一类捕食者-食饵趋化模型解的整体性态 Alternative Title Global Behavior of Solutions for a Predator-Prey Chemotaxis Model 许生虎 Thesis Advisor 李万同 2015-12-04 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword Keller-Segel模型 捕食者-食饵 整体存在性 斑图 非线性不稳定性 稳定性 渐近性 Abstract 本文主要分四部分,着重研究趋化对上述模型整体性态的影响: 整体解的存在性以及有界性;解的长时间行为,包括长时间收敛性以及收敛速率;斑图的定量刻画问题。首先,运用算子半群理论和 Banach 压缩映像原理证明三种群捕食者-食饵趋化模型局部解的存在唯一性。 接着讨论二维空间中整体解的有界性。结果表明食饵趋化或者第一捕食者初值不太大时,模型解整体存在且有界。其次,三维空间上,讨论不等扩散情形下,三种群捕食者-食饵趋化模型的不稳定正常数平衡解附近的非线性动力学性态。 首先讨论正常数平衡解不稳定条件。结果表明两捕食者都存在食饵趋化时不能导致不稳定,只有存在捕食者趋化项且趋化很大时才可以出现斑图。然后,利用嵌入定理、能量估计以及 bootstrap 技巧,对正常数平衡解失稳初期时空斑图的非线性演化给出了严格的定量刻画。接着,高维空间中考虑了不带反应项的三种群捕食者-食饵趋化模型的整体性态。结果表明在光滑有界区域上如果对于充分光滑的初值它的范数充分小,甚至在最优空间中的范数足够小,那么整体解存在并且指数收敛到常数稳态解;当区域是全空间时,柯西问题的整体解存在。最后,高维空间中考虑了带反应项三种群捕食者-食饵趋化模型的整体性态。首先借助 Maximal Sobolev Regularity 引理, 在比率~$a_{1}/\chi_{1}$ 和~$a_{2}/\chi_{3}$ 适当大的情况下,得到整体解存在且一致有界性。进一步,证得正常数平衡解全局渐近稳定。结果表明如果捕食者 Logistic 增长系数相比趋化灵敏度适当大时,三种群共存。 Other Abstract This paper is mainly divided into four parts, which are devoted to studying the impact of chemotaxis on the global existence and long time behavior of solutions and pattern formation. There are three kinds of topics: Global existence and global boundedness of solutions; Long time behavior, including long time convergence and convergence rate, to the global solutions; and pattern formation.Firstly, by using the operator semigroups theory and Banach contraction mapping principle, we prove the existence and uniqueness of local solutions for the three-species chemotaxis model for reasonably regular initial values. Afterwards, global bounded solutions under small inital data condition in a two dimension predator-prey chemotaxis system which has the defensive switching property of predation-avoidance are shown.Secondly, based on the unequal diffusion coefficients, nonlinear dynamics near an unstable constant equilibrium for the three-species predator-prey chemotaxis model in a three-dimensional is discussed by applying the embedding theorem, the energy estimates and the bootstrap arguments. Our results indeed provide a rigorous quantitative characterization for the nonlinear evolution of early spatiotemporal pattern formation on the unstable positive constant equilibrium.Thirdly, this paper considers global behavior of solutions in the higher-dimensional a three-species predator-prey chemotaxis model without reaction term for the predator. The result reveals that under small initial data condition, the solutions of the system is global in time and bounded and approaches the steady state solution exponentially as time tends to infinity in the smooth bounded domain; Moreover, we shall that under small initial data condition the global existence of the solutions on the whole space.Finally, this paper considers global behavior of solutions for a three-species predator-prey chemotaxis model with reaction term.To begin with, we obtain the existence and uniform boundedness of the global solutions based on Maximal Sobolev Regularity. Next, if the ratio~$a_{1}/\chi_{1}$,~$a_{2}/\chi_{3}$ is suitably large, then the unique positive spatially homogeneous equilibrium is globally asymptotically stable. The result implies that three species can coexist. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224523 Collection 数学与统计学院 Recommended CitationGB/T 7714 许生虎. 一类捕食者-食饵趋化模型解的整体性态[D]. 兰州. 兰州大学,2015.
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