We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We show that the periodic fluctuations in the carrying capacity always have a deleterious effect on the average population, thus answering in the affirmative the second of the conjectures. Independently Ryusuke Kon ,  discovered a solution to the second conjecture and in fact proved the result for a wider class of difference equations including the Beverton-Holt equation. The work of Davydova, Diekmann and van Gils,  should also be noted. There they study nonlinear Leslie matrix models describing the population dynamics of an age-structured semelparous species, a species whose individuals reproduce only once and die afterwards. See also the work of N.V. Davydova,  where the notion of families of single year class maps is introduced.
Document Object Identifier (DOI)
Elaydi, S., & Sacker, R. J. (2006). Periodic difference equations, population biology and the Cushing-Henson conjectures. Mathematical Biosciences, 201, 195-207. doi: 10.1016/j.mbs.2005.12.021