非局部扩散DS-I-A模型的行波解和渐近传播问题 Alternative Title Spreading speeds and traveling waves of nonlocal dispersal DS-I-A models 邓熙霖 Subtype 硕士 Thesis Advisor 林国 2021-05-21 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 理学硕士 Degree Discipline 应用数学 Keyword 非局部扩散 非单调 行波解 渐近传播速度 传染病模型 Abstract 本文主要考虑了如下具有非局部扩散的DS-I-A模型的空间传播动力学, 其中所有参数非负, 在该模型中易感者根据其特性的不同被区分为不同的组群. 从单调动力系统的角度来看, 当互不相同时, 该模型一般不具有比较原理. 本文利用行波解和初值问题渐近传播速度描述了上述模型的空间传播动力学, 借此刻画了疾病从无到有最终成为地方病的传播过程中的阈值行为. 本文首先考虑了行波解的最小波速问题. 在单调性缺失的情况下证明波速不小于某个确定阈值时行波解的存在性, 主要通过构造一对广义上下解并使用Schauder不动点定理得到了非常数行波解的存在性, 进而借助渐近传播理论得到了行波解的渐近行为. 最后还说明了当波速小于该阈值时非平凡行波解的不存在性, 进而得到了行波解的最小波速. 接下来研究了当感染者对应的初值具有非空紧支集时, 初值问题的渐近传播速度. 首先通过一对广义上下解证明了感染者传播速度的上界不会超过行波解的最小波速, 其次通过构造适当的辅助方程证明了感染者的传播速度不会慢于最小波速, 从而证明了初值问题的渐近传播速度与行波解的最小波速相同这一结论. Other Abstract This paper studies the spatial propagation dynamics of the following DS-I-A model with non-local diffusion，in which all the parameters are nonnegative. The susceptible is divided into different groups according to their characteristics. From the viewpoint of monotonic dynamic systems, this system does not satisfy the comparison principle when are different. We formulate the propagation dynamics of this system by traveling wave solutions and asymptotic spreading in the corresponding initial value problems, which models the transition process from disease free steady state to epidemic state. We first consider the minimal wave speed of traveling wave solutions. Without the monotonicity, we show the existence of nontrivial traveling wave solutions if the wave speed is not less than a given threshold, which is finished by generalized upper-lower solutions and Schauder's fixed point theorem. Then the asymptotic behavior is obtained by the theory of asymptotic spreading. When the wave speed is less than the threshold, we confirm the nonexistence of traveling wave solutions, which implies that the minimal wave speed equals to the threshold. We then study the asymptotic spreading of initial value problem. When the initial condition of the infected admits nonempty compact support. By constructing a pair of generalized upper-lower solutions, the upper bounds of spreading speed is obtained, which is not larger than the minimal wave speed. Moreover, the lower bounds of spreading speed is given by showing proper auxiliary equations, which indicates the invasion speed of the infected is not less than the minimal wave speed. Thus, the minimal wave speed of traveling wave solutions equals to the spreading speed in this model. Pages 48 URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/460870 Collection 数学与统计学院 Affiliation 数学与统计学院 First Author Affilication School of Mathematics and Statistics Recommended CitationGB/T 7714 邓熙霖. 非局部扩散DS-I-A模型的行波解和渐近传播问题[D]. 兰州. 兰州大学,2021.
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