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Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson’s blowflies model. (English) Zbl 1117.34065

Consider the Nicholson’s blowflies equation with periodic coefficents $\delta \left(t\right),p\left(t\right),\alpha \left(t\right)$:

${x}^{\text{'}}\left(t\right)=-\delta \left(t\right)x\left(t\right)+p\left(t\right)x\left(t-m\omega \right)exp\left(-\alpha \left(t\right)x\left(t-m\omega \right)\right)·$

For $\alpha \left(t\right)$ constant, sufficient conditions for the existence of a globally attracting periodic solution to this equation were given by S. H. Saker and S. Agarwal [“Oscillation and global attractivity in a periodic Nicholson’s blowflies model”, Math. Comput. Modelling 35, No. 7–8, 719–731 (2002; Zbl 1012.34067)].

In the paper under review, these results are generalized and somehow improved. Moreover, an impulsive related problem that can be reduced to the nonimpulsive one is also considered.

It should be noticed that the discussion in Section 4 for the autonomous case is misleading. Indeed, the authors seem to conclude that the positive equilibrium of the Nicholson’s blowflies equation is globally attracting under the condition $p>\delta$, which is clearly false.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses