positive periodic solutions of the following generalized A. J. Nicholson’s [An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–25 (1954)] blowflies model:

${x}^{\text{'}}\left(t\right)=-\delta \left(t\right)x\left(t\right)+\sum _{i=1}^{m}{p}_{i}\left(t\right)x\left(t-{\tau }_{i}\left(t\right)\right){e}^{-{q}_{i}\left(t\right)x\left(t-{\tau }_{i}\left(t\right)\right)},\phantom{\rule{1.em}{0ex}}t\ge 0·$

In the degenerate case, i.e., where the coefficients and delays of the above equation are all constants, a sufficient and necessary condition for the existence of positive periodic solutions is obtained. Our results are completely new, and generalize and improve some results from the literature.

MSC:
 92D25 Population dynamics (general) 34K13 Periodic solutions of functional differential equations 34K60 Qualitative investigation and simulation of models
Keywords:
cone fixed point theorem