两类反应扩散模型的动力学行为

发布时间：2018-03-31 12:31

本文选题：反应扩散方程　切入点：平衡态　出处：《陕西师范大学》2016年博士论文

【摘要】：自然科学的发展很大程度上依赖于物理、化学、生命科学等方面的进展情况,这些具体问题的数学化对它们的进一步研究是很重要的.许多数学模型可以归为反应扩散模型.近几十年来,反应扩散模型的研究已取得了很大进展.随着研究的不断深入,反应扩散模型被广泛用来探讨大量的带有扩散的动力系统。本文利用非线性分析和非线性偏微分方程理论研究了两类反应扩散模型的动力学行为.研究的主要内容包括模型平衡态正解的先验估计、不存在性、存在性(分歧结构)、唯一性、稳定性和渐进行为.所涉及的主要数学理论有最大值原理、能量方法、隐函数定理、分歧理论、拓扑度理论、稳定性理论、正则化理论、扰动理论以及Lyapunov-Schmidt约化方法.本文内容包括以下三个方面:第一章首先介绍带有Degn-Harrison反应项的反应扩散模型和带有交叉扩散和保护区域的Leslie捕食食饵模型的研究背景及研究现状,然后介绍本文的主体工作.第二章考虑了一类带有Degn-Harrison反应项的反应扩散模型.非常数正解的一些基本性质首先被得到,然后,研究了 ODE和PDE系统的常数稳态解的稳定性.同时,我们的结果表明:如果反应器的体积或是有效扩散系数足够大时,系统不存在非常数正解.最后,利用简单特征值处的分歧理论得到了简单特征值处分歧曲线的全局结构,同时,利用Lyapunov-Schmidt约化方法和隐函数定理得到了双重特征值处分歧曲线的局部结构.第三章分析了一类带有保护区域和交叉扩散的Leslie捕食-食饵模型正解行为的变化.首先分析了保护区域和交叉扩散对正解分歧曲线结构的影响.而且,研究了当某些参数充分大或充分小时,模型正解的渐近行为.最后,当两物种的自身生长率充分小和交叉扩散系数充分大时,正解的详细结构和稳定性被建立.我们的结果表明:空间非均匀性和交叉扩散能够产生更加复杂的动力学行为,而且,这些结果明显不同与Lotka-Volterra模型的结论。
[Abstract]:The development of natural science depends to a large extent on the progress of physics, chemistry, life sciences, etc. The mathematics of these specific problems is very important for their further study. Many mathematical models can be classified as reaction-diffusion models. In recent decades, great progress has been made in the study of reaction-diffusion models. The reaction-diffusion model is widely used to study a large number of dynamic systems with diffusion. In this paper, the dynamic behaviors of two kinds of reaction-diffusion models are studied by using nonlinear analysis and nonlinear partial differential equation theory. A priori estimate of the positive solution of the equilibrium state of the model is included. Nonexistence, existence (bifurcation structure, uniqueness, stability and asymptotic behavior.) the main mathematical theories involved are maximum principle, energy method, implicit function theorem, bifurcation theory, topological degree theory, stability theory, Regularization theory, Perturbation theory and Lyapunov-Schmidt reduction method. The main contents of this paper are as follows: in Chapter 1, the background and research status of the Reaction-Diffusion Model with Degn-Harrison reaction term and the Leslie Predator-prey Model with Cross Diffusion and Protection region are introduced. In chapter 2, we consider a class of reaction-diffusion models with Degn-Harrison reaction term. The stability of constant steady-state solutions of ODE and PDE systems is studied. At the same time, our results show that if the reactor volume or effective diffusion coefficient is large enough, there is no non-constant positive solution in the system. By using the bifurcation theory of simple eigenvalues, the global structure of bifurcation curves at simple eigenvalues is obtained. By using Lyapunov-Schmidt reduction method and implicit function theorem, the local structure of bifurcation curves at double eigenvalues is obtained. In chapter 3, the change of positive solution behavior of a class of Leslie predator-prey model with protected region and cross diffusion is analyzed. The influence of protected region and cross diffusion on the structure of forward bifurcation curve is analyzed. The asymptotic behavior of the positive solution of the model is studied when some parameters are sufficiently large or small. Finally, when the growth rate of the two species is sufficiently small and the cross-diffusion coefficient is sufficiently large, The detailed structure and stability of the positive solutions have been established. Our results show that the spatial inhomogeneity and cross-diffusion can produce more complex dynamic behaviors, and these results are obviously different from the conclusions of the Lotka-Volterra model.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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