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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:
\[ x'(t)=- \delta(t)x(t)+ \sum_{i=1}^m p_i(t)x(t-\tau_i(t)) e^{-q_i(t)x(t-\tau_i(t))}, \quad t\geq 0. \]
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 to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
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