## Permanence for a discrete Nicholson’s blowflies model with feedback control and delay.(English)Zbl 1163.39011

The authors discuss the discrete Nicholson’s blowflies model with feedback control, which is a discrete form of its continuous model considered by W. S. C. Gurney et al. in [Nature 287, 17–21 (1980), doi:10.1038/287017a0], together with a feedback control as considered by K. Gopalsamy and P. Weng [Int. J. Math. Sci. 16, No. 1, 177–192 (1993; Zbl 0765.34058)]. The authors consider the solution $$(x(k), \mu(k))$$ associated with the initial condition $$x(-m), x(-m+1),\dots, x(-1)\geq 0$$, $$x(0)$$ and $$\mu(0)>0$$. The main result (Theorem 2.4) gives a sufficient condition for the permanence, i.e., both $$x(k)$$ and $$\mu(k)$$ are bounded below and above by two positive constants.

### MSC:

 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general) 93B52 Feedback control 39A22 Growth, boundedness, comparison of solutions to difference equations 39A30 Stability theory for difference equations

Zbl 0765.34058
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### References:

 [1] DOI: 10.1038/287017a0 [2] Gopalsamy K., Int. J. Math. Sci. 1 pp 177– [3] Lalli B. S., Dynam. Syst. Appl. 5 pp 117– [4] DOI: 10.1006/jmaa.2000.7420 · Zbl 1003.34069 [5] DOI: 10.1016/j.cam.2004.11.002 · Zbl 1069.34100 [6] Wang L. L., J. Lanzhou Univ. (Nat. Sci.) 42 pp 114– [7] DOI: 10.1016/j.jmaa.2005.04.036 · Zbl 1107.39017 [8] Chen F. D., Appl. Math. Comput. 186 pp 23– [9] DOI: 10.1016/j.aml.2006.08.023 · Zbl 1128.92029
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