The Classification on the Global Phase Portraits of Two-dimensional Lotka–Volterra System
The Classification on the Global Phase Portraits of Two-dimensional Lotka–Volterra System
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DOI：
10.1007/s10884-008-9122-5
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发表时间：
2008-07
影响因子：
1.3
通讯作者：
Feng Cao;Jifa Jiang
中科院分区：
文献类型：
--
作者：
Feng Cao;Jifa Jiang
给出了二次Lotka-Volterra系统{\dot{x}=x（a_{0}+a_{1}x+a_{2}y），\，\dot{y}=y（B_{0}+B_{1}x+B_{2}y）}全局相图的一个完整分类.在庞加莱圆盘中有143个不同的全局相图，直到所有轨道的方向反转。首先给出了系统具有闭轨的充要条件。然后讨论了无穷远临界点和有限远临界点的性质。所有全局相位图最终通过结合所有局部信息进行分类。
This paper provides a complete classification on the global phase portraits of quadratic Lotka–Volterra system $${\dot{x}=x(a_{0}+a_{1}x+a_{2}y),\, \dot{y}=y(b_{0}+b_{1}x+b_{2}y)}$$ . There are 143 different global phase portraits in the Poincaré disc up to a reversal of sense of all orbits. Firstly the sufficient and necessary conditions of the system with closed orbits are given. Then the properties on infinite and finite critical points are discussed. All global phase portraits are finally classified by combining all local information.