摘要
随着科学和技术的发展,生物数学已经被广泛的应用于各个领域,例如：生物技术、药物动力学、物理、经济、种群动力学、流行病学等领域.微分方程数学模型在描述生物动力学性质中起到了非常重要的作用.人们用它不仅能够从数学的角度解释各种生物学行为,而且还能够用它来解释一些复杂的生物现象,从而达到对某些生物的相互作用进行有目的地控制.自然界的许多变化规律呈现出脉冲效应,用脉冲微分方程描述某些运动状态在固定时刻或不固定时刻的快速变化或跳跃更为切合实际.由于脉冲动力系统的解在脉冲时刻之间具有连续性,而在脉冲时刻具有间断性,使得脉冲动力系统比相应连续动力系统理论更为复杂.本文根据生化反应动力学、种群动力学、脉冲微分方程、时滞微分方程基本理论研究了脉冲效应在微生物发酵和阶段结构模型中的动力学性质,包括平衡点的存在性和稳定性、周期解的存在性和吸引性、系统的持久性与灭绝等.
全文分五章,主要结构概括如下：
前两章分别给出微生物发酵与阶段结构种群模型的研究背景和有关脉冲微分方程动力系统的一些基本理论,包括解存在性、连续性和Foquet定理等.
由于乳酸是一种非常重要的有机酸,在工农业生产中有很多重要的用途,本文第三章首先利用微分方程定性分析理论研究连续输入营养基的乳酸发酵模型,得到了正平衡点全局渐近稳定性和发酵失败微生物灭绝平衡点的稳定性的条件.接着研究具有周期脉冲输入乳酸发酵模型,通过Floquet乘子理论得到微生物灭绝周期解的稳定性条件.用周期脉冲分支理论得到系统正周期解的存在性,最后通过计算机进行数值分析并得出生物结论.
恒化器是一种最简单的实验装置,能够广泛的用于各种各样的微生物培养.因此第四章我们研究微生物培养中恒化器模型的性质,以便为微生物的培养提供理论依据和指导.本章首先给出带时滞和脉冲输入两种互补营养液恒化器模型、利用脉冲微分方程和时滞微分方程比较定理,得到了微生物灭绝周期解全局吸引的充分条件,同时也得到了系统持续生存的充分条件.接着研究在污染环境下带周期脉冲输入恒化器模型的持续和灭绝.最后一节研究带脉冲输入具有环状结构恒化器模型动力学性质.通过脉冲微分方程和差分方程的有关理论得到边界周期解的局部稳定性,通过计算机进行数值分析当边界周期解失去稳定性时系统出现复杂的动力学性质包括周期解、倍周期分支、混沌等.
第五章假设捕食者具有阶段结构而害虫具有一般的增长率,利用离散动力系统的频闪映射得到捕食者灭绝周期解,并利用脉冲微分方程和时滞微分方程的有关理论,得到了捕食者灭绝周期解全局吸引和系统持续生存的充分条件.数值模拟验证了理论结果,最后通过系统中某些参数的变化研究了脉冲扰动对种群复杂性质的影响.
With the development of science and technology, biomathematics has been used in many domains such as biological technology, medicine dynamics, physics, economy, popu-lation dynamics and epidemiology. Mathematical models of differential equations play an important role in describing biological dynamics. Mathematically, these models explain all kinds of biological behaviors, which allows people to understand biological complexity scientifically so that some interactions of population can be intend to control. Impulsive differential equations are suitable for the mathematical simulation of the evolutionary process in which the parameters undergo relatively long period of smooth variation fol-lowed by a short-term rapid change in their values. The solutions of the impulsive systems are continuous between two impulses and discontinuous at the impulse, which makes the theory of the impulsive systems more complicated than that of the corresponding contin-uous systems. Based on the basic theory of the biochemical reaction kinetics、population dynamics、impulsive differential equations and delay differential equations, microorgan-ism fermentation and stage-structured population models have been established to study the effects of impulses on the these models including the existence and stability of equi-libria, the existence of periodic solution and its global attractivity, the permanence and extinction of system.
The dissertation has five chapters and the main results of this dissertation may be summarized as follows:
Chapter 1 and Chapter 2 give the biological backgrounds of the microorganism fer-mentation and pest control, basic theories and preliminaries of impulsive differential equa-tions, including the existence of the solution, the continuity of the solution and Floquet's theory and so on.
Lactic acid is one of the organic acids, which has many applications in various types of industry and agriculture. Chapter 3 studies the dynamical behaviors of the lactic acid fermentation. Firstly, we present the lactic acid fermentation model with the continuous input. The globally asymptotical stability of the positive equilibrium has been obtained by using the qualitative analysis method. Secondly, we give the lactic acid fermentation model with impulsive input. The sufficient condition for the existence and stability of the biomass-free periodic solution is gotten by using the Floquet's theory. There exists a unique positive periodic solution via bifurcation theory, which implies the substrate, biomass and lactic acid oscillate with a positive amplitude. The numerical simulations verify the theoretical results and give some biological explanation.
The chemostat can be used for representing all kinds of microorganism systems, mostly because these are the simplest and most easily constructed types of continuous cultures. In Chapter 4, we investigate the dynamics of the chemostat model during the microorganism culture, which can give a theoretical guidance. Firstly, we present dynamics of two-nutrient and one-microorganism chemostat model with time delay and pulsed input. By means of the principle of the impulsive differential equations and delay differential equations, the sufficient condition for the existence and attractivity of the microorganism-free periodic solution is obtained. In addition, the sufficient condition for the permanence of the system is also obtained. Next, we discuss the extinction and permanence of chemostat model with pulsed input in a polluted environment. Finally, The dynamical behavior of the annular chemostat model including competition relation and predator-prey relation is investigated. By using the theories about the impulsive differential equations and difference equations, the sufficient condition for the existence and stability of the boundary periodic solution is obtained. When the boundary periodic solution losses its stability, numerical simulation shows there is a characteristic sequence of bifurcation, leading to a chaotic dynamics, which implies that this system has more complex dynamics including period-doubling bifurcation, chaos and strange attractors.
In Chapter 5, It is assumed that the predator population has two stages including immaturity and maturity and prey has a generic growth rare. Using the discrete dynam-ical system determined by the stroboscopic map, we obtain predator-extinction periodic solution. By use of the basic theorem related to impulsive differential equations and delay differential equations, sufficient condition for the global attractivity of predator-extinction periodic solution and permanence of the system is obtained. Furthermore, the results are confirmed by numeric simulations and complex dynamics is also investigated in view of the some parameter variations.
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