Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

中文翻译：

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May's Host-Parasitoid 模型的稳定性与寄生蜂的可变放养

使用 Kolmogorov-Arnold-Mozer (KAM) 理论，我们研究了 May 的寄主-寄生蜂模型解决方案在寄生蜂种群上按比例放养的稳定性。我们展示了灭绝、边界和内部平衡点的存在。当宿主种群的内在增长率和释放系数小于1时，两个种群都灭绝了。有无数个边界平衡点，它们是非双曲线的且稳定的。在某些条件下，会出现 1:1 的非孤立共振不动点，我们对此进行了详尽的描述动力学。关于内部平衡点，我们使用 KAM 理论来证明其稳定性。我们给出了所得结果的生物学意义。使用软件包 Mathematica，我们生成数值模拟来支持我们的发现。