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A note on the nonautonomous Beverton-Holt model. (English) Zbl 1084.39007
The author considers the nonautonomous difference equation \[ x_{n+1}=\frac{rK_nx_n}{K_n+(r-1)x_n}, \quad n=0,1,\ldots \] where \(r > 1\), \(\{K_n\}\) is a positive persistent and bounded sequence. He establishes that all positive solutions \(\{x_n\}\) satisfy the inequality \[ \lim\sup\limits_{n\to\infty}\frac{1}{n}\sum^{n-1}_{k=0}x_k \leq \lim\sup\limits_{n\to\infty}\frac{1}{n}\sum^{n-1}_{k=0}K_k. \] In the case when \(\{K_n\}\) is a periodic sequence with prime period \(p\) the author gives some applications including the proof of the conjecture formulated by J. M. Cushing and S. M. Henson [J. Difference Equ. Appl. 8, No. 12, 1119-1120 (2002; Zbl 1023.39013)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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[1] DOI: 10.1016/0025-5564(90)90057-6 · Zbl 0712.39014 · doi:10.1016/0025-5564(90)90057-6
[2] DOI: 10.1080/1023619021000053980 · Zbl 1023.39013 · doi:10.1080/1023619021000053980
[3] Elaydi S, Proceedings of 8th International Conference on Difference Equations and Applications (2003)
[4] Elaydi S, Journal of Difference Equations and Applications (2004)
[5] Kocic VL, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993)
[6] Kocic VL, Journal of Difference Equations and Applications (2004)
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