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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 $$ x_{n+1}=F(x_n),\ n=0,1,2,\dots. $$ {\it S. Elaydi} and {\it 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 {\it 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 {\it J. M. Cushing} and {\it 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.

39A11Stability of difference equations (MSC2000)
92D25Population dynamics (general)
39A12Discrete version of topics in analysis
37C27Periodic orbits of vector fields and flows
Full Text: DOI
[1] Cushing, J.; Henson, S.: Global dynamics of some periodically forced, monotone difference equations. J. differential equations appl 7, No. 6, 859-872 (2001) · Zbl 1002.39003
[2] Cushing, J.; Henson, S.: A periodically forced beverton--Holt equation. J. differential equations appl 8, No. 12, 1119-1120 (2002) · Zbl 1023.39013
[3] Elaydi, S.: An introduction to difference equations. (1999) · Zbl 0930.39001
[4] Elaydi, S.: Discrete chaos. (2000) · Zbl 0945.37010
[5] Elaydi, S.; Yakubu, A.: Global stability of cycleslotka--Volterra competition model with stocking. J. differential equations appl 8, No. 6, 537-549 (2002) · Zbl 1048.39002
[6] Sacker, R.: The splitting index for linear differential systems. J. differential equations 33, No. 3, 368-405 (1979) · Zbl 0438.34008
[7] Sacker, R.: Dedication to george R. Sell. J. differential equations appl 9, No. 5, 437-440 (2003) · Zbl 1039.01530
[8] Sacker, R.; Sell, G.: Skew-product flows, finite extensions of minimal transformation groups and almost periodic differential equations. Bull. amer. Math. soc 79, No. 4, 802-805 (1973) · Zbl 0265.54044
[9] Sacker, R.; Sell, G.: Lifting properties in skew-product flows with applications to differential equations. Mem. amer. Math. soc 11, No. 190, 1-67 (1977) · Zbl 0374.34031
[10] Sell, G.: Topological dynamics and differential equations. (1971) · Zbl 0212.29202