具有时滞的比率型三种群捕食模型的分支分析
BIFURCATION ANALYSIS FOR A THREE-SPECIES RATIO-DEPENDENT PREDATOR-PREY MODEL WITH DELAY

本文研究了一类具有时滞的比率型三种群捕食模型.通过分析该模型的特征方程，证明了该模型在正平衡点的稳定性.通过选择时滞τ为分支参数，得到了当时滞τ通过一系列的临界值时，Hopf分支产生.应用中心流形和规范型理论，得到了关于确定Hopf分支特性的计算公式.最后进行数值模拟验证了我们所得结果的正确性.所得结果是对前人工作的补充.
In this paper, a class of three-species ratio-dependent predator-prey model is investigated. By analyzing the characteristic equation of the model, the stability at the positive equilibrium is proved. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur when the delay τ passes a sequence of critical values. Meanwhile, using the center manifold theory and normal form approach, we derive the formulae for determining the properties of Hopf bifurcating periodic orbit, such as the direction of Hopf bifurcation, the stability of Hopf bifurcating periodic solution and the periodic of Hopf bifurcating periodic solution. Finally, numerical simulations are carried out to illustrate the analytical results. Our results complement some previously known ones