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Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures. (English) Zbl 1087.39504
Elaydi, Saber (ed.) et al., Proceedings of the 8th international conference on difference equations and applications (ICDEA 2003), Masaryk University, Brno, Czech Republic, July 28–August 1, 2003. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-536-X/hbk). 113-126 (2005).
Summary: S. Elaydi and R. J. Sacker [J. Differ. Equations 208, No. 1, 258–273 (2005; Zbl 1067.39003)] 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. M. Cushing and S. M. 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 deleterions effect on the average population, thus answering in the affirmative the second of the conjectures.
For the entire collection see [Zbl 1072.39001].

##### MSC:
 39A11 Stability of difference equations (MSC2000) 92D25 Population dynamics (general)