The Behavior Of Several Types Of Food Chain Model With Time Delay Analysis

In the studies of population ecology,a lot of the time delayed phenomenon exist inthe world,which lead to the loss of stability and get the periodic solutions,and better re-?ect the actual growth of population, so play an important role in the growth of pop-ulation.Therefore,it is necessary to consider the pregnancy reaction of population in thestudy,and further study the population ecology model of time-delay.In this paper,according to the study of the predecessors,we mainly introduce the follow-ing questions:First,a prey with a normal birth and death rates of the delayed food chain model.Byregarding the delay as the bifurcation parameter,the stability of the positive equilibrium andthe existence of Hopf bifurcation are investigated.Then,using the normal form theory andthe center manifold reduction,the direction of bifurcation and the stability of the bifurcatingperiodic solutions are studied. Finally,numerical simulation results are given to support thetheoretical results.Then on this basis,a prey with a sparse e?ect and a shelter e?ect of food chain model.Thesparse e?ect and the shelter e?ect of the population have a direct impact on the extinctionand existence of species,so there is a very important research significance.For the sparse e?ectof the model,using the extension theorem of Mawhin the continuation theorem based oncoincidence degree theory,existence of positive periodic solutions of system are proved,andby constructing appropriate Lyapunov functions uniqueness of periodic solutions and theglobal stability of the equilibria are investigated.For the shelter e?ect of the model,basingon the first model,we study the stability of the equilibrium and bifurcation.The results have a certain reference value for the biological control in the theory, theyalso practically have a certain guiding significance for the protection and preventing theextinction context of rare animals.