Permanence for a delayed discrete ratio-dependent predator–prey system with Holling type functional response ✩

Sufficient conditions are established for the permanence in a delayed discrete predator–prey model with Holling type III functional response:    N1(k +1) = N1(k) exp b1(k)− a1(k)N1(k − [τ1])− α1(k)N1(k)N2(k) N2 1 (k)+m2N2 2 (k) , N2(k +1) = N2(k) exp −b2(k)+ α2(k)N2 1 (k−[τ2]) N2 1 (k−[τ2])+m2N2 2 (k−[τ2]) . Our investigation confirms that when the death rate of the predator is rather small as well as the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run.  2004 Elsevier Inc. All rights reserved.