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

Copyright © 2004 Lin-Lin Wang and Wan-Tong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.