摘要
生物数学的研究起始于Lotka-Voltcrra年代：20世纪70年代May的《理论生态学》、Smith的《生态学模型》等专著的出版推动了生物数学的迅速发展：近些年来生物数学的研究范围也得以扩展,已经应用于很多领域,例如：种群动力学、传染病动力学、药代动力学、化学反应模型、微生物培养等领域.生物种群、微生物及化学反应物的浓度是否持续生存是生物数学研究的一个重要方面.本文应用泛函微分方程理论、常微分方程定性理论、脉冲微分方程理论来研究种群动力学模型、微生物培养模型及化学反应模型的一些动力学性质.
全文共分五章,基本内容概要如下：
前两章分别介绍了阶段结构比率依赖种群模型、微生物培养模型、化学分子反应模型的研究背景和有关泛函微分方程、脉冲微分方程的一些基本概念理论等.
第三章主要考虑具有时滞比率依赖的阶段结构捕食模型.第一节考虑一类食饵分散捕食者具有阶段结构的比率依赖的Holling-Ⅲ类功能反应的捕食模型.应用泛函微分方程的理论得到了系统持续生存的充分条件,也得到了系统成年捕食者趋于灭绝的充分条件.通过应用Gaines和Mawhin的重合度理论得到了系统存在正周期解的充分条件并给出了数值模拟.第二节我们考虑了一类捕食者具有阶段结构食饵具有脉冲投放的比率依赖Holling-Ⅲ类功能反应的捕食模型.应用泛函微分方程和脉冲微分方程的相关理论得到了系统捕食者灭绝剧期解的全局吸引的充分条件,同时这个充分条件也是系统不带脉冲效应的捕食者灭绝周期解的全局吸引的充分条件.我们也获得了系统持续生存的充分条件.
第四章讨论状态脉冲在微生物培养方面的应用.我们建立了一类具有Beddington-DeAnglis吸收功能和脉冲状态控制的恒化器模型,应用常微分方程定性理论研究了系统不带脉冲控制部分的定性特征.获得了系统全局渐近稳定的充分条件.由脉冲微分方程的有关定理得到了该脉冲系统阶一周期解的存在性和稳定性的充分条件,也对系统阶二周期解的存在可能性进行了分析.通过数值模拟进一步证实了我们的理论结果.
第五章研究了脉冲微分方程在化学分子反应模型中的应用.第一节研究一类具有脉冲输入的三分子反应模型,应用常微分方程定性理论对模型进行了简单分析,得到了系统全局渐近稳定的充分条件.应用脉冲微分方程的理论对具有脉冲输入的模型进行了分析,也得到了脉冲三分子反应系统持续生存的充分条件.通过数值模拟展示了系统的丰富的动力行为.第二节考虑一类具有脉冲输入的饱和三分子反应模型.我们应用常微分方程定性理论对不带脉冲输入的饱和三分了反应模型进行简单的定性分析,获得了有极限环的充分条件,应用脉冲微分方程的理论对脉冲输入的饱和三分子反应模型进行了分析,得到了脉冲饱和三分子反应模型周期解渐近稳定性的充分条件：应用脉冲微分方程分支理论得到了脉冲输入的饱和三分子反应模型存在稳定的非平凡正周期解的条件.我们通过数值模拟进一步证实了我们的结论也展示这个模型的丰富动力行为.
The research of Biomathematics dated from the Lotka-Volterra age, and the pub-lication of monographs such as May's"theoretical ecology", Smith's "ecological model" promoted the rapid development of Biomathematics in 1970's. The research's range of Biomathematics has been expended and applied in many domains such as population dy-namics, epidemic dynamics, pharmacokinetics, chemical reaction model, microbiological culture and so on. The research of permanence for biological populations, microorganisms and chemical reactants is the important aspect of the biomathematics'research. Based on the theory of functional differential equations, ordinary differential equations and im-pulsive differential equations, we study the kinetic properties of the population dynamics, microbial culture models and chemical reaction models.
The dissertation has five chapters and the main results of the dissertation are sum-marized as follows:
The first two chapters introduce the biological backgrounds of the ratio-dependent population model with stage structure, microbial culture model, the chemical reaction model and the relevant theories and concepts of functional differential equations, impulsive differential equations, respectively.
In Chapter 3. the ratio-dependent population models with stage structure are con-sidered. We discuss a delayed ratio-dependent prcdator-prcy with prey dispersal and stage structure for predator in Section 3.1. Using the theory of functional differential equations, we obtain sufficient conditions for permanence of the system and sufficient condition for adult predators of the system tending to extinct. By Gaines and Mawhin's coincidence degree theory, we obtain sufficient conditions that the system exists posi-tive periodic solutions. The theoretical results are verified by numerical simulations. In Section 3.2, we consider a delayed ratio-dependent Holling-Ⅲpredator-prey system with stage-structured and impulsive stocking on prey and continuous harvesting on predator. Using the theory of functional differential equations and impulsive differential equations, we obtain sufficient conditions of the global attractivity of predator-extinction periodic solution, while the sufficient conditions are also sufficient conditions of the global attrac-tivity of predator-extinction periodic solution of the system without impulsive input. We also obtain sufficient conditions for permanence of the system.
In Chapter 4, we discuss the application of state control impulsive equations in mi- crobiological culture. We establish a chemostat model with Bcddington-DcAnglis uptake function and impulsive state feedback control. By the qualitative theory of ordinary differ-ential equations, we sufficient conditions of the global asymptotic stability of the system without impulsive input. Using the relevant theory of impulsive differential equations, we obtain sufficient conditions for existence and stability of order one periodic solutions of impulsive system, and also analyze the existence of order two periodic solutions. The numerical simulation further confirms our theoretical results.
In Chapter 5, we consider applications of impulsive differential equations in chemical reaction models. Section 5.1 studies a trimolecular response model with impulsive input. By qualitative theory of ordinary differential equations and a simple analysis of the system without impulsive input, sufficient conditions for global asymptotic stability are obtained. By the theory of impulsive differential equations, we obtain conditions for permanence of the trimolecular response model with impulsive input. By numerical analysis, we demonstrate complex phenomena such as limit cycles, periodic solutions, and chaos. In Section 5.2, we consider a irreversible three molecular reaction model with impulsive effect. By qualitative theory of ordinary differential equations, we obtain sufficient conditions that the system without impulsive input exists a limit cycle by a simple analysis of the system. Using the theory of impulsive differential equations, we obtain sufficient conditions for asymptotic stability of system with impulse effect. By bifurcation theory of impulsive differential equations, we obtain sufficient conditions that the system with impulse effect exists a stable positive periodic solutions. We further confirm our results and also show that the system has rich dynamic behaviors by numerical simulations.
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