In this paper, we consider a predator–prey system with Holling type III ratio-dependent functional response. Such a system can exhibit complex dynamical behavior such as bistable and tristable phenomena which contain equilibria and oscillating motions for certain parameter values. In particular, we show that the ratio-dependent predator–prey system can exhibit multiple limit cycles due to Hopf bifurcation, giving rise to coexistence of stable equilibria and stable periodic solutions. These solutions may reveal some new type of patterns of complex dynamical behaviors in predator–prey systems.

中文翻译：

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多极限环分岔引起的捕食者-猎物系统中的三稳态现象

在本文中，我们考虑了具有 Holling III 型比率依赖性功能响应的捕食者 - 猎物系统。这样的系统可以表现出复杂的动力学行为，例如双稳态和三稳态现象，其中包含某些参数值的平衡和振荡运动。特别是，我们表明，由于 Hopf 分岔，比率依赖的捕食者 - 猎物系统可以表现出多个极限环，从而导致稳定平衡和稳定周期解的共存。这些解决方案可能会揭示捕食者-猎物系统中复杂动态行为的一些新型模式。