**Volume 10, Issue 12, pp 6177--6191**

**Publication Date**: 2017-12-06

http://dx.doi.org/10.22436/jnsa.010.12.05

Huanyan Xi - Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China

Lihong Huang - Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China

Yuncheng Qiao - Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China

Huaiyu Li - Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China

Chuangxia Huang - Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China

By taking full consideration of maturity (\(\tau_{1}\) represents the maturity of predator and \(\tau_{2}\) 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 \(\tau_{2}\in(n, +\infty)\), then the system is partially extinct: on one hand, if \(\tau_{1}\in(0,n_{1})\) and \(\tau_{2}\in(n, +\infty)\), then the prey and predator species will coexist, i.e., both the prey and predator species are always permanent, yet the top predator species will go extinct eventually. On the other hand, if \(\tau_{1}\in(n_{4},+\infty)\) and \(\tau_{2}\in(n, +\infty)\), where \(n_{4}\) is greater than \(n_{1}\), then all predator species will become extinct eventually. Numerical simulations are great well agreement with the theoretical results.

Food chain model, delay, stage structure, permanence, extinction