对非均匀环境下带自由边界条件的反应扩散方程的定性研究

发布时间：2018-05-02 10:51

本文选题：反应扩散模型 + 自由边界　；参考：《中国科学技术大学》2017年博士论文

【摘要】：随着全球化进程的不断加快,特别是对外贸易以及出国旅游的快速增长,外来物种入境变得更加频繁,这样往往会形成生物入侵.生物入侵不但对当地生物多样性构成了巨大威胁,破坏了生态平衡,造成了难以估量的经济损失,还直接威胁到人类的健康.同时,由于全球气候的变化引起的动植物分布范围发生改变,可能也会影响到生物入侵的进程.为了能够很好地理解由环境迁移导致的自然资源分布不均对外来物种扩散的影响,生物数学家尝试应用简单的反应扩散模型来模拟外来物种在新栖息地的传播情况.生物数学家通过数学模型分析物种在新环境中的扩散速度和传播形式,这是他们研究生物入侵进程的两大方向.本文我们将考虑用具有自由边界条件的反应扩散方程来分析预测单个物种在非均匀环境中的入侵过程.我们将根据所建立的数学模型利用零点数理论、比较原理等基本方法分析入侵物种在非均匀环境中的传播速度和扩散形式.此博士论文中我们将紧紧围绕非均匀环境讨论两类具有自由边界条件的反应扩散问题.对于第一类自由边界问题,在高维径向对称情形下,我们将非均匀环境分为好环境和坏环境,进而讨论环境的好坏这种外因及物种的扩散率这种内因对外来物种入侵过程的影响,并且分析探讨入侵物种长时间的渐近行为.为了更好地考察非均匀环境对物种渐近传播行为的影响,我们首先给出所研究自由边界问题解的适定性结果;其次,我们给出所研究自由边界问题相应的阈值定义及其性质;之后,我们根据所定义的阈值给出判断物种成功入侵或者消亡的充分条件.之后,再利用物种的扩散率、初始占有区域面积及初始密度来考察物种最终的传播情况并给出"扩张-消亡"二择一定理.最后,我们通过构造合适的上下解并结合比较原理得到物种入侵前沿的渐近扩张速度.对于第二类自由边界问题,在一维情形下,当气候变化速度小于某临界速度时,在物种成功入侵的情况下,我们考察外来物种精细的长时间动力学行为.即对于气候变化速度小于某常值速度的情形,当物种成功入侵时我们可以得到物种的最终扩散形式及扩张前沿的最终传播形态和渐近扩张速度.因此,我们将紧紧围绕物种成功入侵的情形,首先通过构造合适的上下解来说明物种的扩张前沿与某条直线关于时间一致有界,再利用零点数方法说明扩张前沿最终将以线性形式向外扩张.也就是说,当入侵成功时,物种最终将以某一直线函数的形式向外传播并且渐进扩张速度是一个固定的常数.进一步地,我们利用比较原理构造上下解给出入侵物种最终详细的传播形式.
[Abstract]:With the accelerating process of globalization, especially the rapid growth of foreign trade and travel abroad, the entry of alien species becomes more frequent, which often forms biological invasion. Biological invasion not only poses a great threat to local biodiversity, destroys the ecological balance, causes incalculable economic losses, but also directly threatens human health. At the same time, changes in the distribution of animals and plants due to global climate change may also affect the process of biological invasion. In order to understand the influence of the uneven distribution of natural resources caused by environmental migration on the diffusion of alien species, biological mathematicians try to simulate the spread of alien species in new habitats by using a simple reactive diffusion model. Biological mathematicians use mathematical models to analyze the diffusion speed and propagation form of species in the new environment, which are their two main directions in the study of biological invasion process. In this paper, we will consider using the reaction diffusion equation with free boundary conditions to analyze and predict the invasion process of a single species in a non-uniform environment. According to the established mathematical model, we will analyze the propagation velocity and diffusion form of invasive species in non-uniform environment by using zero number theory, comparison principle and other basic methods. In this dissertation, we will discuss two kinds of reaction-diffusion problems with free boundary conditions. For the first kind of free boundary problem, in the case of high dimensional radial symmetry, we divide the non-uniform environment into good environment and bad environment. The influence of the external cause of environment and the diffusion rate of species on the invasive process of alien species is discussed, and the asymptotic behavior of invasive species over a long period of time is analyzed. In order to better investigate the effect of non-uniform environment on the asymptotic propagation behavior of species, we first give the suitable qualitative results of the solution of the free boundary problem studied, secondly, we give the corresponding threshold definition and properties of the free boundary problem studied. Then we give sufficient conditions for the species to succeed in invading or disappearing according to the defined threshold. After that, the diffusion rate of species, the area of initial occupied area and the initial density of species are used to investigate the final propagation of species and to give the "expansion-extinction" alternative theory. Finally, by constructing appropriate upper and lower solutions and combining the principle of comparison, we obtain the asymptotic expansion rate of the species invasion front. For the second kind of free boundary problem, in one dimensional case, when the climate change velocity is less than a certain critical velocity, and in the case of successful species invasion, we investigate the fine long-term dynamic behavior of alien species. In other words, when the climate change speed is less than a constant velocity, we can obtain the final diffusion form of species and the final propagation form and asymptotic expansion velocity of the expansion frontier when species invades successfully. Therefore, we will focus on the successful invasion of species, first of all, by constructing appropriate upper and lower solutions, we will show that the expansion front of the species is consistent with the time bound of a straight line. Then the zero point method is used to show that the extension front will eventually expand outwards in a linear form. That is to say when the invasion is successful the species will eventually propagate outwards in the form of a linear function and the rate of gradual expansion is a fixed constant. Furthermore, we use the comparison principle to construct the upper and lower solutions for the final detailed propagation of invasive species.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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