关键词: 双时滞, Allee效应, Lotka-Volterra捕食-食饵系统, Hopf分支, 周期解

Abstract: Aiming at the problem that organism populations in nature were not able to react quickly to environmental changes or  interactions amongst populations. By introducing two time delays  as branching parameters, we analyzed the corresponding characteristic equations and  discussed  the local stability of the system at each equilibrium point and the existence of Hopf bifurcation. Firstly, we obtained explicit formulas that determined  the direction of Hopf bifurcation  and the stability of periodic solutions  when two time delays equal to  τ by using the central manifold theorem and canonical type theory. Secondly,  numerical simulation was used to verify the  accuracy of theoretical analysis. The results show that the stability of the system changes  and  a Hopf bifurcation is  generated when the time delay surpasses a critical value. Time delay is introduced  into biological models can help predict population dynamics more accurately.

Key words: two time delays, Allee effect, Lotka-Volterra predator-prey system, Hopf bifurcation, periodic solution