In this paper under some conditions on the constants A , B ∈ ( 0 , ∞ ) we study the existence of positive solutions, the existence of a unique nonnegative equilibrium and the convergence of the positive solutions to the nonnegative equilibrium of the system of difference equations x n + 1 = ( 1 − y n − y n − 1 ) ( 1 − e − A y n ) , y n + 1 = ( 1 − x n − x n − 1 ) ( 1 − e − B x n ) where A , B ∈ ( 0 , ∞ ) and the initial values x − 1 , x 0 , y − 1 , y 0 are positive numbers which satisfy the relations x 0 + x − 1 < 1 , y 0 + y − 1 < 1 , 1 − y 0 > ( 1 − x 0 − x − 1 ) ( 1 − e − B x 0 ) , 1 − x 0 > ( 1 − y 0 − y − 1 ) ( 1 − e − A y 0 ) .