几类非局部时滞种群扩散模型的空间动力学

兰州大学机构库 >数学与统计学院

	几类非局部时滞种群扩散模型的空间动力学
Alternative Title	Spatial Dynamics of Several Classes of Population Diffusion Models with Nolocal Delay
	田歌
Subtype	博士
Thesis Advisor	王智诚
	2021-05-21
Degree Grantor	兰州大学
Place of Conferral	兰州
Degree Name	理学博士
Degree Discipline	应用数学
Keyword	反应扩散方程非局部时滞行波解渐近传播速度界面生成
Abstract	反应扩散方程常常被用于解释和预测一些具体学科中遇到的问题, 例如数学生态学中新物种的入侵, 传染病的传播;化学反应中的酶促反应, 低温等离子体烟气脱硫反应;物理学中的热传导现象, 流体的运动规律等等. 由于生物个体和环境因子是相互依存的, 空间扩散和时间滞后的协同作用在数学生态学科的研究中不容忽视.基于这种相互作用, 研究者在非线性项中引入了空间和时间滞后的加权平均, 得到了非局部时滞反应扩散方程. 相比于传统模型, 非局部时滞反应扩散方程会带来更多的研究困难, 但同时也揭示了更为丰富的动力学行为, 因此得到了学者们的广泛关注和研究, 并取得了一些研究成果. 本文主要研究非局部时滞种群扩散模型的行波解和渐近传播速度问题, 具体的研究内容如下: 第二章考虑一类非局部Fisher-KPP 方程的行波解(单调或者非单调) 的稳定性. 此时非线性项导致比较原理的缺失, 本章使用反加权的思想, 通过能量估计方法和一些精细技巧处理扰动方程的解, 最终建立了该模型的行波解在大波速情形下的全局稳定性. 第三章研究一类非单调无穷维时滞格微分方程行波解的全局稳定性. 通过加权能量和Fourier 变换的方法建立扰动方程的解的有界性估计, 进一步得到: 在一个加权的Sobolev 空间中, 非临界行波解( >) 是全局稳定的, 并以指数收敛速率 −1/−( >0 且0 <≤2) 收敛;临界行波解( = ) 是全局稳定的, 并以代数收敛速率−1/ 收敛. 第四章研究一类非局部时滞单种群模型的渐近传播速度. 运用Banach 不动点定理和延拓方法最先得到这类方程初值问题解的全局存在性. 关于渐近传播速度的研究, 由于所选取的参数以及核函数的不同, 处理方法不兼容, 因此本章分别给出相应的证明. 首先, 对于带有时空时滞的Food-Limited 模型, 借助核函数的显式结构得到解的一致有界性. 接下来通过一系列比较原理证明了带有紧支集初值解的渐近传播速度. 其次, 对于带有固定时滞的Food-Limited 模型, 运用Harnack 不等式得到带有紧支集初值解的渐近传播速度. 最后, 对于带有紧支集初值的非局部时滞Fisher-KPP 模型解的渐近传播速度, 可以采用反证法得到. 此外, 本章通过有限差分法给出数值模拟, 不仅验证了理论结果, 而且表明方程在时滞充分大时会产生类似时间周期解的正稳态. 第五章考虑一类具有分布时滞的Nicholson 方程的界面生成. 当出生函数满足拟单调条件时, 利用单稳问题的非标准双稳近似构造合适的下解, 然后用单稳行波解构造合适的上解, 最终得到解收敛到一个传播界面. 在此基础上, 进一步讨论不满足拟单调条件的情形, 此时由于方程缺少单调性, 上述方法不再适用. 因此首先构造了两个辅助的拟单调系统, 继而由夹逼近方法和柯西问题的比较原理得到原方程解的极限行为. 结果表明, 无论出生函数是否满足拟单调条件, 行波解的最小波速和界面传播的速度在数值上是相等的, 从而可以从一个新的视角去观察行波解的最小波速.
Other Abstract	Reaction-diffusion equation is often used to explain and predict the problems encountered in some specific disciplines, such as the invasion of new species and the spread of infectious diseases in mathematical ecology, enzyme catalysis and low-temperature plasma flue gas desulfurization reactions in chemical reactions, the changes of electron and hole concentrations in semiconductors, the movement of fluids in physics, etc. Since individual organisms and environmental factors are interdependent, the interaction between spatial diffusion and time delay cannot be ignored in the study of mathematical ecology. On the basis of this interaction, the weighted average of the whole space and temporal delay is introduced into the nonlinear term, i.e., the nonlocal delay reaction-diffusion equation. Compared with the traditional model, the nonlocal delay reaction-diffusion equation will bring more difficulties. At the same time, it also reveals more dynamic behaviors, many scholars have been concerned about this model and obtained some results. In this paper, we study the traveling wave solutions and asymptotic spreading speed of the following population diffusion models with nonlocal delay: In Chapter 2, we study the stability of traveling wave solutions (monotone or non-monotone) of nonlocal Fisher-KPP equation. Since the nonlinear term leads to the lack of the comparison principle, we use the idea of anti-weighted and estimate the solution of the perturbation equation through the energy estimation and some technical methods. Finally, the global stability of the traveling wave solution of the model in the case of large wave speed is proved. In Chapter 3, we investigate the global stability of traveling waves