具有时滞和线性收获项的三维Lotka-Volterra合作系统的Hopf分支  

Hopf bifurcation of a three-dimensional Lotka-Volterra cooperative system with time delay and linear harvesting terms

作　　者：武微 吕堂红[1] WU Wei;LYU Tanghong(School of Mathematics and Statistics,Changchun University of Science and Technology,Changchun 130022,China)

机构地区：[1]长春理工大学数学与统计学院,吉林长春130022

出　　处：《陕西理工大学学报（自然科学版）》2024年第3期84-94,共11页Journal of Shaanxi University of Technology:Natural Science Edition

基　　金：国家自然科学基金项目(11426045);吉林省教育厅科学技术研究项目(JJKH20240891KJ)。

摘　　要：针对三维Lotka-Volterra合作系统的Hopf分支进行了研究。首先,在三维Lotka-Volterra合作系统的基础上,引入时滞项及线性收获项对该系统进行改进;其次,以时滞τ作为分支参数,对改进后系统的局部Hopf分支的存在性进行分析,给出使系统在正平衡点处产生Hopf分支的临界值;然后,利用中心流形定理和规范型理论计算出Hopf分支方向及周期解稳定性的公式;最后,进行数值模拟,根据数值模拟的结果,验证了理论的可行性。结果表明,当时滞τ从0一直增加到超过临界值时,系统的正平衡点由稳定状态变为不稳定状态,且在正平衡点处产生Hopf分支。说明时滞影响了系统的动力学行为,使其动力学性质更加复杂。This article mainly studies the Hopf bifurcation of the three-dimensional Lotka-Volterra cooperative system.Firstly,the time delay term and linear harvesting term are introduced to improve the three-dimensional 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 simulation,the feasibility of theoretical analysis conclusions is verified.The results show that whenτincreases 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 affects the dynamic behavior of the system,making its dynamic properties more complex.

关 键 词：Lotka-Volterra合作系统 时滞 HOPF分支 稳定性 

分 类 号：O175[理学—数学]