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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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