In this paper, we study the dynamic behaviors of a predator–prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [ Zhu et al., 2002 ; Xiao & Ruan, 2001 ]. Finally, we give a brief discussion and numerical simulation.

中文翻译：

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具有非单调函数响应的捕食者-猎物系统的高维分岔分析

在本文中，我们研究了具有一般形式的非单调函数响应的捕食者 - 猎物系统的动态行为。通过分析发现，系统存在消光平衡、边界平衡和两个正平衡，一个或没有一个正平衡。此外，还给出了边界平衡为余维数为 1、2 或 3 的鞍点、节点或鞍-节点点的条件。然后，得到了一些条件，使得系统的唯一正平衡点为余维2、3及以上或余维1或3中的一个鞍节点，或余维4的幂零鞍节点。当系统中有两个正平衡时，有大量猎物的平衡是鞍点. 另一方面，系统可能会经历 Hopf 分岔。为了验证我们的结论，我们考虑了文献中的功能反应函数 [Zhu et al., 2002; 肖、阮，2001]. 最后，我们给出一个简短的讨论和数值模拟。