种群控制中的脉冲和时滞效应
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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种群间及种群与环境间的动力学行为,从而使人们对某些种群之间以及种群与环境之间的相互作用进行有目的地控制。时滞脉冲微分方程不但考虑到瞬间变化对事物状态的影响,而且考虑到过去状态对事物变化的影响,所以能够更合理、更精确地反映种群变化的规律。本文针对种群控制和微生物病毒控制的几个问题,利用时滞泛函微分方程和脉冲微分方程的相关理论和方法建立了动力学模型,并研究它的相关的动力学行为,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。同时借助计算机数值分析了部分模型的动力学行为,并讨论其生物意义,本文的主要内容概括如下:
第二章简要介绍了时滞微分方程和脉冲微分方程的基本知识。
第三章讨论了三个具有功能性反应的时滞捕食模型,第一节研究了一个定期释放天敌来控制害虫的具Ivlev功能性反应的时滞捕食模型。在模型中,假设食饵分为幼年食饵和成年食饵,捕食者种群只捕食成年食饵,幼年食饵到成年食饵的转化期是一个常数,用一个常数时滞表示。捕食者对食饵的功能函数是Ivlev型的。得到了害虫灭绝周期解全局吸引和系统持久生存的充分条件,第二节研究了一个具有阶段结构和Ivlev功能反应的捕食模型的收获控制策略,对食饵脉冲式的投放,而对捕食者连续地收获。并假设捕食者分为幼年和成年捕食者,幼年捕食者不会捕食食饵种群。得到了捕食者灭绝周期解的全局吸引的条件,并且利用比较方法分析了脉冲投放食饵和适度收获捕食者的行为,可保证捕食者种群持续生存。第三节研究了一个脉冲干扰食饵并且具有阶段结构和时滞的Gomportz捕食模型。捕食者种群分为幼年捕食者和成年捕食者,只有成年捕食者具有捕获食饵的能力。食饵种群的增长方式是Gomportz型的,捕食功能函数是HollingⅡ型。研究了当脉冲式的捕获食饵时对天敌生存的影响,得到了捕食者灭绝周期解全局吸引的充分条件和系统持续生存的条件。并且数值模拟了种群的动力学行为,分析了在害虫的种群密度低于经济危害水平的前提下,需要害虫和天敌共存,以维护生态平衡。
第四章基于当前“以虫治虫,以菌治虫”的可持续发展思想,讨论了害虫管理策略的数学模型。第一节研究了一个SI流行病模型。具体地研究了一个连续控制模型和一个脉冲控制模型,目的是利用流行病来控制害虫的数量。对于连续控制模型研究了系统的平衡点的存在性和全局渐近稳定性;对于脉冲控制模型研究了易感害虫灭绝周期解的存在条件和系统的持久性。并且利用数值模拟验证了所得结果,第二节研究一个利用投放病毒来控制害虫数量的害虫-病毒模型。害虫分为两类:易感害虫和染病害虫,染病害虫可以释放病毒细胞,从而感染更多的易感害虫,建立了在染病害虫中引入有限染病年龄结构的害虫-病毒模型,利用合理的假设将偏微分模型转化成了相应的具有分布时滞的常微分方程模型,利用分析的技巧和初始条件的非负性,研究系统的渐近行为和线性化的稳定性,得到了易感害虫灭绝周期解全局吸引的充分条件。
第五章讨论一个具有阶段结构和周期时间依赖的种群模型。由于自然界中很多因素都是呈现周期变化的,因此在这一章讨论了周期变化环境下非自治种群模型的动力学行为,食饵种群分为两个阶段:幼年和成年,捕食者种群是杂食的,并且按不同的捕食率捕获幼年食饵和成年食饵,同时对捕食者种群进行脉冲式的收获,脉冲时间和脉冲量也是周期变化的,得到了系统持久的充要条件,并且得到了因过度捕获使捕食者种群灭绝的阈值。
Mathematical models of differential equations play an important role in describing population dynamic behavior. Mathematically, these models explain all kinds of population dynamic behavior, which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control. Delayed impulsive equations, which consider the effects of both the present state and the passed state on the behavior of dynamical system, is more suitable to describe the population dynamical system. In this dissertation, population dynamic models are established to consider several problems in pathogen controls and population controls by means of the theory and method of delayed functional differential equations and impulsive differential equations. We investigate dynamic behavior including the stability of equilibrium, the existence of periodic solution, the permanence and extinction of system. The main results of this dissertation may be summarized as follows:
In Chapter 2, some basic theories are provided for delayed equations and impulsive equations.
