We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of singular Hopf bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from geometric singular perturbation theory. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.

中文翻译：

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两时间尺度生态系统中奇异 Hopf 分岔附近的复杂振荡模式

我们考虑一个由两种捕食者组成的生态模型，它们通过明确的干扰竞争来争夺共同的猎物。通过适当的重新缩放，该模型被描绘成一个奇异扰动系统，具有一个快速变量（猎物动态）和两个慢变量（捕食者动态）。该模型展示了各种丰富而有趣的动力学，包括但不限于混合模式振荡 (MMO)，具有小振幅和大振幅振荡的串联、弛豫振荡以及半平凡平衡状态和共存振荡状态之间的双稳态. 更有趣的是，在奇异的 Hopf附近当系统分别接近由超临界 Hopf 分叉或半平凡平衡状态产生的周期性吸引子时，观察到分叉、混沌 MMO 形式或弛豫振荡形式的持久瞬态动力学。在生态系统经历制度转变之前，瞬态动态可能会持续数百或数千代。该模型中出现的具有不同类型不规则振荡的种群周期时间序列源于生物现实特征，即捕食者之间种内竞争的变化。为了解释这些振荡，我们使用分岔分析和几何奇异摄动理论的方法. 数值连续研究揭示了系统中丰富的分岔结构，包括折叠-Hopf和广义Hopf分岔等余维二分岔的存在。