脉冲方程在微生物培养和种群控制中的应用
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摘要
微分方程数学模型在描述种群动力学行为中起到了非常重要的作用。它从数学的角度解释各种种群动力学行为，使人们科学地认识种群动力学，从而对某些种群相互作用进行有目的地控制。特别是用脉冲微分方程来描述种群动力学模型能够更合理、更精确地反映各种变化规律，因为现实世界中的许多生命现象和人类的开发行为几乎都是脉冲的。本文针对微生物培养和种群控制的几个问题利用脉冲微分方程的相关理论和方法建立并研究了相应的动力学模型，同时借助计算机模拟讨论了所提模型的各种动力学行为，包括平衡点的稳定性、周期解的存在性、系统的持久性与灭绝以及系统动力学的复杂性。本文的主要结果概括如下：
第二章讨论微生物培养。第一节研究具有变消耗率的比率确定型Chemostat模型的渐近行为。模型假定了消耗率是营养基的线性函数而且增长率是比率确定型函数，推广了经典的Monod模型。利用常微分方程定性理论证明了只要正平衡点存在系统就是持续生存的，同时也给出了极限环存在和正平衡点全局渐近稳定的充分条件。第二节研究脉冲输入营养基的Monod型Chemostat模型的动力学行为。利用Floquet理论和小振幅干扰的方法证明脉冲周期满足一定条件时，微生物灭绝周期解是渐近稳定的。然后用分析的方法讨论了系统的持久性。最后用数值模拟验证了主要结论。第三节利用类似于第二节的方法讨论了具有变消耗率和脉冲干扰营养基的Chemostat模型的复杂动力学。
第三章讨论了非自治周期系统周期解的存在性。第一节研究一个具有Beddington-DeAngelis功能反应和脉冲干扰的捕食系统。利用拓扑度理论的连续性定理给出系统正周期解存在的充分条件，并给出例子借助计算机模拟说明脉冲对种群动力学的影响。第二节利用k-集压缩理论研究脉冲时滞Logistic模型正周期解的存在性，给出正周期解存在的充分条件。第三节利用拓扑度理论研究一个具有功能反应的捕食系统正周期解的存在性问题。给出系统正周期解存在的充分条件并模拟了主要结论。
第四章讨论脉冲干扰对捕食模型的影响。第一节针对传染病控制害虫的理论，研究投放染病害虫控制害虫增长的优化控制。我们假定投放染病害虫的方式有连续的和脉冲的两种。因此，相应的模型分别是常微分方程模型和脉冲微分方程模型。利用常微分方程定性理论和脉冲微分方程理论分析了两个模型。从数学的角度给出在综合害虫管理中利用染病害虫控制害虫增长的一个理论依据。第二节研究一个捕食者具有脉冲迁入的Holling型捕食系统的动力学行为。用分析的方法证明了该系统是持续生存的，通过数值模拟显示了捕食者迁入对系统动力学的影响。
Mathematical models of differential equations play an important role in describing popula-tion dynamic behavior. Mathematically, these models explain all kinds of population dynamicbehavior, which allows people to understand population dynamics scientifically so that someinteractions of population can be intend to control. Especially, impulsive differential equationsdescribe population dynamic models, which is more reasonable and precise on reflecting all kindsof change orderliness, since many life phenomena and human exploitation are almost impulsivein the natural world. In this dissertation, population dynamic models are established to con-sider several problems in microorganism cultures and population controls by means of the theoryand method of impulsive differential equations. Numerical simulations are used to investigatedynamic behavior including the stability of equilibrium, the existence of periodic solution, thepermanence and extinction of system and the complexity of system. The main results of thisdissertation mav be summarized as follows:
In Chapter 2, microorganism cultures are investigated. Asymptotic behavior in the ratio-dependent chemostat model with variable yield is studied in Section 2.1. In the model, weassume that the yield is a linear function of the nutrient concentration and the microbial growthrate is a ratio-dependent type function. Thus, we have developed the classical Monod model. Itis shown that system is permanent if and only if it has a positive equilibrium by the qualitativetheory of ordinary differential equation. The sufficient conditions of existence of limit cyclesand globally asymptotic stability of the positive equilibrium for the model are given. In Section2.2, dynamic behaviors of Monod type chemostat model with impulsive input on the nutrientconcentration. Using Floquet theory and small amplitude perturbation method, we prove thatthe microorganism-eradication periodic solution is asymptotically stable if the impulsive periodsatisfies some conditions. Moreover, the permanence of the system is discussed in analyticalmethod. Finally, we verify the main results by numerical simulation. Using the similar methodof Section 2.2, complex dynamics of a chemostat with variable yield and periodically impulsiveperturbation on the substrate is studied in Section 2.3.
