Global stability of periodic orbits of non-autonomous difference equations and population biology.

*(English)*Zbl 1067.39003Let \(X\) 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.

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.

Reviewer: Jurang Yan (Taiyuan)

##### 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 |

##### Keywords:

population biology; skew-product dynamical system; global stability; periodic nonautonomous difference equation; periodic orbit; minimum period; Beverton-Holt equation; metric space
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\textit{S. Elaydi} and \textit{R. J. Sacker}, J. Differ. Equations 208, No. 1, 258--273 (2005; Zbl 1067.39003)

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##### References:

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