With Much Of The Food Chain Model With Time-delay Stability And Hopf Bifurcation Analysis

In this paper, Stability and Hopf bifurcation for a class of mixed food chain system with multiple delays are studied.Firstly, The food chain system of different increasing function for prey and predator is established, and proves the positive solution of the system is the uniform bounded, with the characteristic root method and Hurwitz criterion, we obtain stability of boundary equilibrium point, moreover, the sufficient conditions of the existence of positive equilibrium point is given.Furthermore, the multiple delays in the system are divided into four cases to discuss. The first three kinds of cases are finally all four time delays into a delay, regarding the different delay as the bifurcation parameter, the sufficient conditions for existence of Hopf bifurcation is given in each case. Through the comparison, the conditions for the existence of Hopf bifurcation in the first two cases are consistent, while, conditions of existence of Hopf bifurcation in the third case are enhanced. The methods of these three cases are searching articles about the food chain system with multiple delays, thus the ways to deal with multiple delays are summed up. For the fourth case, the four time delays is regarded as two separate delay, the conditions for the existence of the Hopf bifurcation are studied. Conditions of existence of Hopf bifurcation are even more enhanced than in the first three cases. This part is equivalent to the food chain with two independent of the delays is studied. It is firstly studied the sufficient conditions of the existence of Hopf bifurcation in that. But the conditions for existence of Hopf bifurcation in these four cases are a certain connection. It is shown that the method of the existence of Hopf bifurcation for the food chain system with multiple delays is not unique. It is shown that the positive equilibrium would loss its stabilities and Hopf bifurcation then occurs the delay surpasses the critical values with critical conditions in four cases. Its biological significance showed that three groups will coexist in the form of periodic oscillations.Finally, by using the normal form method and center manifold theorem, the formula to determine the bifurcation direction, the stability of bifurcation periodic solutions, and period are studied in the second case, The research of the Hopf direction in this part is based on the conditions for the existence of Hopf bifurcation in Wang Xi, Li Wenling, so its conclusion have been improved and generalized by our results in multiple delays food chain.