摘要
生物数学中以生物动力系统为基础的研究近年来得到了长足的发展,其中以微分方程为模型的研究工作主要集中在连续动力系统和脉冲动力系统上.特别是脉冲动力系统对于在瞬间干扰下状态发生突变的演变过程提供了有力的自然描述.脉冲的出现使得系统具有混合性,既有连续的特点,又有离散的特点,因而脉冲动力系统的理论比相应的连续动力系统的理论更加丰富和复杂.它被广泛应用于生物技术、物理、经济、种群动力学、流行病学等领域.本文考虑了害虫治理中的传染病模型和微生物培养模型,并利用脉冲微分方程的的相关理论和方法研究了相应的动力学模型,讨论了所提模型的各种动力学行为,包括平衡点的存在性和稳定性、周期解的存在性和吸引性、系统持久性和灭绝等.全文共分五章：
第1章是绪论,介绍了对害虫进行控制和管理的研究意义及背景,并简要叙述了脉冲微分方程的研究现状及本文的主要工作.
第2章给出脉冲微分方程的基本理论和相关知识.
第3章讨论治理害虫的病毒感染动力学模型.第一节研究连续和脉冲投放病毒治理害虫的模型.在连续投放病毒模型中,应用常微分方程的分析方法,得到系统存在唯一全局渐近稳定平衡点的充分条件.在脉冲投放病毒的模型中,根据Floquet乘子定理,小振幅干扰的方法以及脉冲微分方程的比较定理,得到害虫根除周期解是全局渐近稳定的以及系统持久的充分条件.最后给出数值模拟并讨论两种方法的有效性.第二节研究状态反馈控制对一类病毒感染SV模型周期解存在性的影响.通过周期解的存在性来讨论把害虫控制在经济危害水平下的周期性.在定性分析的基础上,利用一般脉冲自治系统周期解的存在准则,获得了阶1周期解的存在条件,并指出系统要么趋于一个稳定状态要么趋于周期解,这依赖于反馈控制状态(h)、控制参数(p：ω)和病毒与害虫的初始浓度.最后利用数值模拟验证了理论结果.第三节研究的是把害虫分为易感害虫和染病害虫的SIV流行病模型.为了控制害虫的数量,在固定时刻脉冲投放病毒颗粒,使易感害虫染病,从而让流行病在害虫种群中传播以达到控制害虫数量的目的,因为染病害虫是不危害农作物的.得到了无易感害虫周期解是全局渐近稳定的和系统持久的充分条件.结论指出脉冲周期T和病毒颗粒的释放量p对系统的动力学行为具有重要的影响.
第4章讨论两个食饵具有流行病的阶段结构种群动力系统.第一节研究了害虫(食饵)具有流行病的阶段结构及成熟时滞的捕食模型.利用离散动力系统的频闪映射获得易感害虫根除周期解,并利用脉冲时滞微分方程的比较定理及不等式的技巧,证明了该周期解的全局吸引性.通过脉冲投放染病害虫和天敌得到了把易感害虫控制在易感害虫的危害程度不超过作物的补偿点,即在经济危害水平之下,染病害虫和天敌的最小投放量和最长投放周期.为合理的利用染病害虫和天敌控制害虫提供了理论依据.第二节研究捕食者具有阶段结构的食饵依赖的且具有流行病的捕食模型.利用Floquct乘子定理和脉冲微分方程的比较定理,通过脉冲释放染病害虫和天敌获得易感害虫灭绝周期解是全局渐近稳定的和系统持续生存的条件.我们的结论表明：染病害虫和天敌的脉冲释放量和脉冲周期对系统的动力学行为有重要的影响.
第5章讨论了微生物培养模型.第一节考虑了具有时滞增长反应及脉冲输入的两种群竞争Monod-Haldane模型.分析了营养基的脉冲输入,时滞增长反应对恒化器系统的影响.结论表明微生物的灭绝与否决定于在每一次nT时刻的营养液的脉冲输入量,脉冲周期的长短以及微生物生长繁殖的迟滞.第二节考虑了在污染环境中具有时滞增长反应及脉冲输入的两种群竞争Monod模型.分析了营养基的脉冲输入,时滞增长反应以及有毒物质的脉冲输入对恒化器系统的影响.所得结果表明微生物的灭绝与否决定于在每一次nT时刻的营养液的脉冲输入量以及同时伴随的有害物质的输入量.分析得出没有污染的恒化器环境有利于微生物的培养,而污染的环境可能导致微生物的灭绝.这表明有害物质的输入对恒化器模型的动力学行为产生了重要的影响.第三节考虑微生物培养的一个应用,气提法乙醇发酵数学模型的研究.分别考虑连续输入与脉冲输入营养基的情形.对连续输入营养基模型,根据Poincare-Bendixson定理,得到正平衡点是全局渐近稳定的充分条件,这意味着我们可以得到稳定的乙醇生产.对脉冲输入营养基的模型,我们得到边界周期解是局部稳定的.进而,证明了在临界的情况下,系统会分支出一个非平凡的周期解.通过数值模拟验证了主要结果并且得到脉冲输入比连续输入更加有效.
