For the predator–prey model with the Sigmoid functional response, the known result is on the global stability of its positive equilibrium when it is locally stable. Here, we characterize existence of particular type of limit cycles using qualitative theory and geometric singular perturbation methods. The main results are as follows. If the positive equilibrium exists and is a weak focus, it is a stable weak focus of order 1. The positive equilibrium is unique, and it is either globally stable or unstable and there exists a limit cycle surrounding it. The limit cycle could be a canard cycle without head, or a canard cycle with head, or a relaxation oscillation. The system could present canard explosions consecutively two times, which first births from one canard point, via relaxation oscillation for a large range of parameter values, and then there exhibits an inverse canard explosion and disappears at another canard point with the parameter variation.

中文翻译：

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具有 Sigmoid 函数响应的捕食者-猎物模型的动力学

对于具有 Sigmoid 函数响应的捕食者-猎物模型，已知结果是其局部稳定时其正平衡的全局稳定性。在这里，我们使用定性理论和几何奇异摄动方法来表征特定类型的极限环的存在。主要结果如下。如果正均衡存在并且是弱焦点，则它是一阶稳定弱焦点。正均衡是唯一的，它要么是全局稳定的，要么是不稳定的，并且存在一个极限环。极限环可以是无头的鸭式循环，或带头的鸭式循环，或弛豫振荡。该系统可以连续两次呈现鸭翼爆炸，首先从一个鸭翼点产生，通过大范围参数值的弛豫振荡，