Permanence for the discrete mutualism model with time delays

Sufficient conditions are obtained for the permanence of the following discrete model of mutualism x1(k+1)=x1(k)exp{r1(k)[K1(k)+α1(k)x2(k−τ2(k))1+x2(k−τ2(k))−x1(k−σ1(k))]},x2(k+1)=x2(k)exp{r2(k)[K2(k)+α2(k)x1(k−τ1(k))1+x1(k−τ1(k))−x2(k−σ2(k))]}, where ri,Ki,αiri,Ki,αi, τiτi and σiσi, with i=1,2i=1,2, are non-negative sequences bounded above and below by positive constants, and αi>Kiαi>Ki, with i=1,2i=1,2.