In Chapter 3, the existence of periodic solutions of nonautonomous periodic systems isconsidered. In Section 3.1, the existence of periodic solutions of a predator-prey system withthe Beddington-DeAngelis functional response and impulsive perturbations is studied. By thecontinuation theorem of the coincidence degree theory, the sufficient conditions of the existenceof positive periodic solution are obtained. Finally, an example is given to illustrate the influence of the impulse on population dynamics by numerical simulation. In Section 3.2, using the con-tinuation theory for k-set contractions, the sufficient conditions of existence of positive periodicsolution of the impulsive delay Logistic model are given. In Section 3.3, the problem of existenceof positive periodic solution of the predator-prey system with functional response is analyzed bythe continuation theorem of the coincidence degree theory, the sufficient conditions of existenceof positive periodic solution are obained and the main result is simulated by computer.
In Chapter 4, the effect of impulsive perturbation on predator-prey models is investigated.On the theory of controlling pests using epidemic, the problem on optimal controlling pestsby infected pests is studied in Section 4.1. We assume that the releases of infected pests arecontinuous and impulsive. Thus, the corresponding models are the ordinary differential equationsand the impulsive differential equations, which are studied by the qualitative theory of ordinarydifferential equations and the theory of impulsive differential equations. Mathematically, thetheoretical evidence of the controlling pests using epidemic in the integrated pest managementis given. In Section 4.2, the dynamic behaviors of a Holling-type predator-prey system withimpulsive immigration on the predator is considered. It is shown that the system is permanent.By means of numerical simulation, we illustrate that with the increasing of the immigrationnumber, the system exhibits complex dynamics.
第二章讨论微生物培养。第一节研究具有变消耗率的比率确定型Chemostat模型的渐近行为。模型假定了消耗率是营养基的线性函数而且增长率是比率确定型函数，推广了经典的Monod模型。利用常微分方程定性理论证明了只要正平衡点存在系统就是持续生存的，同时也给出了极限环存在和正平衡点全局渐近稳定的充分条件。第二节研究脉冲输入营养基的Monod型Chemostat模型的动力学行为。利用Floquet理论和小振幅干扰的方法证明脉冲周期满足一定条件时，微生物灭绝周期解是渐近稳定的。然后用分析的方法讨论了系统的持久性。最后用数值模拟验证了主要结论。第三节利用类似于第二节的方法讨论了具有变消耗率和脉冲干扰营养基的Chemostat模型的复杂动力学。
第三章讨论了非自治周期系统周期解的存在性。第一节研究一个具有Beddington-DeAngelis功能反应和脉冲干扰的捕食系统。利用拓扑度理论的连续性定理给出系统正周期解存在的充分条件，并给出例子借助计算机模拟说明脉冲对种群动力学的影响。第二节利用k-集压缩理论研究脉冲时滞Logistic模型正周期解的存在性，给出正周期解存在的充分条件。第三节利用拓扑度理论研究一个具有功能反应的捕食系统正周期解的存在性问题。给出系统正周期解存在的充分条件并模拟了主要结论。
第四章讨论脉冲干扰对捕食模型的影响。第一节针对传染病控制害虫的理论，研究投放染病害虫控制害虫增长的优化控制。我们假定投放染病害虫的方式有连续的和脉冲的两种。因此，相应的模型分别是常微分方程模型和脉冲微分方程模型。利用常微分方程定性理论和脉冲微分方程理论分析了两个模型。从数学的角度给出在综合害虫管理中利用染病害虫控制害虫增长的一个理论依据。第二节研究一个捕食者具有脉冲迁入的Holling型捕食系统的动力学行为。用分析的方法证明了该系统是持续生存的，通过数值模拟显示了捕食者迁入对系统动力学的影响。
Mathematical models of differential equations play an important role in describing popula-tion dynamic behavior. Mathematically, these models explain all kinds of population dynamicbehavior, which allows people to understand population dynamics scientifically so that someinteractions of population can be intend to control. Especially, impulsive differential equationsdescribe population dynamic models, which is more reasonable and precise on reflecting all kindsof change orderliness, since many life phenomena and human exploitation are almost impulsivein the natural world. In this dissertation, population dynamic models are established to con-sider several problems in microorganism cultures and population controls by means of the theoryand method of impulsive differential equations. Numerical simulations are used to investigatedynamic behavior including the stability of equilibrium, the existence of periodic solution, thepermanence and extinction of system and the complexity of system. The main results of thisdissertation mav be summarized as follows:
