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Global stability of periodic orbits of non-autonomous difference equations and population biology. (English) Zbl 1067.39003
Let $$X$$ by a connected metric space and $$F: X\to X$$ be a continuous map. Consider an autonomous difference equation $x_{n+1}=F(x_n),\;n=0,1,2,\dots.$ S. Elaydi and A.Yakubu [J. Difference Equ. Appl. 8, No. 6, 537–549 (2002; Zbl 1048.39002)] showed that if a $$k$$-cycle $$c_k$$ is globally asymptotically stable(GAS), then $$c_k$$ must be a fixed point. In this paper, the authors extend this result to periodic nonautonomous difference equation $x_{n+1}=F(n,x_n),\;n=0,1,2,\dots, \tag{$$*$$}$ via the concept of skew-product dynamical systems which comes from G. R. Sell [Topological Dynamics and ordinary differential equations (Van Nostrand- Reinhold, London) (1971; Zbl 0212.29202)]. The authors show that for a $$k$$-periodic differential 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 $$k$$ they construct a non-autonomous dynamical system having minimum period $$k$$ and which has a GAS periodic orbit with minimum period $$r$$.
Then they apply these methods to prove a conjecture by J. M. Cushing and S. M. Henson [J. Differential Equations Appl. 8, No. 12, 1119–1120 (2002; Zbl 1023.39013)] concerning a non-autonmous 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.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 92D25 Population dynamics (general) 39A12 Discrete version of topics in analysis 37C27 Periodic orbits of vector fields and flows
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##### References:
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