# zbMATH — the first resource for mathematics

On the Cushing-Henson conjecture, delay difference equations and attenuant cycles. (English) Zbl 1158.39004
Consider a difference equation of the form
$x_{n+1}=f(K_{n},x_{n-h_{1}(n)},\dots,x_{n-h_{r}(n)}), \quad n\in \{0,1,2,\dots\}, \tag{1}$ where $$r$$ is a fixed positive integer, $$\{ K_{n}\} _{n=0}^\infty$$ is a positive $$k$$-periodic sequence, each $$h_{q}$$ is a nonnegative sequence with period $$l$$ such that the function $$H_{q}$$ defined by $$H_{q}(n)= (n-h_{q}(n))\bmod l$$ is a one to one map of the set $$\{ 0,1,\dots,l-1\}$$ onto itself, and $$f:[0,\infty)^{r+1}\rightarrow [0,\infty)$$ is a continuous function satisfying
$f(K_n,K_n,\dots,K_n)= K_n,$ and
$f(K_n,x,\dots,x)\begin{cases} >x &\text{if }0<x<K_n,\\ <K_n &\text{if }x>K_n.\end{cases}$ A sufficient condition (based on a concavity assumption on $$f$$) is obtained for a positive $$m$$-cycle of (1) to be attenuant, where an $$m$$-cycle of a periodic solution $$\left\{ x_{n}\right\}$$ of (1) is said to be attenuant if
$\frac{1}{m}\sum_{j=1}^{m}x_{j}<\frac{1}{k}\sum_{j=1}^{k}K_{j}.$ A sufficient condition for the global attractivity of the positive $$p$$-periodic solution of the so called Beverton-Holt equation
$x_{n+1}= \frac{\lambda x_{n}}{1+(\lambda -1)(x_{n-p}/K_n)}, \quad n\in \{0,1,2,\dots\},$ where $$\lambda >1$$ and $$\{K_n\}$$ is a positive $$p$$-periodic sequence, is also obtained.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] DOI: 10.1080/10236190108808308 · Zbl 1002.39003 · doi:10.1080/10236190108808308 [2] Cushing J.M., Journal of Difference Equations and Applications 12 pp 1119– (2002) · Zbl 1023.39013 · doi:10.1080/1023619021000053980 [3] Elaydi S., Undergraduate Text in Mathematics, 3. ed. (2005) [4] DOI: 10.1080/10236190412331335418 · Zbl 1084.39005 · doi:10.1080/10236190412331335418 [5] DOI: 10.1016/j.jde.2003.10.024 · Zbl 1067.39003 · doi:10.1016/j.jde.2003.10.024 [6] Elaydi S., Mathematical Biosciences pp 195– (2006) [7] Erbe L., Journal of Differential Equations pp 151– (1995) [8] Graef J.R., Dynamic System and Applications 2 pp 165– (1996) [9] Györi I., Oscillation Theory of Delay Differential Equations (1991) · Zbl 0780.34048 [10] Györi, I. and Pituk, M., 1997, Asymptotic stability in a linear delay difference equation, In Proceedings of SICDEA, Veszprem, Hungary, August 6-11, 1995, Gordon and Breach Science, Langhorne, PA. · Zbl 0846.39003 [11] Kocic V.L., Journal of Difference Equations and Applications 4 pp 415– (2005) · Zbl 1084.39007 · doi:10.1080/10236190412331335463 [12] Kocic V.L., Journal of Difference Equations and Applications 13 pp 1267– (2004) · Zbl 1072.39016 · doi:10.1080/10236190410001652766 [13] Kocić V.L., Mathematics and Its Applications (1993) [14] Kon R., Journal of Difference Equations and Applications 8 pp 791– (2004) · doi:10.1080/10236190410001703949 [15] Kon R., Journal of Difference Equations and Applications 4 pp 423– (2005) · Zbl 1067.92048 · doi:10.1080/10236190412331335472 [16] DOI: 10.1201/9781420035384 · doi:10.1201/9781420035384 [17] Pielou E.C., An Introduction to Mathematical Ecology (1969) · Zbl 0259.92001 [18] Pielou E.C., Population and Community Ecology (1974) · Zbl 0349.92024 [19] Stević, S., 2006, A short proof of the Cushing–Henson conjecture, Discrete Dynamics in Nature and Society, (2006), Article ID 37264, 1–5, DOI 10.1155/DDNS/2006/37264. · Zbl 1149.39300 [20] DOI: 10.1016/S0893-9659(03)80027-7 · Zbl 1049.39017 · doi:10.1016/S0893-9659(03)80027-7
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.