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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 ,,x n-k+1 ),(*)

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(0,1),

f(x,y)=px+(1-p)y-K 2 (x,y)-K 3 (x,y)+o((x 2 +y 2 ) 3 2 )asx 2 +y 2 0

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 ) 3 2 )asx 2 +y 2 +z 2 0

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 =c n+blnn n 2 +olnn n 2 ,

where c and b are constants.

MSC:
39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis
92D25Population dynamics (general)