非局部扩散捕食-食饵模型的行波解及渐近行为 Alternative Title Traveling Wave Solutions and Asymptotic Behavior for Predator-prey Model with Nonlocal Dispersal 唐婉悦 Subtype 硕士 Thesis Advisor 杨飞英 2023-05-17 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 理学硕士 Degree Discipline 应用数学 Keyword 捕食-食饵模型 Predator-prey model 非局部扩散 Nonlocal dispersal 行波解 Traveling wave solutions Schauder不动点定理 Schauder's fixed-point theorem. Abstract 众所周知, 自然界中的许多现象都是个体之间的相互作用导致的, 而研究种群间相互关系(合作、竞争、捕食等)的种群生态学已经是目前数学上最为成熟的生态领域. 近期, 非局部扩散的引入可以有效的描述种群间在相邻空间位置的相互作用, 而行波解作为一类特解是研究扩散系统时空动力学非常重要的概念. 本文主要考虑两类具有非局部扩散三种群捕食-食饵模型的行波解. 首先, 本文考虑了一类具有非局部扩散捕食-食饵模型的行波解, 其中捕食者为外来入侵种群, 两个食饵在无捕食者情况下为弱竞争关系. 通过构造合适的上下解并结合Schauder不动点定理, 得到当波速大于等于临界波速时行波解的存在性, 并充分利用非局部扩散算子的性质, 构造适当的Lyapunov函数, 证明行波解连接由初始无捕食者状态到最终三物种共存状态. 这里需要特别指出, 由于非局部扩散算子的影响, 对于临界波速上下解的构造与相应局部问题存在不同. 之后利用非局部算子对应的特征值问题讨论行波解的不存在性, 最后利用比较原理讨论了带有初值条件的渐近传播速度. 其次, 本文考虑了具有非局部扩散三种群捕食-食饵模型的行波解的存在性与不存在性, 其中行波解的存在性表明一个捕食者与两个食饵同时入侵成功. 利用Schauder不动点定理结合上下解方法, 讨论行波解的存在性. 进一步, 通过渐近传播理论和压缩矩形的方法证明行波解的渐近行为, 并给出行波解的不存在性. 结果表明, 极小波速为三物种的入侵波速提供一个估计. Other Abstract It is well known that many phenomena in nature are caused by the interaction between individuals, and the study of the population ecology of inter-species relationship (cooperation, competition, predation, etc.) is one of the most mature field in mathematics. Recently, the introduction of nonlocal dispersal can effectively describe the interaction between the species in spatial, and the traveling wave solutions as a type of special solution is a very important concept of studying time and spatial dynamics of the diffusion system. This paper mainly considers two types of the traveling wave solutions for three species predator-prey model with nonlocal dispersal. First of all, this paper considers the traveling wave solutions for predator-prey system with nonlocal dispersal. In which the predator is an alien invasion species, and the two aboriginal preys are weakly competitive without predator. By constructing the suitable upper lower solutions and with the help of Schauder's fixed theorem, we get the existence of the traveling wave solutions when the speed of the traveling wave is greater than or equal to the speed of the critical wave speed, and make full use of the properties of the nonlocal dispersal operator to construct the appropriate Lyapunov function to show that the traveling wave solutions connect the predator-free state to the three species coexistence state. In particular, it is necessary to point out that due to the influence of nonlocal dispersal operator, the construction of the upper lower solutions of the critical wave speed is different from the corresponding local problems. Then, we use the eigenvalue of the nonlocal operator to discuss the nonexistence of traveling wave solutions. Finally, we use the comparison principal to study the asymptotic spreading speed of the system with initial value. Secondly, this paper considers the existence and nonexistence of the traveling wave solutions for three species predator-prey model with nonlocal dispersal. In which the existence of the traveling wave solutions indicates that one predator and two preys can invade successfully at the same time. We use the upper lower solutions and the Schauder's fixed-point theorem to discuss the existence of the traveling wave solutions. Furthermore, the asymptotic behavior of the traveling wave solutions is proved by the asymptotic propagation theory and the contracting rectangles, and then we give the nonexistence of the traveling wave solutions. The results show that the minimum wave speed gives an estimate for the invasion of three species. Subject Area 微分方程与动力系统 MOST Discipline Catalogue 理学 - 数学 - 应用数学 URL 查看原文 Language 中文 Other Code 262010_220200931660 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/536271 Collection 数学与统计学院 Affiliation 兰州大学数学与统计学院 Recommended CitationGB/T 7714 唐婉悦. 非局部扩散捕食-食饵模型的行波解及渐近行为[D]. 兰州. 兰州大学,2023.
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