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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 ' (t)=-δ(t)x(t)+ i=1 m p i (t)x(t-τ i (t))e -q i (t)x(t-τ i (t)) ,t0·

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:
92D25Population dynamics (general)
34K13Periodic solutions of functional differential equations
34K60Qualitative investigation and simulation of models
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