Basins of attraction of an anti-competitive discrete rational system

C Dedicated to Professor Mustafa Kulenovic on the occasion of his 60th birthday Abstract. We investigate the global asymptotic behavior of solutions of the following anti-competitive system of difference equations xn+1 = 1yn A1 + xn ; yn +1 = 2xn + 2yn yn ; n = 0; 1;:::; where the parameters 1; 2; 2;A1 are positive numbers and the initial conditions x0 0;y0 > 0. We find the basins of attraction of all at- tractors of the system, which are the equilibrium point and period-two solutions.