Qualitative behavior of two systems of higher-order difference equations

In this paper, we study the qualitative behavior of following two systems of higher-order difference equations: x(n+1) = alpha x(n-k)/beta + gamma gamma(2)(n-k+1), y(n+1) = alpha(1)y(n-k)/beta(1) + gamma(1)x(n-k+1)(2), n = 0, 1, . . . , and x(n+1) = alpha gamma(n-k)/b + cx(n-k+1)(2), y(n+1) = alpha(1)x(n-k)/b(1) + c(1)y(n-k+1)(2), n = 0, 1, . . . , where the parameters alpha, beta, alpha(1), beta(1), gamma(1), a, b, c, a(1), b(1), and c(1) and the initial conditions x(0), x(-1), . . . , x(-k), y(0), y(-1), . . . , y(-k) are positive real numbers. More precisely, we study the equilibrium points, local asymptotic stability, instability, global asymptotic stability of equilibrium points, and rate of convergence of positive solutions that converges to the equilibrium point P-0 = (0,0) of these systems. Some numerical examples are given to verify our theoretical results. These examples are experimental verification of our theoretical discussions. Copyright (C) 2015 JohnWiley & Sons, Ltd.