非局部反应扩散方程的空间动力学研究

发布时间：2018-06-13 01:05

本文选题：反应扩散方程 + 行波解　；参考：《兰州大学》2017年博士论文

【摘要】：非局部反应扩散方程被认为可以更加准确地描述物理、化学、生态学中的自然现象,所以受到越来越多的关注.但是随着非局部时滞的引入,使得原有的许多关于反应扩散方程的研究方法受到了挑战,同时在研究过程中也发现了许多由非局部时滞作用引起的动力学行为方面的本质变化.目前关于非局部反应扩散方程行波解的研究大都考虑的是非局部时滞充分弱或反应项满足某些条件,如拟单调、指数拟单调、弱拟单调以及指数弱拟单调等等.关于非局部时滞没有限制时行波解的相关研究很少,而且这些研究结果不能充分揭示非局部反应扩散方程的许多重要性质.另外关于无界区域上的初值问题以及系统的斑图生成等问题的研究目前也很少,而这些都是反应扩散方程中的重要问题,因此本文将致力于研究几类非局部反应扩散方程的行波解、初值问题以及斑图生成等等.主要内容将分五部分进行阐述.本文首先研究了一类具有Allee效应的非局部反应扩散单种群模型的行波解.由于比较原理不成立,从而基于比较原理的经典方法,如上下解方法、移动平面法等都不能应用.因此我们应用Leray-Schauder度理论等方法证得当且仅当波速c≥2r~(1/2)(其中r0是物种的内禀增长率)时,模型存在连接平衡点0到未知正稳态的行波解.进一步利用常数变易法、柯西-施瓦兹不等式以及一系列分析讨论说明了当波速c充分大时,这个未知的正稳态恰好就是方程唯一的正平衡点.此外,针对两类特殊的核函数,我们还讨论了随着非局部性增强行波解性质的变化,并说明前面所说的未知的正稳态也可能是周期稳态.其次研究了带有聚集项的非局部反应扩散方程的行波解.由于聚集项的出现,使模型的解不能被其在零平衡点处的线性化方程所控制.因此,我们借助于一个辅助方程来构造合适的上解,进而证明了连接0到未知正稳态的行波解的存在性.对充分大的波速,我们也证明了未知的正稳态解就是正平衡点.另外,我们还应用上下解方法证明了该模型存在连接0到正平衡点的单调行波.最后,取特殊的核函数,通过数值模拟的办法,我们说明随着非局部性的增强,方程的行波解可能连接0到一个周期稳态.进一步借助于稳定性分析我们解释了为什么以及什么时候出现周期稳态.第三部分考虑了一类带有积分项的捕食-食饵模型的初值问题.通过重新定义问题的上下解,并借助于一些辅助函数,我们建立了比较原理,从而构造单调序列并以此给出了初值问题解的存在性和唯一性证明.紧接着借助于辅助方程证明了解的一致有界性.最后,我们给出了初值问题出现Turing分支的条件并通过数值模拟验证了这些条件.本文第四部分研究了具有非局部项的Lotka-Volterra竞争系统的行波解.借助于两点边值问题和Schauder不动点定理,我们证明了当波速cc*=max{2,2dr~(1/2)}(其中d和r分别是扩散系数和物种的内禀增长率)时,系统存在连接平衡点(0,0)到未知正稳态的行波解;而当波速cc*时不存在这样的行波解.最后,针对特殊的核函数,通过数值模拟的办法,我们发现随着非局部的增强,系统的行波解可能连接平衡点(0,0)到一个周期稳态.最后,我们探讨了具有非局部项的Lotka-Volterra竞争系统的动力学行为.通过稳定性分析,建立了系统出现Turing分支的条件.然后根据这些条件并结合多尺度分析,得到了关于不同Turing斑图的振幅方程.接着,通过分析振幅方程的稳定性给出了系统出现不同斑图(包括点状斑图和条状斑图)的条件.最后,通过数值模拟结果验证了我们的理论结果.
[Abstract]:The non local reaction diffusion equation is considered to be more accurate in describing physical, chemical and ecological natural phenomena, so it has attracted more and more attention. However, with the introduction of non local time delay, many existing research methods about the reaction diffusion equation have been challenged, and many of them have been found in the process of research. The essential changes in the dynamic behavior caused by non local delay action. At present, most of the study on the traveling wave solutions of non local reaction diffusion equations is that the nonlocal time delay is fully weak or the reaction term satisfies some conditions, such as quasi monotone, exponential quasi monotone, weakly quasi monotone and weakly quasi monotone, etc. There are few related studies on the time traveling wave solutions, and these results can not fully reveal the many important properties of the nonlocal reaction diffusion equation. In addition, there are few studies on the initial value problem on the unbounded region and the formation of the system speckle patterns. These are all important problems in the inverse diffusion equation. The traveling wave solution, initial value problem and speckle pattern generation of several non local reaction diffusion equations are studied. The main content will be divided into five parts. First, the traveling wave solution of a class of non local reaction diffusion single population model with Allee effect is studied. The method, such as the method of upper and lower solutions and the moving plane method, can not be applied. Therefore, we apply the Leray-Schauder degree theory and other methods to prove that the model has a traveling wave solution that connects the equilibrium point 0 to the unknown steady state when the wave velocity is C > 2r~ (1/2) (and R0 is the intrinsic growth rate of the species). The Cauchy Schwartz inequality is further used by the constant variation method. And a series of analysis and discussion shows that the unknown positive steady state is the only positive equilibrium point of the equation when the wave velocity C is sufficiently large. In addition, we also discuss the changes in the properties of the traveling wave solutions with the non local enhancement for the two class of special kernel functions, and show that the unknown positive steady state mentioned above may also be a periodic steady state. The traveling wave solution of a nonlocal reaction diffusion equation with an aggregation term is studied. Due to the appearance of the aggregation term, the solution of the model can not be controlled by the linearized equation at the zero equilibrium point. Therefore, we construct a suitable upper solution with the aid of an auxiliary equation, and then prove the existence of the traveling wave solution of the 0 to the unknown positive steady state. For the full wave velocity, we also prove that the unknown positive steady state solution is the positive equilibrium point. In addition, we also use the upper and lower solutions to prove that the model has the monotone traveling wave between 0 and the positive equilibrium points. Finally, we take the special kernel function, and through the numerical simulation, we show that the traveling wave solution of the equation with the non local enhancement. We can connect 0 to one periodic homeostasis. Further by the stability analysis we explain why and when the periodic steady state appears. The third part considers the initial value problem of a predator-prey model with integral terms. By redefining the upper and down solutions of the problem and using some auxiliary functions, we have established a comparative original. The existence and uniqueness of the solution of initial value problem are proved by constructing the monotone sequence, and the uniform boundedness of the understanding is proved by the aid of the auxiliary equation. Finally, we give the condition of the Turing bifurcation of the initial value problem and verify these conditions by numerical simulation. The fourth part of this paper studies the non local conditions. With the help of the two point boundary value problem and the Schauder fixed point theorem, we prove that when the wave velocity cc*=max{2,2dr~ (1/2)} (of the D and R is the intrinsic growth rate of the diffusion coefficient and the species), the system has a traveling wave solution that connects the equilibrium point (0,0) to the unknown positive steady state when the wave velocity cc*=max{2,2dr~ (cc*=max{2,2dr~)} (and D and R are respectively); but when the wave velocity cc* does not exist, there is no existence. In the end, by means of numerical simulation, we find that the traveling wave solution of the system may connect the equilibrium point (0,0) to a periodic steady state with the non local enhancement. Finally, we discuss the dynamic behavior of the Lotka-Volterra competitive system with non local terms. The condition of the Turing branch of the system is presented. Then, according to these conditions and combined with multiscale analysis, the amplitude equation about different Turing speckle patterns is obtained. Then, the conditions of the system appearance of different speckle patterns (including spot pattern and bar pattern) are given by analyzing the stability of the amplitude equation. Finally, the numerical simulation results are used to verify that The results of our theory.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

【参考文献】