Hopf bifurcation of a three-dimensional Lotka-Volterra cooperative system with time delay and linear harvesting terms
This article mainly studies the Hopf bifurcation of the three-dimensional Lotka-Volterra coop-erative system.Firstly,the time delay term and linear harvesting term are introduced to improve the three-di-mensional Lotka-Volterra cooperative system.Secondly,taking the time delay τ as the bifurcation parameter,the existence of local Hopf bifurcation of the improved system is analyzed and discussed,and the critical value for the system to produce Hopf bifurcation at the positive equilibrium point is given.Then,using the center manifold theorem and normal form theory,the formula for Hopf bifurcation direction and stability of periodic solutions is calculated.Finally,numerical simulation is performed.According to the results of numerical sim-ulation,the feasibility of theoretical analysis conclusions is verified.The results show that when τ increases from 0 to pass the critical value,the positive equilibrium point of the system changes from a stable state to an unstable state,and Hopf bifurcation occurs at the positive equilibrium point.This indicates that time delay af-fects the dynamic behavior of the system,making its dynamic properties more complex.
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