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).
Reviewer: Gennadij Demidenko (Novosibirsk)
MSC:
39A11 | Stability of difference equations (MSC2000) |
39A12 | Discrete version of topics in analysis |
39A20 | Multiplicative and other generalized difference equations |
Keywords:
Beverton-Holt difference equation; periodic solution; Cushing-Henson’s conjecture; positive solution; rational difference equation; minimal period; geometric cycleCitations:
Zbl 1023.39013References:
[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 |
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