Permanence for the Michaelis–Menten type discrete three-species ratio-dependent food chain model with delay ✩

A discrete three trophic level food chain model with ratio-dependent Michaelis–Menten type functional response is investigated. It is shown that under some appropriate conditions the system is permanent. The results indicate that, to make the species coexist in the long run, it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large. © 2005 Elsevier Inc. All rights reserved