具有双时滞及Holling Ⅱ型功能反应捕食模型的Hopf分支

Hopf Bifurcation in Predator-prey Model with Two Time Delays and Holling Ⅱ Type Functional Response

探讨一类有双时滞项和Holling Ⅱ型功能反应函数的三维捕食模型,讨论该模型在平衡点处的稳定性和Hopf分支的存在性;以正平衡点作为研究对象,分析系统的特征方程,当时滞项不存在时,根据Hurwitz判据得到该模型在正平衡点的渐近稳定性条件;当时滞项存在时,由Bulter引理判定系统在正平衡点的稳定性;以双时滞为Hopf分支参数,得到系统在正平衡点处发生Hopf分支的临界值,当时滞超过临界值并分支出周期解时,取适当的参数和不同的时滞值对该模型进行数值模拟,得到系统在临界值附近的各分量变化图和解曲线走势图。结果表明,随着分支参数值的变化,系统的稳定性会发生变化,同时系统也会产生Hopf分支。

A three-species predator-prey model with two time delays and Holling type-II functional response was ana-lyzed. The stability of model at positive equilibrium point and the existence of Hopf bifurcation were discussed. The equi-librium point was taken as the research object, and the characteristic equation of the system was analyzed. When the de-lay was equal to zero, the asymptotic stability condition at the positive equilibrium point was obtained according to Hur-witz criterion. When the delay existed, the stability of the system was judged at the positive equilibrium point according to Bulter lemma. Taking two time delays as Hopf bifurcation parameters, the critical value of the system existing Hopf bifur-cation at the positive equilibrium point was obtained. When the delay was greater than the critical value and the periodic solution existed, the model was numerically simulated by taking appropriate parameters and different time delay values, and diagrams of all components change and solution curves were given around the critical value. The results show that with the variation of branch parameters, the stability of the system changes and Hopf bifurcation is produced.