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Discrete Dynamics in Nature and Society
Volume 2006, Article ID 12176, 21 pages

Harmless delays in a discrete ratio-dependent periodic predator-prey system

1School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 18 December 2005; Accepted 13 February 2006

Copyright © 2006 Yong-Hong Fan 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.


Verifiable criteria are established for the existence of positive periodic solutions and permanence of a delayed discrete periodic predator-prey model with Holling-type II functional response N1(k+1)=N1(k)exp{b1(k)a1(k)N1(k[τ1])α1(k)N2(k)/(N1(k)+m(k)N2(k))} and N2(k+1)=N2(k)exp{b2(k)+α2(k)N1(k[τ2])/(N1(k[τ2])+m(k)N2(k[τ2]))}. Our results show that the delays in the system are harmless for the existence of positive periodic solutions and permanence of the system. In particular our investigation confirms that if the death rate of the predator is rather small as well as the intrinsic growth rate of the prey is relatively large, then the species could coexist in the long run.