捕食者-食饵系统的行波解与渐近传播

项目名称： 捕食者-食饵系统的行波解与渐近传播

项目编号： No.11461040

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 潘书霞

作者单位： 兰州理工大学

项目金额： 36万元

中文摘要： 捕食者-食饵系统是描述自然现象的重要模型，与竞争系统、合作系统相比较，此类系统因单调性条件特殊使得相关研究有其自身的复杂之处，许多重要模型的空间传播性质还未被研究，对一些系统的探讨主要集中在行波解存在性因而时空传播阈值理论也不够完善。本项目以建立两种群及多种群系统行波解的最小波速、对捕食者和食饵的渐近传播速度进行估计为具体目标。在行波解研究中，将使用广义上下解解决存在性。对于渐近行为以及最小波速，将在使用比较原理的基础上，结合动力系统理论及经典渐近传播理论进行研究。在渐近传播研究中，将结合算子半群理论、捕食者-食饵系统的比较原理、非自治方程的传播理论、经典渐近传播理论对于不同未知函数进行估计，并着力体现其中非线性项的非平凡作用。在本项目中，对多种群耦合系统时空传播阈值理论的研究以及要体现的非线性项复杂性是项目的特色。

中文关键词： 行波解；渐近传播；反应扩散方程；动力系统；单调半流

英文摘要： In applied mathematics, predator-prey systems are very important to model the enery transmission, and they admit more complex dynamical behavior than the competitive and cooperative systems. In particular, the special monotone conditions of predator-prey systems lead to much difficulty in formulating the spatial propagation of them. The goal of this proposal is to investigate the spatial propagation of predator-prey systems by combining the theory of partial differential equations with the method of monotone dynamical systems. We will establish the minimal wave speed of traveling wave solutions and estimate the asymptotic speed of spreading of predator-prey systems with two or multi species. To obtain the existence of traveling wave solutions, the generalized upper and lower solutions will be utilized. In the study of asymptotic behavior and minimal wave speed of traveling wave solutions, we will apply the method of monotone dynamical systems and the classical theory of asymptotic spreading of KPP equation. The asymptotic spreading will be investigated by the theory of semigroups of linear operators, comparison principle appealing to the predator-prey systems and the asymptotic spreading of nonautonomous equations. The proposed work will lead to a better understanding of the dynamical behavior of predator-prey systems from the viewpoint of spatial propagation, especially to the dynamics of multi species modles and the role of coupled nonlinearities.

英文关键词： Traveling Wave Solutions;Asymptotic Spreading;Reaction-Diffusion Equations;Dynamical Systems;Monotone Semiflows