A periodically forced Beverton-Holt equation. (English) Zbl 1023.39013

Cushing, J. M.; Henson, Shandelle M.

J. Difference Equ. Appl. 8, No. 12, 1119-1120 (2002).

The authors present an open problem: to prove or disprove the following assertions for \(p\geq 3\).
(a) The periodically forced Beverton-Holt equation \[ x_{n+1}=\frac{rK_n}{K_n+(r-1)x_n}x_n, \quad n=0, 1, 2, \ldots, \] has a positive \(p\)-periodic solution \(\{y_n\}\), and it is globally attracting for \(x_0>0\), where \(r>1\) and \(\{K_n\}\) is a sequence of positive numbers with a base period \(p\geq 1\).
(b) If \(p>2\), the strict inequality \(av(y_n)<av(K_n)\) holds. Here \(av(y_n)=\frac{1}{p}\sum_{i=0}^{p-1}y_i\).

Reviewer: Xianhua Tang (Changsha)

MSC:

39B05	General theory of functional equations and inequalities
39A11	Stability of difference equations (MSC2000)

Keywords:

Beverton-Holt equation; periodic solution

References:

[1]	DOI: 10.1080/10236190108808308 · Zbl 1002.39003 · doi:10.1080/10236190108808308