#### Title

Global Stability of Periodic Orbits of Non-Autonomous Difference Equations and Population Biology

#### Document Type

Post-Print

#### Publication Date

1-2005

#### 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 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 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.

#### Document Object Identifier (DOI)

10.1016/j.jde.2003.10.024

#### Repository Citation

Elaydi, S., & Sacker, R. J. (2005). Global stability of periodic orbits of non-autonomous difference equations and population biology. *Journal of Differential Equations*, 208, 258-273. doi: 10.1016/j.jde.2003.10.024

#### Publication Information

Journal of Differential Equations