Permanence and partial extinction in a delayed threespecies food chain model with stage structure and timevarying coefficients

By taking full consideration of maturity (τ1 represents the maturity of predator and τ2 represents the maturity of top predator) and the effects of environmental parameters, a new delayed three-species food chain model with stage structure and time-varying coefficients is established. With the help of the comparison theorem and the technique of mathematical analysis, the positivity and boundedness of solutions of the model are investigated. Furthermore, some sufficient conditions on the permanence and partial extinction of the system are derived. Some interesting findings show that the delays have great impacts on the permanence for the system. More precisely, if τ2 ∈ (n, +∞ , then the system is partially extinct: on one hand, if τ1 ∈ (0, n1) and τ2 ∈ (n, + ), then the prey and predator species will coexist, i.e., both the prey and predator species are always permanent, yet the top pr∞ species will go extinct eventually. On the other hand, if τ1 ∈ (n4 , +∞ ) and τ2 ∈ (n, +∞ ), where n4 is edator n1, then all predator species will become extinct eventually. Numerical simulations are great well agreement with the theoretical results.