1. Global stability of periodic orbits of non-autonomous difference equations and population biology
- Author
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Elaydi, Saber and Sacker, Robert J.
- Subjects
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EQUATIONS , *VIBRATION (Mechanics) , *OSCILLATIONS , *RESEARCH - 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 periodr is GAS, thenr must be a divisor ofk . In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, ifr dividesk we construct a non-autonomous dynamical system having minimum periodk and which has a GAS periodic orbit with minimum periodr . Our methods are then applied to prove a conjecture by 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. [Copyright &y& Elsevier]- Published
- 2005
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