一类具有相互干扰和Holling-III功能反应模型的定性分析

Qualitative Analysis on a Class of Holling-III Function Response Models with Mutual Interference

摘要: 通过分析模型在平衡点处的线性近似系统，发现该模型在某些条件下具有独特的正平衡点。当给定模型具有正平衡点时，运用Bendixson-Dulac定理获得系统在第一象限中的闭合轨道。借助Liénard变换，利用张芷芬的唯一性定理得到该系统极限环的存在性和唯一性。最后，利用Mathematica软件对模型进行了四组数值模拟，验证了所得结论的合理性。

Abstract: In this paper, by analyzing the linear approximation system of the model at the equilibrium point, we first find that the model has a unique positive equilibrium point under some conditions. When the given model has positive equilibrium point, the closed orbit of the system in the first quadrant is obtained via Bendixson-Dulac theorem. By using the Liénard transformation and the uniqueness theorem of Zhang Zhifen, the existence and uniqueness of the limit cycle of the system is obtained. Finally, four numerical simulations of the model are given to illustrate the validity of our results by using Mathematica software.