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Asymptotic behavior of a class of nonlinear difference equations. (English) Zbl 1121.39006
The author investigates a class of difference equations of type $x_{n+1}=f(x_n,\dots,x_{n-k+1}),\tag{$$*$$}$ for $$k=2,3$$, which includes a large class of mathematical biology models, such as the generalized Beverton-Holt stock recruitment model, the flour beetle population model, a mosquito population equation, and a discrete delay logistic difference equation. The main result shows that for $$p,q,p+q\in(0,1)$$, $f(x,y)=px+(1-p)y-K_2(x,y)-K_3(x,y) +o((x^2+y^2)^\frac{3}{2}) \text{ as }x^2+y^2\rightarrow0$ for $$k=2$$ or
$f(x,y,z)=px+qy+(1-p-q)z -K_2(x,y,z)-K_3(x,y,z) +o((x^2+y^2+z^2)^\frac{3}{2}) \text{ as }x^2+y^2+z^2 \rightarrow0$ for $$k=3$$, where $$K_2$$, $$K_3$$ are homogeneous polynomials of second and third order respectively, and moreover $$K_2$$ is a positive definite form. Then there exists a positive solution $$(x_n)$$ of $$(*)$$ with the following asymptotics: $x_n=\frac{c}{n}+b \frac{\ln n}{n^2}+o\left (\frac{\ln n}{n^2}\right),$ where $$c$$ and $$b$$ are constants.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general)