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Asymptotic behavior of a sequence defined by iteration with applications. (English) Zbl 1029.39006

The main result of the paper provides an asymptotic formula for the positive solutions of the second order nonlinear difference equation \[ x_{n+1}=f(x_n,x_{n-1}),\qquad n=1,2,\dots, \] where \(f:(0,\infty)^2\to(0,\infty)\) is a continuous function such that for some \(p\in(0,1)\) and \(\alpha>0\), \[ 0<f(x,y)<px+(1-p)y,\qquad x, y\in(0,\alpha), \] and \[ f(x,y)=px+(1-p)y-\sum_{s=m}^\infty{\mathcal K}_s(x,y) \] uniformly in a neighborhood of the origin, where \(m\geq 1\), \({\mathcal K}_s(x,y)=\sum_{i=0}^s a_{i,s}x^{s-i}y^i\), and \({\mathcal K}_m(1,1)>0\). The result is applied to some particular model equations from biology and ecology.

MSC:

39A11 Stability of difference equations (MSC2000)
92D40 Ecology
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