摘要
恒化器模型是生物和数学中非常重要的模型之一。利用恒化器连续培养微生物已是微生物学研究中的一项重要的研究手段；是原理和应用之间的一个极其重要的中介。它已广泛的应用于研究微生物的种群增长和相互作用规律，也应用于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。
本论文基于当前生物学模型，特别是恒化器模型的研究现状，深入系统的研究了时滞和扩散方程描述的几类恒化器系统的渐近性态，本文的主要内容包括以下几个方面：
一、研究了具有Beddington-DeAngelies功能性反应函数的时滞恒化器模型，利用无穷维连续动力系统的一致持续生存的理论给出了两竞争种群一致持续生存的充分条件，利用单调动力学系统得到了系统的全局渐近稳定性。
二、研究了无种内竞争和有种内竞争的具有阶段结构的时滞恒化器模型的渐近性态，对于两类模型，都在正平衡点存在性的条件下证明了该系统的一致持续生存，对于两类相应的常微系统的模型，均在正平衡点存在性的条件下证明了该正平衡点的全局稳定性。
三、研究了单营养食物链的恒化器模型的渐近性态，利用波动引理给出了边界平衡点全局吸引性的充分条件。然后利用无穷维动力系统一致持续生存的理论给出了该系统一致持续生存和绝灭的充分条件。
四、在周期环境中研究了扩散双营养恒化器系统的一致持续生存和周期解的存在性。利用无穷离散动力系统的一致持续生存的理论给出了该系统一致持续生存的充分条件。然后在一致持续生存的条件下得到了该系统周期解的存在性。
五、研究了一般的具有周期环境扩散种群模型的渐近性态。利用反应扩散方程的比较原理给出了系统存在周期解的充分条件。然后利用单调正、凹算子理论，给出了该扩散种群模型周期解全局吸引的充分条件。从而把有关时滞系统的相关结果推广到了扩散系统。并给出了具体的应用。然后进一步研究具有周期环境的双营养扩散恒化器模型的渐近性态，在周期解存在唯一的条件下证明了该周期解的全局吸引性。
六、研究了一类生物反应器中双营养扩散模型的渐近性态。在该生物反应器系统中引入了系统本身存在的流速，并考虑了系统中营养和种群的不同扩散率和种群在反应器中的死亡率。首先考虑了具有互补营养的扩散模型，得到了该系统中种群
绝灭和一致持续生存的充分条件；并对营养和种群具有相同的扩散系数和种群零死
亡率的模型，证明了该系统存在唯一的正平衡解，并证明了该平衡解的全局吸引性。
然后研究了具有可替代营养的扩散模型，给出了系统中种群绝灭和一致持续生存的
充分条件；并进一步研究了营养和种群具有相同的扩散系数和种群零死亡率的模型
唯一正平衡解的全局吸引的充分条件。
Chemostat model is one of the most significant models in Mathematical biology. The Chemostat is an important device used for growing micro-organisms in a continuous cultured environment, and a medium of great importance between principles and applications. It has been widely applied to the study of the increase in different populations of micro-organisms and their interactive law. In addition, it has also been applied to the management and prediction of the ecology system, especially the marine ecology, and the control of the environment pollution.
In the light of the recent work in biological models, especially in the chemostat models, the dissertation provides a systematic study on the asymptotical behaviour of some chemostat models built by delay or diffusion differential equations. The main contents and results in this dissertation are as follows:
i) The global asymptotic behavior of the Chemostat model with the Beddington-DeAngelies functional responses and time delays is studied. The conditions for the uniform persistence of the competing populations are obtained via uniform persistence of infinite dimensional systems. Then the global asymptotical stability of the positive equilibrium of the model with time delays is proved via monotone dynamical systems. Our results imply that mutual interference in a species may result in coexistence of the two competing species and demonstrate that those time delays do not influence the competitive outcome of the organisms.
ii) The asymptotic behaviour of the Chemostat model with mutual interference or without mutual interference is studied. For the two models with delay, the uniform persistence of the models are both proved under the conditions of the existence of the positive equilibrium. Moreover, under those conditions, the global stability of the positive equilibrium is proved for the two models without delays.
iii) The asymptotic behaviour of the Chemostat model with predator-prey populations and delays is studied. Sufficient conditions for the global attractivity of boundary equilibrium are obtained via fluctuation lemma, and sufficient conditions for uniform persistence of this model are obtained via uniform persistence of infinite dimensional systems.
iv) The Chemostat model with diffusion and two-nutrients is considered. Sufficient conditions for uniform persistence of this model are obtained via uniform persistence of infinite dimensional discrete dynamical systems. Then one can easily obtain the existence of periodic solution of the Chemostat model.
v) The asymptotical behavior of population models with diffusion is studied. Firstly sufficient conditions for the existence of periodic solution are obtained by comparison theory of reaction-diffusion differential equations; secondly sufficient conditions are established, under which the models admits a positive periodic solution which attracts all positive solutions. Then we apply the general theory to some types of population models with diffusion and periodic coefficients. Thus some earlier results of population models with delays are extended to diffusion population models.
Finalh. the asymptotic behaviour of the Chemostat model with two-nutrient and diffusion is further studied. The global attractivity of the periodic solution is proved under the unique existence of the periodic solution.
vi) The asymptotic behavior of flow reactor models with two-nutrient are considered. Different diffusion coefficients of the population and nutrients, the death rates of the population and the velocity exist in the flow reactor are introduced in these models. In complementary case, sufficient conditions for uniform persistence and extinction of the population are obtained by the theory of uniform persistence of infinite dimensional dynamical systems. Especially for the model with equal diffusion coefficients and zero death rates, the global attractivity of the unique positive steady-state solution is proved. In substitutable case, sufficient conditions for uniform persistence and extinction of population
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