The need for describing more actual natural system impels the evolution of mathe-matical biological models. In recent years, the researchs in mathematical biology which models by normal differential equations are mainly concerntrated on two branches:con-tinuous dynamical systems and impulsive semi-dynamical systems. Especially, the im-pulsive dynamical systems are suitable for the mathematical modelling of evolutionary processes which experience a change of state abruptly owing to instantaneous perturba-tions. The presence of impulses gives the system a mixed nature, both continuous and discrete. Therefore the theory of impulsive dynamical systems is much richer than the corresponding theory of dynamical systems without impulsive effects. In recent years, im-pulsive dynamical systems, have been widely used in biotechnology, physics, economics, population dynamics, epidemiology and so on. In this thesis, the infectious diseases mod-els and microbial culture models in pest management are established to consider several problems by means of the theory and method of impulsive differential equations. Dy-namic behaviors, including the existence and stability of equilibriums, the existence of periodic solution and its global attractivity, the permanence and extinction of system, are investigated. The thesis has five chapters:
In Chapter 1，the backgrounds of the system investigated in the thesis are given. The relative researches are stated briefly.
In Chapter 2, some preliminaries and the relative results of impulsive differential equations are introduced.
In Chapter 3, viral infection dynamical models in pest management are formulated and investigated. In section 3.1, two models of continuous and impulsive release viruses are studied. In the case in which a continuous control is used, it is shown that the system admits a globally asymptotically stable positive equilibrium by means of analytical methods of ordinary differential equations. In the case in which an impulsive control is used, it is observed that the pest-eradication periodic solution is globally asymptotically stable by using Floquet theorem and comparison results of impulsive differential equations. Finally, the efficiency of continuous and impulsive control policies is compared by means of numerical simulation. Section 3.2 discusses the effects of impulsive state feedback control on the viral infection SV model. Based on the qualitative analysis, the conditions for the existence of periodic solution of order one are obtained by the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the viruses and the initial concentrations of viruses and pests. Finally, the theoretical results are verified by numerical simulations. In section 3.3, the SIV epidemic model is constructed and studied. It is assumed that the pests has two classes, susceptible pests and infected pests. In order to control the number of pest, virus particles are impulsively released at fixed time. So as to the susceptible pests can be infected, the number of the pests can be controlled effectively. The sufficient conditions of the susceptible pest-eradication periodic solution and the permanence of the system are obtained. Our results indicate that impulsive period and the release amount of the virus particles have great effects on the dynamics of our system.
In Chapter 4, two stage-structured predator-prey models with infectious disease in the prey are studied. In section 4.1, a stage-structure delay prey-predator model with infectious disease in the prey is investigated. Using the discrete dynamical system de-termined by the stroboscopic map, we obtain the susceptible pest-eradication periodic solution. By use of comparison theorem and differential inequalities for delay impul-sive differential equations, we show that the period solution is globally attractive. By impulsive releasing infected pests and natural enemies, we obtain the minimal releasing infected pests and natural enemies and maximum impulsive period, the susceptible pests are controlled under the economic threshold level, that is the harm of the susceptible pests are no more than the super compensation point of the crops. The results pro-vide a reliable theoretical tactics for pest management and also indicate the important influence of time delay on population dynamics. In section 4.2, a prey-dependent con-sumption predator-prey(natural enemy-pest) model with age structure for the predators and infectious disease in the prey is studied. By using Floquet theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. Our results indicate that impulsive period and the release amount of infected pests and natural enemies have great effects on the dynamics of our system.
In Chapter 5, microbial culture models are investigated. Section 5.1 consider a Monod-Haldane competitive chemostat model with delayed growth response and impul-sive input nutrient. The effect of impulsive input of the nutrient, time delay for growth response on dynamic behaviors of chemostat model is analyzed. Whether the microor-ganisms is extinct or not is determined completely by the input amount of the substrate, the length of impulsive period and the time delay of microbial growth and reproduction at fixed impulsive period nT. Section 5.2 consider a Monod competitive chemostat model with delayed growth response and impulsive input nutrient in a polluted environment. We also analyze the effect of impulsive input of the nutrient, time delay for growth response and impulsive input of the toxicant on dynamic behaviors of chemostat model. Whether the microorganism is extinct or not is determined completely by the input amount of the substrate and concentration of the toxicant at fixed impulsive period nT. The results show that the environment without pollution conducive to microbial culture and polluted environment may lead to the extinction of microorganism. This shows that the input con-centration of the toxicant greatly affects the dynamics behaviors of the model. Section 5.3 consider an application of microorganism cultured-the study of mathmatical model of ethanol fermentation with gas stripping. We consider continuous input and pulse input of nutrition respectively. For the case of continuous input substrate, we study the existence and local stability of two equilibria. According to Poincare-Bendixson Theorem, the suf-ficient condition of the globally asymptotical stability of positive equilibria is obtained. Which implies we can get stable ethanol product. For the case of impulsive input sub-strate, we obtain the sufficient condition of the local stability of cell-free periodic solution by using the Floquet's theory of impulsive equation and small amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges. Further, our main results are verified by means of numerical simulation and obtain the the impulsive input is more effective than continuous input.
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