在本文中,我们开发了一种在一类捕食者-食饵模型中分析长瞬态动力学的方法,其中两种捕食者明确竞争它们的共同食饵,猎物比捕食者进化得更快。在共存平衡状态的奇异零Hopf分叉附近的参数状态中,我们假设所研究的系统在从奇异Hopf点分叉的周期吸引子和另一个吸引子之间表现出双稳态,可以是周期性吸引子或点吸引子,这样,共存平衡点的不变流形在组织动力学中起着核心作用。要确定在共存平衡附近开始的解是否接近周期性吸引子或其他吸引子,我们将方程简化为合适的正规形式,并检查奇异Hopf点附近的盆地边界。我们研究的一个关键组成部分包括对长瞬态动力学的分析,其特征是振幅缓慢变化的快速振荡,通过应用移动平均技术。我们在共存平衡附近的解的初始值上获得了一组必要和充分条件,以确定它是否位于周期性吸引子的吸引盆地中。根据我们的分析,我们设计了一种识别预警信号的方法,明显提前,未来的危机可能导致其中一只捕食者的灭绝。该分析应用于Sadhu(DiscreteContinDynSystB26:5251-5279,2021)中考虑的捕食者-食饵模型,我们发现我们的理论与该模型进行的数值模拟非常吻合。
In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.