兰州大学机构库 >数学与统计学院
一类非局部扩散捕食-食饵模型的行波解
Alternative TitleTraveling Wave Solutions of a Predator-Prey Model with Nonlocal Dispersal
郝玉霞
Thesis Advisor李万同
2018-04-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword捕食-食饵模型 非局部扩散 行波解 Schauder's不动点定理 渐近行为 上下解方法 Schauder不动点定理.
Abstract

非局部算子相较经典的\,Laplace\,算子而言, 更有助于精确地描述种群在空间上的非局部作用, 越来越多的非局部扩散模型被用于模拟生物模型的扩散. 由于行波解可以很好地描述自然界中大量有限速度传播问题及振荡现象, 近年来, 非局部扩散生物模型行波解的研究得到了广泛的关注. 本文主要讨论一类非局部扩散捕食-食饵模型的行波解.

首先, 考虑捕食者与食饵采取不同的扩散策略情况下行波解的存在性与不存在性. 当食饵采取随机扩散, 捕食者采取非局部扩散时, 利用上下解方法结合Schauder's不动点定理, 证明了当c>c*时系统行波解的存在性(c*为临界波速). 同时, 通过基本的分析理论, 讨论了行波解的边界渐近行为以及c=c*时行波解的存在性. 进一步, 对捕食者所满足的方程进行线性化, 利用特征值理论证明了当0*时行波解的不存在性.

其次, 考虑捕食者与食饵同时采取非局部扩散策略时行波解的存在性与不存在性. 在本文中, 主要采用截断的方法, 结合Schauder's不动点定理证明了行波解的存在性. 同时, 通过细致的分析, 证明了临界波速下行波解的存在性以及小于临界波速下行波解的不存在性.

最后, 比较这两种不同扩散策略的情况, 我们得到结论: 食饵的扩散形式不影响行波解的存在性与不存在性.

Other Abstract

Nonlocal operator is more helpful to accurately describe the nonlocal interaction of the population in the space than  the classical Laplace operator, and more and more nonlocal diffusive equations is used to model the dispersal of the biological models.
Since traveling waves can represent a large number of finite velocity propagation problems and the oscillation phenomenons in nature.
In recent years, the research on traveling wave solutions of nonlocal diffusion biological model has attracted wide attention. In this paper, we mainly discuss traveling wave solutions for a class of nonlocal dispersal predator-prey model.

Firstly, we shall consider the existence and non-existence of traveling wave solutions with different diffusion strategies of predator and prey. When the prey adopts the random diffusion and the predator adopts the nonlocal diffusion, the existence of traveling wave solutions as c>c*is proved by using the upper-lower solution technique and the Schauder's fixed point theorem (where c* is critical wave speed). At the same time, through the basic analytical theory, we discuss the boundary asymptotic behavior of traveling wave solutions and the existence of traveling wave solutions with c=c*. Furthermore, we linearize the equation of the predator satisfied, and prove the non-existence of traveling wave solutions by the eigenvalue theory.

Secondly, the existence and non-existence of traveling wave solutions for the non local diffusion strategy of predator and prey are considered. In current paper, the existence of traveling wave solutions is proved mainly by adopting the truncation method, combined with the Schauder's fixed point theorem.  By careful analysis, we get the existence of traveling wave solutions of the critical wave speed and the non-existence of traveling wave solutions with speed less than the critical speed.

Finally, by comparing the two different diffusion strategies. We conclude that the diffusion form of prey does not affect the existence and non-existence of traveling wave solutions.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224505
Collection数学与统计学院
Recommended Citation
GB/T 7714
郝玉霞. 一类非局部扩散捕食-食饵模型的行波解[D]. 兰州. 兰州大学,2018.
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