Periodic solutions to nonautonomous difference equations. (English) Zbl 0712.39014

Clark, Mark E.; Gross, Louis J.

Math. Biosci. 102, No. 1, 105-119 (1990).

The equation \(x_{n+1}=f_ n(x_ n)\), \(n=0,1,..\). with \(f_{n+T}(x)=f_ n(x)\) for all n is discussed. A special class C of functions is introduced for which a periodic solution of the equation exists and also the asymptotics of all other solutions can be characterized.
The second part of the paper analyses in detail the equation \(x_{n+1}=a_ nx_ n/(1+b_ nx_ n),\) with \(a_ n,b_ n\) positive, bounded and periodic of integer period T, which belongs to the described class for \(a_ n>1\). The relevance of these results to some models in mathematical biology is discussed.

Reviewer: J.Gregor

MSC:

39A10	Additive difference equations
39A11	Stability of difference equations (MSC2000)
92B05	General biology and biomathematics

Keywords:

nonautonomous difference equations; periodic solution; asymptotics; models in mathematical biology

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