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ourn~~l OJ UtJjercmc L .~u # J ! u~~ 0#8d A~,~L u~~L v I~. IW8. Vol. 4. pp. 315-323Rcprinls available directly from the publisher Photocopying permitted by license only I IV'\ l l t ! l ,,, ,, \ "In this paper we investigate the persistence, the oscillatory behavior and the periodic nature of solutions of a system of two nonlinear difference equations.

We consider the following system of Lyness-type difference equations: x 1 (n + 1) = (a k x k (n) + b k)/x k−1 (n − 1), x 2 (n + 1) = (a 1 x 1 (n) + b 1)/x k (n − 1), x i (n + 1) = (a i−1 x i−1 (n) + b i−1)/x i−2 (n − 1), i = 3,4,...,k, where a i , b i , i = 1,2,...,k, are positive constants, k ≥ 3 is an integer, and the initial values are positive real numbers. We study the existence of invariants, the boundedness, the persistence, and the periodicity of the positive solutions of this system.

We present some invariants for difference equations and systems of difference equations of rational form. We examine two cases. In the first case the coefficients are constant and in the second are positive periodic sequences of certain periods.ᮊ 1997 Academic Press

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In this paper, we study the stability of the zero equilibria of the following systems of difference equations: xn+1=axn+byne−xn,yn+1=cyn+dxne−yn and xn+1=ayn+bxne−yn,yn+1=cxn+dyne−xn where a, b, c and d are positive constants and the initial conditions x0 and y0 are positive numbers. We study the stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. Copyright © 2016 John Wiley & Sons, Ltd.