In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

中文翻译：

---

具有简化 Holling IV 型函数响应的非线性捕食者-猎物模型的稳定性和分岔分析

在本文中，通过添加非线性 Michaelis-Menten 型猎物捕获来探索捕食者 - 猎物系统的动力学，改进了一种具有简化 Holling IV 型功能响应的捕食者 - 猎物模型。首先分析了不同平衡点存在的条件，考察了可能平衡点的稳定性，以预测系统的最终状态。其次，研究了该系统的分岔行为，利用索托马约尔定理，发现在某些参数值的条件下会出现鞍节点分岔和跨临界分岔；计算第一个 Lyapunov 常数以确定 Hopf 分岔的分岔极限环的稳定性；基于二参数分岔分析定理，通过计算尖点附近的普遍展开来探索余维2的排斥和吸引Bogdanov-Takens分岔，因此模型具有极限环或同宿环的不同参数值；通过分析改进系统的等斜线，还预测随着参数值的变化，可能会出现异宿分叉。最后通过数值模拟验证了理论分析。