zbMATH — the first resource for mathematics

Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures. (English) Zbl 1084.39005
The authors consider the nonautonomous Beverton–Holt equation \[ x_{n+1}=\frac{\mu_n K_nx_n}{K_n+(\mu_n-1)x_n}, \quad n=0,1,\ldots \eqno(1) \] where \(\{\mu_n\}\) and \(\{K_n\}\) are sequences with minimal common period \(p \geq 2\). For a positive \(p\)-periodic solution \(\{x_n\}\) the authors establish the inequality \[ \frac{1}{p}\sum^{p-1}_{k=0}x_k < \frac{\mu^*(\mu^*-1)}{\mu_*(\mu_*-1)} \frac{1}{p}\sum^{p-1}_{k=0}K_k, \eqno(2) \] where \(\mu^* = \max(\mu_n)\), \(\mu_* = \min(\mu_n)\). The obtained result is an extension 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)] in the case when \(\mu_n\) is constant. In the present paper a refinement of (2) is given for \(p=2\). The authors construct an example having a geometric cycle with minimal period \(r < p\). In the case of (1) with constant \(\mu_n\) the authors prove that any geometric cycle must have the same minimal period \(p\) as (1).

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
Full Text: DOI
[1] Adler A, The Theory of Numbers (1995)
[2] DOI: 10.1080/10236190108808308 · Zbl 1002.39003 · doi:10.1080/10236190108808308
[3] DOI: 10.1080/1023619021000053980 · Zbl 1023.39013 · doi:10.1080/1023619021000053980
[4] Elaydi, S. and Sacker, R.J., Global stability of periodic orbits of nonautonomous difference equations and population biology,Journal of Differential Equations,208(11), 258–273. · Zbl 1067.39003
[5] Elaydi S, Proceedings of the 8th International Conference on Difference Equations (2003)
[6] DOI: 10.1016/S0022-247X(03)00417-7 · Zbl 1035.37020 · doi:10.1016/S0022-247X(03)00417-7
[7] Kocic, V.L.A note on the Cushing-Henson conjectures(this issue).
[8] DOI: 10.1080/10236190410001703949 · doi:10.1080/10236190410001703949
[9] Kon, R.Attenuant cycles of population models with periodic carrying capacity(this issue). · Zbl 1067.92048
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.