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Periodic difference equations, population biology and the Cushing–Henson conjectures. (English) Zbl 1105.39006
Authors’ abstract: We show that for a \(k\)-periodic difference equation, if a periodic orbit of period \(r\) is globally asymptotic stable (GAS), then \(r\) must be a divisor of \(k\). Moreover, if \(r\) divides \(k\) we construct a non-autonomous dynamical system having minimum period \(k\) and which has a GAS periodic orbit with minimum period \(r\).
Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of J. M. Cushing and S. M. Henson [J. Difference Equ. Appl. 8, No. 12, 1119–1120 (2002; Zbl 1023.39013)] concerning a non-autonomous 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. We give an equality linking the average population with the growth rates and carrying capacity (in the \(2\)-periodic case) which shows that out-of-phase oscillations in these quantities always have a deleterious effect on the average population. We give an example where in-phase oscillations cause the opposite to occur.

MSC:
39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
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