Permanence for the discrete mutualism model with time delays

Sufficient conditions are obtained for the permanence of the following discrete model of mutualism x 1 ( k + 1 ) = x 1 ( k ) exp { r 1 ( k ) [ K 1 ( k ) + α 1 ( k ) x 2 ( k − τ 2 ( k ) ) 1 + x 2 ( k − τ 2 ( k ) ) − x 1 ( k − σ 1 ( k ) ) ] } , x 2 ( k + 1 ) = x 2 ( k ) exp { r 2 ( k ) [ K 2 ( k ) + α 2 ( k ) x 1 ( k − τ 1 ( k ) ) 1 + x 1 ( k − τ 1 ( k ) ) − x 2 ( k − σ 2 ( k ) ) ] } , where r i , K i , α i , τ i and σ i , with i = 1 , 2 , are non-negative sequences bounded above and below by positive constants, and α i > K i , with i = 1 , 2 .