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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.

MSC:
39A11Stability of difference equations (MSC2000)
92D25Population dynamics (general)
39A12Discrete version of topics in analysis
37C27Periodic orbits of vector fields and flows
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Full Text: DOI
References:
[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