×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Braverman E., J. Difference Equ. Appl. 14 pp 275– (2008) · Zbl 1158.39004 · doi:10.1080/10236190701565511
[2] Bullen P. S., Handbook of Means and Their Inequalities (2003) · Zbl 1035.26024
[3] DOI: 10.1016/j.jmaa.2006.10.096 · Zbl 1120.39003 · doi:10.1016/j.jmaa.2006.10.096
[4] DOI: 10.1016/0025-5564(90)90057-6 · Zbl 0712.39014 · doi:10.1016/0025-5564(90)90057-6
[5] DOI: 10.1080/1023619021000053980 · Zbl 1023.39013 · doi:10.1080/1023619021000053980
[6] DOI: 10.1201/9781420034905 · doi:10.1201/9781420034905
[7] DOI: 10.1080/10236190412331335418 · Zbl 1084.39005 · doi:10.1080/10236190412331335418
[8] DOI: 10.1016/j.jde.2003.10.024 · Zbl 1067.39003 · doi:10.1016/j.jde.2003.10.024
[9] DOI: 10.1080/10236190412331335463 · Zbl 1084.39007 · doi:10.1080/10236190412331335463
[10] V. L. Kocic, Global behaviour of solutions of a nonautonomous delay logistic difference equation, II, J. Difference Equ. Appl. (to appear) · Zbl 1215.39017
[11] Kocic V. L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 · doi:10.1007/978-94-017-1703-8
[12] DOI: 10.1080/10236190410001652766 · Zbl 1072.39016 · doi:10.1080/10236190410001652766
[13] DOI: 10.1080/10236190410001703949 · doi:10.1080/10236190410001703949
[14] DOI: 10.1080/10236190412331335472 · Zbl 1067.92048 · doi:10.1080/10236190412331335472
[15] DOI: 10.1080/10236190601045929 · Zbl 1123.39004 · doi:10.1080/10236190601045929
[16] Royama T., Analytical Population Dynamics (1992) · doi:10.1007/978-94-011-2916-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.