一类具有恐惧效应和额外食物项的捕食生态模型的分岔与多稳态研究

Bifurcation and Multi Steady State of a Predatory Ecological Model with Fear Effect and Additional Food Item

摘要: 本文提出了一类具有恐惧效应和额外食物项的捕食者–食饵生态模型，对其特定动力学行为进行理论分析与数值模拟。数学理论工作讨论了模型解的正性、一致有界性、平衡点的存在性和稳定性，分析模型发生跨临界分岔、鞍结分岔和霍普夫分岔的充分条件。数值模拟结果发现：额外食物项参数和恐惧效应强度是调控系统动力学拓扑行为的关键因素。同时，从一维全局分岔及相平面轨迹的角度，揭示了种群生长共存模式、多稳态转换与灾难性相变的内在驱动机制。这些研究结果为拓宽捕食生态模型复杂非线性动力学问题的研究框架提供一定的理论支撑。

Abstract: This paper proposed a predator-prey ecological model with fear effect and additional food item, and conducted theoretical analysis and numerical simulation of its specific dynamic behavior. The mathematical theory work discussed the positivity and consistent boundedness of model solutions, the existence and stability of equilibrium points, and analyzed the sufficient conditions for the occurrence of transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation. The numerical simulation results found that additional food item parameter and fear effect intensity were key factors for regulating the topological behavior of the system dynamics. Meanwhile, from the perspectives of one-dimensional global bifurcation and phase plane trajectory, the intrinsic driving mechanisms of population growth coexistence modes, multi steady state transition, and catastrophic phase transition were revealed. These research results could provide theoretical support for expanding the research framework of complex nonlinear dynamics problems in predator-prey ecological models.