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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.

92D25 Population dynamics (general)
39A12 Discrete version of topics in analysis
Full Text: DOI
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