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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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##### References:
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