摘要
Lotka-Volterra模型和恒化器模型是两类重要的生物数学模型。Lotka-Volterra模型是种群动力学研究的核心内容，它在生态学，特别是动植物保护和生态环境的治理与开发等领域中有着非常重要的作用。恒化器是用于微生物连续培养的一种实验装置。它不仅是一个简化了的湖泊模型，可用于模拟湖泊和海洋中单细胞藻类浮游生物的生长，而且它已被广泛地应用于微生物的生产、生物制药、食品加工及生态系统尤其是水生生态系统的管理、预测和环境污染的控制。
本文基于这两类生物模型的研究现状，主要运用非线性分析和非线性偏微分方程工具，特别是反应扩散方程(组)和对应椭圆方程(组)的理论和方法，深入系统地研究了具有抑制剂的非均匀恒化器模型和具有非单调转换率的Lotka-Volterra模型的动力学行为，包括正平衡态解的存在性、多解性、稳定性以及长时行为。所涉及的数学理论包括：上下解方法、比较原理、单调动力系统理论、全局分歧理论、拓扑不动点理论、Lyapunov-Schmidt过程和扰动理论等。本文的主要内容包括以下几个方面：
一、研究了基本的非均匀恒化器模型，利用比较原理和上下解方法得到了模型正平衡解的全局吸引性。而且，采用上下解方法、Sobolev嵌入定理并结合特征值性质，详细分析了单物种模型的正解同物种生长率的关系。
二、考察了一类具有内部抑制剂的非均匀恒化器模型。首先分析了平凡的、半平凡的非负解的稳定性，得到了系统解的一些渐近行为，并根据单调动力系统理论得到了正平衡解的存在性。然后，利用度理论、分歧理论以及摄动理论，重点分析了抑制剂对系统正平衡态解及渐近行为的影响。结果表明体现抑制作用的参数μ在决定模型解的稳定性和长时行为时起了重要作用。当参数μ充分大时，如果物种μ的生长率适当大，则此模型没有正解，且其中一个半平凡的非负解是全局吸引的；如果物种μ的生长率满足一定条件，则此模型的所有正解由一个极限问题决定，且两个半平凡的非负解是双稳定的。
三、讨论了一类具有外加抑制剂的非均匀恒化器模型，利用分歧理论分析了共存解的全局结构和局部稳定性，采用单调方法研究了系统的渐近行为，并用数值模拟的方法说明了竞争物种灭绝或共存以及正平衡态解全局稳定的可能性，讨论了物种振荡与模型各参数的关系。
四、研究了一类具有内部抑制剂的质体负载(plasmid-bearing)与质体自由(plasmid-free)的物种相互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、分歧理论以
Lotka-Volterra model and chemostat model are two kinds of the most significant models in Mathematical biology. Lotka-Volterra model is the nuclear contents of population dynamics. This model plays a very important role in ecology, especially in protection of plants and creatures and in the control and exploiture of environment. The chemostat is a laboratory apparatus used for the continuous culture of microorganisms. It is used as an ecological model of a simple lake, as a model of the growth of unicellular phytoplankton in lake and sea. Moreover, it has been widely applied to the commercial production of microorganisms, biological pharmacy, food manufacture and the management and prediction of the ecology system, particularly the marine ecology, and the control of the environment pollution.
In the light of the recent work in these two kinds of biological models, mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of reaction-diffusion equations and corresponding elliptic equations, we have systematically studied the dynamical behavior of the unstirred chemostat model with inhibitor and Lotka-Volterra model with nonmonotonic conversion rate, such as coexistence, multiplicity, stability of positive steady states and the longtime behavior of species. The tools used here include super-sub solutions method, comparison principle, monotone system theory, global bifurcation theory, fixed-point theory of topology, Lyapunov-Schmidt procedure and perturbation technique. The main contents and results in this dissertation are as follows:
i) The standard unstirred chemostat model is studied. The global attractivity of the positive steady-state solutions of the original system is established by the comparison principle and super-sub solutions method. Moreover, the effects of the growth rate on the unique positive equilibrium of the single population model are studied in detail by means of super-sub solutions method, Sobolev embedding theorem and the properties of eigenvalues.
ii) An unstirred chemostat model with an internal inhibitor is discussed. First, the elementary stability and asymptotic behavior of solutions of the system are determined. The existence of positive steady-state solutions is given by monotone system theory. Second, the effects of the inhibitor are considered carefully by making use of the degree theory,
引文
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