摘要
恒化器(chemostat)模型和Lotka-Volterra模型是生态数学研究领域中两类非常重要的生态数学模型.这两类模型分别在微生物学与生态学中有着广泛的应用,因此受到了国内外专家与学者的广泛关注.近多半个世纪,相关的研究取得许多有意义的成果.本文基于这两类生物模型的研究现状,主要利用反应扩散方程及对应椭圆方程的理论和方法,深入地研究了两种具有Beddington-DeAngelis功能反应函数的非搅拌的恒化器(chemostat)模型和两种具有功能反应项的Lotka-Volterra模型平衡态解及动力学行为.本文主要应用上下解方法、比较原理、单调动力系统理论、全局分歧理论、拓扑度理论、Lyapunov-Schmidt过程、扰动理论和持续性理论等,研究了模型的平衡态解的存在性、多重性、稳定性、动力系统的长时行为以及参数对模型解的影响.
本文的主要内容共分五章：
第一章首先介绍了恒化器(chemostat)模型和Lotka-Volterra模型的研究背景与发展现状.然后列举了本文要用到的一些基本理论,主要包括锥映射不动点指标理论,特征值问题及分歧理论等.
第二章研究了一类带B-D反应项的具有质粒载体的微生物(plasmid-bearing or-ganism)与质粒自由的微生物(plasmid-free organism)之间竞争的非均匀恒化器模型.首先,采用通常的锥映射的不动点指标理论得到了物种共存的充分条件.然后,利用度理论、分歧理论、摄动理论、比较原理及持续性理论,主要讨论了模型的分歧解全局走向及稳定性,参数ki(i=1,2)及q对解的多重性及稳定性的影响和解的动力学行为.结果得到了分歧解的全局分支走向及稳定性；还得到了当参数k1特别小k2特别大且q>1/2时,模型至少存在两个正解,而当参数k1充分大时,模型存在唯一且稳定的正解；最后得到了解的长时行为,包括非负解的渐近稳定性和正解的一致持续性.
第三章研究了一类具有B-D反应项的食物链恒化器模型的正平衡态解的存在性.首先分析了平凡解、弱半平凡解及强半平凡解的存在性及稳定性.然后利用度理论计算平凡解、弱半平凡解及强半平凡解处的指标数.最后利用指标可加性得到了该模型正平衡态解存在的充分条件.
第四章研究了一类具有交叉扩散项的捕食模型的平衡态解.首先利用极大值原理讨论并得到了模型的正解的先验估计；然后给出了正解存在的充分条件,并利用介值定理、隐函数定理及Dini's定理等数学分析理论讨论了正解的共存区域,得到了正解的共存区域与两物种的生长率a,b的关系,并且发现正解的共存区域随着交叉扩散系数β(α)趋于无穷大而扩张(紧缩)的变化规律；最后利用分歧理论讨论了局部分歧解的存在性、全局分歧解的存在性及解的结构.
第五章研究了一类具有饱和竞争项的捕食模型.采用度理论、分歧理论、Lyapunov-Schmidt过程、扰动理论、比较原理及持续性理论分析了此模型平衡态系统的全局分歧解的结构及稳定性,二重特征值处分歧解的存在唯一性及稳定性,参数引起解的多重性及稳定性的变化和系统的动力学行为.结果如下：(ⅰ)得到了全局分歧解的具体走向及稳定性；(ⅱ)得到了二重特征值处的分歧解是唯一存在且稳定的；(ⅲ)如果参数d适当小,那么系统的至少存在两个正解,而且当参数m或k充分大时,模型至少存在两个正解；特别地,得到了参数m充分大且参数α满足一定条件时,模型正好存在两个确定的解,且解的类型正好有两种,一种是渐进稳定性,另一种不稳定；(ⅳ)得到了平凡、半平凡解一致渐近稳定性和正解的一致持续存在性.
Chemostat model and Lotka-Volterra model are two kinds of the most significant models in mathematical biology. The two kinds of models play a very important role in microbiology and ecology, respectively. Therefore, it has aroused extensive concern to domestic and overseas experts and scholars, and many significant research results have been obtained. Base on the recent work on these two kinds of biological models, mainly applying the theory and method of reaction-diffusion equations and correspond-ing elliptic equations, we have profoundly investigated the equilibrium state and the dynamical behavior of two unstirred chemostat models with Beddington-DeAngelis functional response and two (cross-) diffusion Lotka-Volterra model with functional response. The main tools used in this thesis include super-sub solutions method, com-parison principle, monotone system of topology, Lyapunov-Schmidt procedure, per-turbation technique and persistence theory. The obtained results include coexistence, multiplicity, uniqueness, stability of positive steady-states and the longtime behavior of species.
The main contents in this dissertation are as follows:
Chapter 1 firstly introduce some research background and current stage of chemo-stat model and Lotka-Volterra model. Secondly, we list some primary theory such as fixed-point index theory, eigenvalue problem and bifurcation theory, etc.
Chapter 2, a competition model between plasmid-bearing and plasmid-free or-ganisms in the unstirred chemostat is researched. Firstly, the sufficient condition of existence on the coexistence solutions are obtained by applying fixed-point index theory. Secondly, global structure of bifurcation solutions and local stability are inves-tigated by global bifurcation theory and eigenvalue perturbation technique. Thirdly, uniqueness, multiplicity and linear stability of coexistence solutions depending on pa-rameter ki(i=1,2) are discussed by fixed-point index theory, perturbation technique and linear stability method. Finally, the longtime behavior of the system is determined by monotone method.
Chapter 3, the positive solutions of a Beddington-DeAngelis food chain unstirred chemostat model is investigated. Firstly, the existence and stability of the trivial、semi-trivial、strongly semi-trivial solutions of the model are obtained. Secondly, the index of the trivial、weakly semi-trivial、strongly semi-trivial solutions are calculated by fixed-point index theory. Finally, the sufficient condition of the existence to positive solutions are given.
Chapter 4 investigate a strongly couple prey-predator model with Michelis-Menten functional response in a bounded domain under Dirichlet boundary condition. some priori estimates for steady-state solutions are obtained by maximum principle. The sufficient condition of the existence for positive solutions is given, next, the coexistence region of positive solutions is analyzed by implicit function theorem, intermediate value theorem and Dini's theorem. The relation of the coexistence regionΣand the parameters a and b is considered. Moreover, we can prove that the coexistence region∑spreads (narrows) asβ(α) increases. Finally, local bifurcation solution and global bifurcation solution are obtained by bifurcation theory.
Chapter 5 deals with the positive solution and asymptotic behavior of a prey-predator model with predator saturation and competition. Firstly, global structure of bifurcation solutions and local stability are investigated by global bifurcation theory and eigenvalue perturbation technique. Secondly, the unique existence and stability of bifurcation solution which bifurcates from double multiplicity eigenvalue are proved by Lyapunov-Schmidt procedure. Thirdly, by fixed-point index theory and perturbation, we may show that if parameter d is sufficient small or m or k is sufficient large, then the model exist at least two positive solutions. Especially, if parameter m is sufficient large and a satisfies some condition, the model have exactly two positive solutions which belongs to two different types, one asymptotically stable and the other unstable. Finally, the asymptotic stability and persistence of positive solutions are obtained by monotone method and persistence theory.
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