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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.

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