具时滞和避难所的Lotka-Volterra竞争模型的稳定性与Hopf分支  

Stability and Hopf bifurcation of the Lotka-Volterra competition model with time delay and refuge

作　　者：吕堂红[1] 衣薪燃 LYU Tang-hong;YI Xin-ran(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：《吉林师范大学学报（自然科学版）》2024年第2期50-59,共10页Journal of Jilin Normal University:Natural Science Edition

基　　金：国家自然科学基金项目(11426045);吉林省科技发展计划项目(20180101229JC)。

摘　　要：以时滞τ为分支参数,研究了时滞对具有避难所和线性收获项的Lotka-Volterra竞争模型的动力学性质产生的影响.首先利用特征根法及霍尔维兹判据,分析了系统正平衡点的稳定性,得到了保证局部渐近稳定的充分条件;其次应用中心流形定理和规范型理论,研究了正平衡点处产生的Hopf分支性质,给出了Hopf分支的方向及周期解稳定性的表达式;最后通过数值模拟验证了理论的可行性.结果表明,时滞会对系统的动态性态产生影响.在Lotka-Volterra模型的基础上考虑时滞效应,有助于更好地理解并预测种群的变化趋势.The effect of the time lag on the nature of the dynamics of the Lotka-Volterra competition model with refuge and linear harvest terms was investigated using the time lagτas a branching parameter.Firstly,the stability analysis of the positive equilibrium point of the system was conducted using the eigenroot method and the Horwitz criterion.This approach yielded adequate conditions to guarantee local asymptotic stability;Secondly,the properties of Hopf branches generated at the positive equilibrium point were studied by applying the central manifold theorem and the normal-type theory,and the expressions of Hopf branch direction and periodic solution stability were provided;Finally,the feasibility of the theory was verified by numerical simulation.The results showed that the time lag could have an effect on the dynamic behaviour of the system.When the time delay was small(less than the critical value),the system maintained a stable state.Hopf branches and periodic solutions occurred when the time lag reached or even exceeded the critical value.Considering time lag effects on the basis of the Lotka-Volterra model might help to understand and predict population trends.

关 键 词：Lotka-Volterra竞争模型 时滞 HOPF分支 周期解 稳定性 

分 类 号：O175.1[理学—数学]