In Chapter 2, microorganism cultures are investigated. Asymptotic behavior in the ratio-dependent chemostat model with variable yield is studied in Section 2.1. In the model, weassume that the yield is a linear function of the nutrient concentration and the microbial growthrate is a ratio-dependent type function. Thus, we have developed the classical Monod model. Itis shown that system is permanent if and only if it has a positive equilibrium by the qualitativetheory of ordinary differential equation. The sufficient conditions of existence of limit cyclesand globally asymptotic stability of the positive equilibrium for the model are given. In Section2.2, dynamic behaviors of Monod type chemostat model with impulsive input on the nutrientconcentration. Using Floquet theory and small amplitude perturbation method, we prove thatthe microorganism-eradication periodic solution is asymptotically stable if the impulsive periodsatisfies some conditions. Moreover, the permanence of the system is discussed in analyticalmethod. Finally, we verify the main results by numerical simulation. Using the similar methodof Section 2.2, complex dynamics of a chemostat with variable yield and periodically impulsiveperturbation on the substrate is studied in Section 2.3.
In Chapter 3, the existence of periodic solutions of nonautonomous periodic systems isconsidered. In Section 3.1, the existence of periodic solutions of a predator-prey system withthe Beddington-DeAngelis functional response and impulsive perturbations is studied. By thecontinuation theorem of the coincidence degree theory, the sufficient conditions of the existenceof positive periodic solution are obtained. Finally, an example is given to illustrate the influence of the impulse on population dynamics by numerical simulation. In Section 3.2, using the con-tinuation theory for k-set contractions, the sufficient conditions of existence of positive periodicsolution of the impulsive delay Logistic model are given. In Section 3.3, the problem of existenceof positive periodic solution of the predator-prey system with functional response is analyzed bythe continuation theorem of the coincidence degree theory, the sufficient conditions of existenceof positive periodic solution are obained and the main result is simulated by computer.
In Chapter 4, the effect of impulsive perturbation on predator-prey models is investigated.On the theory of controlling pests using epidemic, the problem on optimal controlling pestsby infected pests is studied in Section 4.1. We assume that the releases of infected pests arecontinuous and impulsive. Thus, the corresponding models are the ordinary differential equationsand the impulsive differential equations, which are studied by the qualitative theory of ordinarydifferential equations and the theory of impulsive differential equations. Mathematically, thetheoretical evidence of the controlling pests using epidemic in the integrated pest managementis given. In Section 4.2, the dynamic behaviors of a Holling-type predator-prey system withimpulsive immigration on the predator is considered. It is shown that the system is permanent.By means of numerical simulation, we illustrate that with the increasing of the immigrationnumber, the system exhibits complex dynamics.
引文
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