A short proof of the Cushing-Henson conjecture.

*(English)*Zbl 1149.39300Summary: We give a short proof of the Cushing-Henson conjecture [cf. J. M. Cushing and S. M. Henson, J. Difference Equ. Appl. 8, No. 12, 1119–1120 (2002; Zbl 1023.39013)] concerning the Beverton-Holt difference equation [cf. R. J. Beverton and S. J. Holt, On the dynamics of exploited fish populations. Fish. Invest. 19, HM50, London (1957)] which is important in theoretical ecology. The main result shows that a periodic environment is always deleterious for populations modeled by the Beverton-Holt difference equation.

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A20 | Multiplicative and other generalized difference equations |

92D25 | Population dynamics (general) |

92D40 | Ecology |

##### Keywords:

positive solution; rational difference equation; Cushing-Henson conjecture; Beverton-Holt difference equation; ecology; population##### Citations:

Zbl 1023.39013
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\textit{S. Stević}, Discrete Dyn. Nat. Soc. 2006, Article ID 37264, 5 p. (2006; Zbl 1149.39300)

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