一类随机捕食–被捕食模型的持久性与灭绝性研究

Research on Persistence and Extinction of a Class of Stochastic Predator-Prey Models

摘要: 为探究环境随机性对种群动态的影响，本文考虑了一类带有Holling II型功能反应项和Leslie-Gower增长项的混合模型，且随机噪声仅作用于捕食过程的随机捕食–被捕食模型。针对所建模型，对其有界性、随即持久性与灭绝性进行研究，通过局部Lipschitz连续性讨论全局正解的唯一性，利用构造合适的Lyapunov函数来证明模型的有界性，并结合Itô公式和布朗运动的性质等，重点分析了模型的随机持久性与灭绝性。结果表明：捕食者种群是随机最终有界的，同时被捕食者种群是随机持久的，且在特定条件下被捕食者种群将以概率1灭绝。

Abstract: To investigate the impact of environmental randomness on population dynamics, this study considers a hybrid stochastic predator-prey model that incorporates a Holling Type II functional response term and a Leslie-Gower growth term, with random noise acting exclusively on the predation process. For the constructed model, research is conducted on its boundedness, stochastic persistence, and extinction. The uniqueness of the global positive solution is discussed by leveraging local Lipschitz continuity; the boundedness of the model is proven through the construction of an appropriate Lyapunov function; and the stochastic persistence and extinction of the model are analyzed in focus by combining Itô’s formula, the properties of Brownian motion, and other relevant methods. The results demonstrate that the predator population is stochastically ultimately bounded, while the prey population exhibits stochastic persistence. Furthermore, under specific conditions, the prey population will go extinct with probability one.