for nonmonotone infinite-dimensional lattice differential equations with time delay. The boundedness estimation of the solution of the perturbation equation is established by the method of weighted energy and Fourier transform, we obtain that: for any initial perturbations around the traveling wave, the noncritical traveling waves ( >) are globally stable with the exponential convergence rate −1/−( >0 and 0 <≤2), and the critical traveling waves ( = ) are globally stable with the algebraic convergence rate −1/ in a weighted Sobolev space. In Chapter 4, we are devoted to studying the asymptotic spreading speed of a kind of single species model with time delay. The global existence of the solutions to the initial value problem of the model is obtained first by Banach fixed point theorem and the extension method. For the research about the spreading speed, due to the different selected parameters and kernel function, the processing methods are not compatible, then different methods are applied to study the spreading speed. Firstly, for the Food-Limited model with spatio-temporal delay, the uniform boundedness of the solution is acquired by means of the explicit structure of the kernel function. Through comparison principle, we establish the asymptotic spreading speed of solutions with compactly supported initial data. Secondly, for the Food-limited model with fixed time delay, the Harnack inequality is used to study the asymptotic spreading speed of the solution with compactly supported initial. Thirdly, for the Fisher-KPP model with nonlocal delay, we study the asymptotic spreading speed of the solution with compactly supported initial data by contradiction. Furthermore, the numerical simulation is demonstrated by the finite difference method, which not only verify the theoretical conclusion, but also show that the equation may produce a positive steady state which is similar to the time periodic solution when the time delay is sufficiently large. In Chapter 5, we consider the generation of interface for a class of Nicholson's blowflies equation with distributed delay. If the birth function satisfies the quasi-monotone condition, we utilize the nonstandard bistable approximation of the monostable problem to establish a suitable lower solution, and then employ the monostable traveling wave solution to construct a suitable upper solution. It is proved that the solution converge to a propagation interface. Furthermore, for the non-quasi-monotone situation, due to the lack of monotonicity of the equation, the above method is no longer applicable. We first construct two auxiliary quasi-monotone systems, then the limit behavior of the solution of equation can be obtained from the sandwich technique and the comparison principle of Cauchy problem. Whether the birth function satisfies the quasi-monotone condition or not, the propagation speed of the interface is proved to be equal to the minimum wave speed of the corresponding traveling waves, which makes it possible to observe the minimum speed of traveling waves from a new perspective.
Pages	140
URL	查看原文
Language	中文
Document Type	学位论文
Identifier	https://ir.lzu.edu.cn/handle/262010/460304
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 Citation GB/T 7714	田歌. 几类非局部时滞种群扩散模型的空间动力学[D]. 兰州. 兰州大学,2021.

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