Oto-Escaff-Cisternas模型的定性分析

The Qualitative Analysis of Oto-Escaff-Cisternas Model

考虑Oto-Escaff-Cisternas植物–食草动物模型的动力学性质。首先,讨论了模型解的正性及有界性,分析对应平衡点处雅可比矩阵的特征值,给出了模型平衡点的拓扑类型;其次,应用分支分析的理论及解析方法,讨论了该模型的鞍结点分支和Hopf分支,得到了正平衡点的稳定性和Hopf分支的存在条件;再次,通过计算第一李雅普洛夫系数,证明了该系统会发生亚临界Hopf分支,并且在正平衡点附近产生了唯一一个不稳定的极限环。最后,对所获得的理论结果给出了数值模拟。

In this paper, the dynamics properties of the Oto-Escaff-Cisternas plant-herbivore model are con-sidered. Firstly, the positiveness and boundedness of the model solution are discussed, the eigen-values of the Jacobian matrix at the corresponding equilibrium point are analyzed, and the topolog-ical type of the model equilibrium point is given. Secondly, via the analytical method of bifurcation analysis, the saddle node bifurcation and Hopf bifurcation of the model are discussed, and the sta-bility of the positive equilibrium point and the existence conditions of the Hopf bifurcation are ob-tained. Furthermore, by calculating the first Lyapunov coefficient, it is proved that the system will have a subcritical Hopf bifurcation and produce the only unstable limit ring near the positive equi-librium point. Finally, some numerical simulations were carried out to illustrate our theoretical re-sults.