## 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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