离散时间反应扩散系统的行波解

发布时间：2018-09-02 05:54

【摘要】：随着科学技术的发展,反应扩散方程在描述时空模式方面发挥重要的作用,其行波解可以解释种群扩散,种群入侵和疾病传播等许多自然现象,因此研究反应扩散方程行波解具有重要的理论和实际意义.在现实世界中,许多事物的变化规律不仅依赖于当前的状态而且还依赖于过去某个时刻或者某段时间内的状态,因此,时间滞后和空间扩散现象都是普遍存在的.由于自然界中的许多物种是世代不重叠的,种群数量的变化是离散的,用离散时间模型更能准确的反映种群发展的规律,从而利用离散时间时滞反应扩散方程来刻画这类物种的发展变化过程更符合其本质属性.基于上述原因,本文主要针对几类离散时间反应扩散方程行波解的存在性进行分析.全文共分为六章,主要内容如下:第一章主要介绍离散时间反应扩散系统的历史背景,研究现状,最新研究进展以及本文的主要研究工作.第二章考虑了一类积分差分方程波前解的存在性.利用比较原理,单调迭代技巧和上下解方法,对增长函数在满足较弱的限制条件下得到单稳情形下波前解的存在性,从理论上完善了前人的结论.第三章研究了离散时间时滞捕食-食饵扩散系统行波解的存在性.通过对反应项引入部分拟单调条件(PQM),利用上下解方法,交错迭代技巧和Schauder不动点定理得到抽象扩散系统在满足一定条件下的行波解的存在性,并将所得结论运用到捕食-食饵系统,证明了该类系统行波解的存在性.对于高维离散时间时滞竞争合作扩散系统,第四章构造了一类新的混合拟单调条件,利用Schauder不动点定理,在适当构造的赋予指数衰减范数的某个子空间上证明了系统行波解的存在性,并将所得结论应用到三维离散时间时滞K-型单调扩散系统,得到该类系统行波解的存在性.对于离散时间时滞K-型竞争扩散系统,第五章研究了三维离散时间时滞K-型竞争扩散系统当反应项满足另一个混合拟单调条件时行波解的存在性.第六章是全文的总结和对未来工作的展望。
[Abstract]:With the development of science and technology, the reaction diffusion equation plays an important role in describing the space-time model. Its traveling wave solution can explain many natural phenomena, such as population diffusion, population invasion and disease transmission. Therefore, it is of great theoretical and practical significance to study the traveling wave solutions of the reaction diffusion equation. In the real world, the law of change of many things depends not only on the current state, but also on the state of a certain time or a certain period of time in the past. Therefore, the phenomenon of time lag and spatial diffusion are ubiquitous. Because many species in nature do not overlap from generation to generation, and the variation of population quantity is discrete, the discrete time model can more accurately reflect the law of population development. Therefore, the discrete time delay reaction diffusion equation is used to describe the evolution process of this kind of species, which is more consistent with its essential properties. For the above reasons, the existence of traveling wave solutions for several discrete time reaction diffusion equations is analyzed in this paper. The thesis is divided into six chapters. The main contents are as follows: the first chapter mainly introduces the historical background, the research status, the latest research progress and the main research work of the discrete time reaction diffusion system. In chapter 2, we consider the existence of wavefront solutions for a class of integral difference equations. By means of comparison principle, monotone iterative technique and upper and lower solution method, the existence of wavefront solution under the condition that the growth function satisfies the weak limit is obtained, and the previous conclusions are improved theoretically. In chapter 3, the existence of traveling wave solutions for predator-prey diffusion systems with discrete time delay is studied. By introducing partial quasi-monotone condition (PQM), to the reaction term, using the method of upper and lower solutions, staggered iterative technique and Schauder fixed point theorem, the existence of traveling wave solutions for abstract diffusion systems under certain conditions is obtained. The results obtained are applied to the predator-prey system and the existence of traveling wave solutions is proved. For high dimensional discrete time delay competitive cooperative diffusion systems, a new class of mixed quasi-monotone conditions is constructed in chapter 4, and the Schauder fixed point theorem is used. In this paper, the existence of traveling wave solutions is proved on a subspace with exponential attenuation norm, and the results are applied to 3-dimensional discrete time delay K-type monotone diffusion systems, and the existence of traveling wave solutions for such systems is obtained. For discrete time delay K- type competitive diffusion systems, in chapter 5, we study the existence of traveling wave solutions for three dimensional discrete time delay K- type competitive diffusion systems when the reaction term satisfies another mixed quasi-monotone condition. The sixth chapter is a summary of the full text and prospects for future work.
【学位授予单位】：广州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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