一类捕食模型的空间动力学研究

Spatial Dynamics of a Class of Diffusive Predator-Prey Models

摘要: 通过构建一类具有空间扩散的捕食模型。首先，结合Neumann边界条件与Banach空间理论，通过Lipschitz连续性理论，获得模型解的适定性，进一步地分析模型平衡点存在性。其次，借助特征方程证明存在的平衡点是局部渐近稳定的；最后，通过构造Lyapunov泛函证明系统存在的平衡点是全局渐近稳定性。研究的结果丰富了捕食模型的空间动力学行为，对种群物种可持续发展。

Abstract: By constructing a class of predator-prey models with spatial diffusion, we first establish the well-posedness of solutions using Neumann boundary conditions, Banach space theory, and Lipschitz continuity theory, and further analyze the existence of equilibria. Next, with the aid of the characteristic equation, the existing equilibria are proved to be locally asymptotically stable. Finally, by constructing a Lyapunov functional, the equilibria of the system are shown to be globally asymptotically stable. The results enrich the spatial dynamics of predator-prey models and contribute to the sustainable development of populations and species.