Attenuant cycles of population models with periodic carrying capacity

This paper considers attenuation of cycles generated by periodic difference equations for population dynamics. This study concerns the second conjecture of Cushing and Henson [A periodically forced Beverton-Holt equation, Journal of Difference Equations and Applications, 8, 2002, pp. 1119 - 1120], which was recently resolved affirmatively by Elaydi and Sacker [ Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures, Proceedings of the 8th International Conference on Difference Equations and Applications, Brno, Czech Republic ( in press)]. We extend their result and obtain a sufficient condition for attenuation of cycles in population models. This sufficient condition is applicable to a wide class of periodic difference equations with arbitrary period. For an illustration, the result is applied to the Beverton-Holt equation and other specific population models.