#### 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.

#### Identifier

10.1016/j.jde.2003.10.024

#### Publisher

Elsevier

#### 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*(1), 258-273. doi:10.1016/j.jde.2003.10.024

#### Publication Information

Journal of Differential Equations