arXiv 1806.044012018-06-12math.DS

具有广义Holling型III功能反应的Leslie型捕食系统的四个极限环

Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response

Yanfei Dai, Yulin Zhao, Bo Sang

备注：: 43 pages, 2 figures. arXiv admin note: text overlap with arXiv:1806.04390

摘要：

本文作为对Hsu et al [SIAM. J. Appl. Math. 55（1995）]和Huang et al [J.微分方程257（2014）]，旨在研究具有广义Holling III型功能反应的Leslie型捕食者-食饵模型在两种情况下的Hopf分支和全局动力学：（A）当它具有唯一的非退化正平衡点时;（B）当它具有三个不同的正平衡点时。对每种情形，研究了平衡点的类型、稳定性、弱焦点处的Hopf分支以及第一象限内极限环的个数和分布.它表明，每个平衡是不是一个中心。对于情形（A），$i$极限环可能出现在唯一正平衡点$i=1, \cdots, 4$附近。对于$i=3$或$4$，该模型存在两个稳定的极限环，从而肯定地回答了Coleman [微分方程模型，1983]提出的公开问题：在任何自然（或实验室）的捕食者-食饵系统中至少找到两个生态稳定的环.对于情形（B），一个正平衡点是鞍点，其他两个都是反鞍点。如果两个反鞍点中的一个是弱焦点而另一个不是，则弱焦点的阶数至多为$3$。如果两个反鞍点都是弱焦点，则它们是一阶不稳定弱焦点。此外，一个极限环可以同时从它们中的每一个分叉。数值模拟结果表明，在这两个极限环之外还存在一个大的稳定极限环。结果表明，这类模型的最大极限环个数至少为$4$，改进了已有的最大极限环个数至少为$2$的结果.

英文摘要：

This paper, as a complement to the works by Hsu et al [SIAM. J. Appl. Math. 55 (1995)] and Huang et al [J. Differential Equations 257 (2014)], aims to examine the Hopf bifurcation and global dynamics of a predator-prey model of Leslie type with generalized Holling type III functional response for the two cases: (A) when it has a unique non-degenerate positive equilibrium; (B) when it has three distinct positive equilibria. For each case, the type and stability of each equilibrium, Hopf bifurcation at each weak focus, and the number and distribution of limit cycles in the first quadrant are studied. It is shown that every equilibrium is not a center. For the case (A), $i$ limit cycle(s) can appear near the unique positive equilibrium, $i=1, \cdots, 4$. For $i=3$ or $4$, the model has two stable limit cycles, which gives a positive answer to the open problem proposed by Coleman [Differential equations model,1983]: finding at least two ecologically stable cycles in any natural (or laboratory) predator-prey system. For the case (B), one positive equilibrium is a saddle and the others are both anti-saddle. If one of the two anti-saddles is weak focus and the other is not, then the order of the weak focus is at most $3$. If both anti-saddles are weak foci, then they are unstable weak foci of order one. Moreover, one limit cycle can bifurcate from each of them simultaneously. Numerical simulations demonstrate that there is also a big stable limit cycle enclosing these two limit cycles. Our results indicate that the maximum number of limit cycles in the model of this kind is at least $4$, which improves the preceding results that this number is at least $2$.

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