Discrete Dynamics in Nature and Society 
Volume 2004 (2004), Issue 2, Pages 325-343
doi:10.1155/S1026022604310010

Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response

Lin-Lin Wang1 and Wan-Tong Li2

 1Department of Mathematics, Tianjing University, Tianjing 300072, China
2Department of Mathematics, Lanzhou University, Gansu, Lanzhou 730000, China
Received 1 October 2003

Abstract

The existence of positive periodic solutions for 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)/(N12(k)+m2N22(k))}, N2(k+1)=N2(k)exp{−b2(k)+α2(k)N12(k−[τ2])/(N12(k−[τ2])+m2N22(k−[τ2]))} is established by using the coincidence degree theory. We also present sufficient conditions for the globally asymptotical stability of this system when all the delays are zero. Our investigation gives an affirmative exemplum for the claim that the ratio-dependent predator-prey theory is more reasonable than the traditional prey-dependent predator-prey theory.