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A note on the nonautonomous delay Beverton-Holt model. (English) Zbl 1345.92118
Summary: It is well known that the periodic cycle $$\{\bar{x}_n\}$$ of a periodically forced nonlinear difference equation is attenuant (resonant) if $$av (\bar{x}_n)<av (K_n)(av (\bar{x}_n)>av (K_n))$$ where $$\{K_n \}$$ is the carrying capacity of the environment and $$av(t_n)=(1/p) \sum_{i=0}^{p-1}t_i$$ (arithmetic mean of the $$p$$-periodic cycle $$\{t_n \}$$). In this article, we extend the concept of attenuance and resonance of periodic cycles using the geometric mean for the average of a periodic cycle. We study the properties of the periodically forced nonautonomous delay Beverton-Holt model
$x_{n+1} = \frac{r_nx_n}{1 + (r_{n-l}-1)x_{n-k}/K_{n-k}}, \quad n = 0,1,\dots,$ where $$\{K_n\}$$ and $$\{r_n\}$$ are positive $$p$$-periodic sequences; ($$K_n>0$$, $$r_n>1$$) as well as $$k$$ and $$l$$ are nonnegative integers. We will show that for all positive solutions $$\{x_n\}$$ of the previous equation
$\limsup_{n \to \infty}\bigg(\prod_{i=0}^{n-1} x_i\bigg)^{1/n} \leq \Bigg(\bigg(\prod_{i=0}^{p-1} r_i\bigg)^{1/p}-1\Bigg) \bigg(\prod_{i=0}^{p-1} (r_i-1)\bigg)^{-1/p} \bigg(\prod_{i=0}^{p-1} K_i\bigg)^{1/p}.$
In particular, in the case where $$\{\bar{x}_n\}$$ is a $$p$$-periodic solution of the above equation (assuming that such solution exists) and $$r_n=r>1$$, the periodic cycle is $$g$$-attenuant, that is
$\bigg(\prod_{i=0}^{p-1} \bar{x}_i \bigg)^{1/p}< \bigg(\prod_{i=0}^{p-1} K_i\bigg)^{1/p}.$ Surprisingly, the obtained results show that the delays $$k$$ and $$l$$ do not play any role.

##### MSC:
 92D25 Population dynamics (general) 39A12 Discrete version of topics in analysis
##### Keywords:
Beverton-Holt model; attenuance
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##### References:
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