ABSTRACT
Abstract Elaydi and Yakubu showed that a globally asymptotically stable (GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular subharmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a nonautonomous dynamical system having a minimum period k and which has a GAS periodic orbit with minimum period r. Our methods are then applied to prove two conjectures of J. Cushing and S. Henson concerning a nonautonomous Beverton-Holt equation that 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.
