Harmless delays in a discrete ratio-dependent periodic predator-prey system (Q871379)

Summary: 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 \[ N_1(k+1)=N_1(k) \exp\biggl\{b_1(k)-a_1(k)N_1\bigl(k-[\tau_1]\bigr)-\alpha_1(k)N_2(k)/ \bigl(N_1(k)+m(k)N_2(k)\bigr) \biggr\} \] and \[ N_2(k+1)=N_2(k)\exp \biggl\{-b_2(k)+\alpha_2(k)N_1\bigl(k-[\tau_2]\bigr)/\biggl(N_1(k-[\tau_2]\bigr)+m(k)N_2\bigl(k-[\tau_2]\bigr)\biggr) \biggr\}. \] 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.