Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response☆
Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response☆
复制标题
DOI：
10.1016/j.jde.2014.04.024
复制
发表时间：
2014-09
影响因子：
2.4
通讯作者：
Jicai Huang;S. Ruan;Jing Song
中科院分区：
文献类型：
--
作者：
Jicai Huang;S. Ruan;Jing Song
我们考虑具有广义 Holling III 型功能响应 p (x)= m x 2 a x 2+ b x+ 1 的 Leslie 型捕食者-被捕食系统。通过允许 b 为负数 (b>− 2 a)，p (x) 对于 b> 0 是单调的，当 x≥ 0 时，对于 b< 0 是非单调的。该模型有两个非双曲正平衡（一个是多重焦点，另一个是多重焦点）对于某些参数值，余维 2 的尖点和对于其他参数值，余维 3 的退化 Bogdanov-Takens 奇点（焦点或中心情况）。当存在重数 1 的多重焦点和余维 2 的尖点时，我们表明该模型分别在两个简并平衡点的相应小邻域中同时表现出亚临界 Hopf 分岔和 Bogdanov-Takens 分岔。通过计算机数值模拟获得了模型的不同相图，表明该模型可以具有：（i）包含两个非双曲正平衡点的稳定极限环；（ii）包含一个不稳定同宿环的稳定极限环；（iii）包含一个双曲正平衡点的两个极限环；（iv）包含三个双曲正平衡点的一个稳定极限环；（v）包含两个非双曲正平衡点的稳定极限环。 (v) 三个稳定状态的共存（两个稳定平衡和一个稳定极限环）。当模型具有余维3的Bogdanov-Takens奇点时，我们证明该模型表现出余维3的简并焦点型Bogdanov-Takens分岔。这些结果不仅证明了当b>− 2 a时该模型的动力学比b> 0时的情况更加复杂和丰富，而且为捕食者-被捕食系统提供了新的分岔现象。
We consider a predator–prey system of Leslie type with generalized Holling type III functional response p (x)= m x 2 a x 2+ b x+ 1. By allowing b to be negative (b>− 2 a), p (x) is monotonic for b> 0 and nonmonotonic for b< 0 when x≥ 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov–Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov–Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have:(i) a stable limit cycle enclosing two non-hyperbolic positive equilibria;(ii) a stable limit cycle enclosing an unstable homoclinic loop;(iii) two limit cycles enclosing a hyperbolic positive equilibrium;(iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov–Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>− 2 a are much more complex and far richer than the case when b> 0 but also provide new bifurcation phenomena for predator–prey systems.