标题： 检测营养盐-浮游植物-浮游动物系统的最可能转变现象 显示英文标题

标题： Detecting the most probable transition phenomenon of a nutrient-phytoplankton-zooplankton system

摘要： 种群生物学模型在生态系统中占据重要地位。 将随机扰动引入模型可以更准确地描述真实的生物过程。 在本文中，我们主要研究三维营养盐-浮游植物-浮游动物（NPZ）浮游生物模型中最可能的转变现象。 在适当的参数值下，系统共存于一个稳定平衡点和一个稳定极限环。 在噪声扰动下，这两个稳态之间会发生转变。 基于Onsager-Machlup作用泛函和神经射击方法，我们研究了NPZ系统的最可能转变时间、最可能转变路径和最可能转变概率。 这些亚稳态之间的转变在随机生态系统中起着关键作用，为更好地理解复杂的生物过程提供了指导。 显示英文摘要

摘要： The population biology model holds a significant position within ecosystems. Introducing stochastic perturbations into the model can more accurately depict real biological processes. In this paper, we primarily investigate the most probable transition phenomenon in a three-dimensional nutrient-phytoplankton-zooplankton (NPZ) plankton model. With appropriate parameter values, the system coexists with a stable equilibrium point and a stable limit cycle. Under noise perturbations, transitions occur between these two steady states. Based on the Onsager-Machlup action functional and the neural shooting method, we have studied the most probable transition time, the most probable transition pathway and the most probable transition probability of the NPZ system. The transition between these metastable states plays a crucial role in stochastic ecosystems, providing guidance for a better understanding of complex biological processes.