Permanence and partial extinction in a delayed threespecies food chain model with stage structure and timevarying coefficients
Authors
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
Abstract
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 threespecies food chain model with stage structure and timevarying 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.
Keywords
 Food chain model
 delay
 stage structure
 permanence
 extinction
MSC
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