We give a short proof of the Cushing-Henson conjecture concerning Beverton-Holt difference equation, 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.
L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 217–250, 2005.
View at: Google Scholar | MathSciNetR. J. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, vol. 19 of Fish. Invest., HMSO, London, 1957.
J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859–872, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. M. Cushing and S. M. Henson, “A periodically forced Beverton-Holt equation,” Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1119–1120, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Elaydi and R. J. Sacker, “Global stability of periodic orbits of non-autonomous difference equations and population biology,” Journal of Differential Equations, vol. 208, no. 1, pp. 258–273, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Elaydi and R. J. Sacker, “Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures,” in Proceedings of the 8th International Conference on Difference Equations and Applications, pp. 113–126, Chapman & Hall/CRC, Florida, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Elaydi and R. J. Sacker, “Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures,” Journal of Difference Equations and Applications, vol. 11, no. 4-5, pp. 337–346, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetH.-F. Huo and W.-T. Li, “Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 135–144, 2005.
View at: Publisher Site | Google Scholar | MathSciNetM. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Florida, 2002.
View at: Zentralblatt MATH | MathSciNetP. Liu and K. Gopalsamy, “On a model of competition in periodic environments,” Applied Mathematics and Computation, vol. 82, no. 2-3, pp. 207–238, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence ,” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631–638, 2006.
View at: Google Scholar | MathSciNetL.-L. Wang and W.-T. Li, “Existence and global attractivity of positive periodic solution for an impulsive delay population model,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, vol. 12, no. 3-4, pp. 457–468, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNetX.-X. Yan, W.-T. Li, and Z. Zhao, “On the recursive sequence ,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 269–282, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet