几类反应扩散生物学模型的动力学研究
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摘要
本文主要研究几类具有生物背景的反应扩散模型解的动力学行为.讨论的问题包括正稳态解的存在性与稳定性,系统的持续生存性质,以及解的渐近行为.首先研究一个非均匀双营养基食物链恒化器模型,证明了系统正稳态解的存在性,并得到系统持续生存的条件.其次考虑一个非均匀双营养基竞争恒化器模型,得到系统正稳态解的存在性与持续生存条件.第三,研究一个三种群捕食者一被捕食者模型,证明了系统在一定条件下具有一个全局吸引子.最后,研究一个Kelle-Segel模型,得到了解的衰减及渐近行为.
第一章概述本文所研究问题的生物背景和国内外发展状况,并简要介绍本文的主要工作.
第二章考虑一个非均匀双营养基食物链恒化器模型.首先利用不动点指数结合特征值理论得到了系统正稳态解存在的条件.然后运用全局分歧理论得到一定条件下系统正稳态解的全局结构,并研究了系统参数对捕食种群和被捕食种群灭绝与持续生存,以及被捕食种群对捕食种群持续生存的影响.
第三章研究一个非均匀双营养基竞争恒化器模型.在证明系统正稳态解存在性的基础上,探究了其中一个种群正稳态解的全局吸引条件,然后得到两种群灭绝与持续生存的条件,并以两种群扩散系数作为参数,讨论种群的稳定性,持续生存及渐近行为.本文证明,在一定参数范围内,当两种群扩散速度都较快时：两者都将灭绝；而一个扩散得较快,另一个较慢时,扩散较慢种群的浓度具有全局稳定性.
第四章考虑一个三种群捕食者一被捕食者模型解的长时间行为.利用无穷维动力系统的理论结合先验估计的方法,得到系统存在全局吸引子条件.改进了相关文献的整体解存在性结果.
第五章研究一个具有体积填充效应的Keller-Segel模型.利用Lp-Lq估计的方法对任意扩散系数ε>0得到了解的衰减估计以及解的渐近行为,去掉了相关文献的过强限制：ε>1/4(或∫Rnρ0(x)dx≤G0(ε,n)).
This thesis deals with dynamics of solutions to some reaction-diffusion models from biology. The topics include the existence and the stability of steady states, the persistence of solutions for two un-stirred chemostat models, and the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion, as well as a Keller-Segel model for chemotaxis with prevention of overcrowding. Firstly, we consider a multiple food chain model for two resources in un-stirred chemostat. We derive the existence of the positive steady solutions and the persistence of solutions.
Secondly, we concern a competition model for two resources in un-stirred chemostat to prove the conditions for the existence of positive steady solutions and the persistence for solutions. Thirdly, we study the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion. We prove the existence of a global attractor to the system. Finally, we investigate a Keller-Segel model. We obtain the decay and the long-time behavior of solutions to the system.
Chapter 1 is to summarize the background of the related issues and to briefly intro-duce the main results of the thesis.
Chapter 2 is concerned with a multiple food chain model for two resources in un-stirred chemostat. Firstly, Combining with the fixed point theory and the eigenvalue theory, we get conditions for the existence of positive solutions, and then find the global structure of positive steady solutions under some conditions by the global bifurcation theory. Meanwhile, we study how the parameters affect the extinction or persistence of the predator and prey, and how the prey affect the predator.
Chapter 3 is devoted to a competition model for two resources in un-stirred chemo-stat. On the basis of the existence of positive solutions, we study the global attracting conditions for one of the populations, and derive the extinct and persistent conditions for the two (or one) populations. Furthermore, regarding the diffusion rates as parameters, we consider the stability, persistence and asymptotic behavior for the two populations. We find that the two populations will go to extinct when both possess large diffusion rate. If just one of them spreads fast with the other one diffusing slower, then the related semitrivial steady state will be global attracting.
Chapter 4 deals with large-time behavior of solutions for a three-species predator-prey model with cross-diffusion. By using the infinite-dimensional dynamical systems theory and the a priori estimate method, we prove that the system admits a global attractor if m,l≥2. This extends the known results on the global existence of solutions.
Chapter 5 studies a Keller-Segel model for chemotaxis with prevention of overcrowd-ing. We prove a crucial decay result for arbitrary diffusion rateε>0, and then obtain the asymptotic behavior of solutions, where the stronger assumptionε>1/4 (or∫nρ0(x)dx≤Co(ε, n)) in the related literature is unnecessary any more.
第一章概述本文所研究问题的生物背景和国内外发展状况,并简要介绍本文的主要工作.
第二章考虑一个非均匀双营养基食物链恒化器模型.首先利用不动点指数结合特征值理论得到了系统正稳态解存在的条件.然后运用全局分歧理论得到一定条件下系统正稳态解的全局结构,并研究了系统参数对捕食种群和被捕食种群灭绝与持续生存,以及被捕食种群对捕食种群持续生存的影响.
第三章研究一个非均匀双营养基竞争恒化器模型.在证明系统正稳态解存在性的基础上,探究了其中一个种群正稳态解的全局吸引条件,然后得到两种群灭绝与持续生存的条件,并以两种群扩散系数作为参数,讨论种群的稳定性,持续生存及渐近行为.本文证明,在一定参数范围内,当两种群扩散速度都较快时：两者都将灭绝；而一个扩散得较快,另一个较慢时,扩散较慢种群的浓度具有全局稳定性.
第四章考虑一个三种群捕食者一被捕食者模型解的长时间行为.利用无穷维动力系统的理论结合先验估计的方法,得到系统存在全局吸引子条件.改进了相关文献的整体解存在性结果.
第五章研究一个具有体积填充效应的Keller-Segel模型.利用Lp-Lq估计的方法对任意扩散系数ε>0得到了解的衰减估计以及解的渐近行为,去掉了相关文献的过强限制：ε>1/4(或∫Rnρ0(x)dx≤G0(ε,n)).
This thesis deals with dynamics of solutions to some reaction-diffusion models from biology. The topics include the existence and the stability of steady states, the persistence of solutions for two un-stirred chemostat models, and the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion, as well as a Keller-Segel model for chemotaxis with prevention of overcrowding. Firstly, we consider a multiple food chain model for two resources in un-stirred chemostat. We derive the existence of the positive steady solutions and the persistence of solutions.
Secondly, we concern a competition model for two resources in un-stirred chemostat to prove the conditions for the existence of positive steady solutions and the persistence for solutions. Thirdly, we study the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion. We prove the existence of a global attractor to the system. Finally, we investigate a Keller-Segel model. We obtain the decay and the long-time behavior of solutions to the system.
Chapter 1 is to summarize the background of the related issues and to briefly intro-duce the main results of the thesis.
Chapter 2 is concerned with a multiple food chain model for two resources in un-stirred chemostat. Firstly, Combining with the fixed point theory and the eigenvalue theory, we get conditions for the existence of positive solutions, and then find the global structure of positive steady solutions under some conditions by the global bifurcation theory. Meanwhile, we study how the parameters affect the extinction or persistence of the predator and prey, and how the prey affect the predator.
Chapter 3 is devoted to a competition model for two resources in un-stirred chemo-stat. On the basis of the existence of positive solutions, we study the global attracting conditions for one of the populations, and derive the extinct and persistent conditions for the two (or one) populations. Furthermore, regarding the diffusion rates as parameters, we consider the stability, persistence and asymptotic behavior for the two populations. We find that the two populations will go to extinct when both possess large diffusion rate. If just one of them spreads fast with the other one diffusing slower, then the related semitrivial steady state will be global attracting.
Chapter 4 deals with large-time behavior of solutions for a three-species predator-prey model with cross-diffusion. By using the infinite-dimensional dynamical systems theory and the a priori estimate method, we prove that the system admits a global attractor if m,l≥2. This extends the known results on the global existence of solutions.
Chapter 5 studies a Keller-Segel model for chemotaxis with prevention of overcrowd-ing. We prove a crucial decay result for arbitrary diffusion rateε>0, and then obtain the asymptotic behavior of solutions, where the stronger assumptionε>1/4 (or∫nρ0(x)dx≤Co(ε, n)) in the related literature is unnecessary any more.
引文
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