In Chapter 3, three predator-prey models with time delay and functional response are investigated. In section 3.1, a Ivlev predator-prey model with time delay and impulsive interruption in the predator is studied. The prey population is divided into two classes, the immature prey and mature prey. The time from immature to mature is a constant, and is expressed with a time delay. The functional response is Ivlev type, and the predator only capture the mature prey. We get the sufficient condition for the global attractivity of the pest-eradication periodic solution, and the condition for the permanence of the system. In section 3.2, a predator-prey model with time delay and impulsive harvesting strategy is studied. We impulsively release the prey at fixed time, and harvest the predator continuously. We get the condition for the global attractivity of the predator-eradication periodic solution, and we also obtain the condition for permanence of the system. Our result provide some theoretical base for exploitation of biological resources. In section 3.3, a stage-structured Gomportz predator-prey model with time delay and impulsive interruption in the prey is studied. The predator population is divided into two classes, the immature predator and the mature predator. Only the mature predator has the ability of capture the prey. We impulsively capture the prey, and the growth rate for the prey is Gomportz type, while the predation response is type Holling II. Our main purpose is to studied the effect of impulsive capturing of prey on the predator population. The sufficient condition for the global attractivity of the predator-eradication periodic solution is obtained, and we also get the condition for the permanence of the system. Some numerical results are done to show the dynamical behaviors of the system.
In Chapter 4, pest management models are studied. In section 4.1, two SI epidemic models are investigated. One continuous control system and one impulsive control system are used to control the number of pest, by using endemic. To the continuous system, we get the sufficient condition for the existence and stability condition for the equilibriums. To the impulsive system, a sufficient condition for the existence and stability of periodic solution are obtained. The results are also verified by simulations. In section 4.2, a pest-pathogen model is formulated. The pest is divided into two classes, the susceptible and infective. Infected pest can release pathogen cells to infect the susceptible. By some suitable assumptions, the partial differential equation system is transformed to ordinary differential equation system. The sufficient condition for the susceptible pest eradication periodic solution is obtained as our main result.
In Chapter 5, a stage-structured and periodic time dependent predator-prey model is studied. Since there are many factors in the natural world are periodic, it is necessary to study the effect of periodic variable environment on the behaviors of population model. In our model, the prey population is divided into two classes, the immature prey and mature prey. The predator is omnivorous, and it can capture both of the immature prey and the mature prey at different predation rate. The predator population is also being impulsively harvested. The condition for the permanence of the system is obtained. We get the threshold, above which the predator population will die out due to excessive harvesting.
第二章简要介绍了时滞微分方程和脉冲微分方程的基本知识。
第三章讨论了三个具有功能性反应的时滞捕食模型,第一节研究了一个定期释放天敌来控制害虫的具Ivlev功能性反应的时滞捕食模型。在模型中,假设食饵分为幼年食饵和成年食饵,捕食者种群只捕食成年食饵,幼年食饵到成年食饵的转化期是一个常数,用一个常数时滞表示。捕食者对食饵的功能函数是Ivlev型的。得到了害虫灭绝周期解全局吸引和系统持久生存的充分条件,第二节研究了一个具有阶段结构和Ivlev功能反应的捕食模型的收获控制策略,对食饵脉冲式的投放,而对捕食者连续地收获。并假设捕食者分为幼年和成年捕食者,幼年捕食者不会捕食食饵种群。得到了捕食者灭绝周期解的全局吸引的条件,并且利用比较方法分析了脉冲投放食饵和适度收获捕食者的行为,可保证捕食者种群持续生存。第三节研究了一个脉冲干扰食饵并且具有阶段结构和时滞的Gomportz捕食模型。捕食者种群分为幼年捕食者和成年捕食者,只有成年捕食者具有捕获食饵的能力。食饵种群的增长方式是Gomportz型的,捕食功能函数是HollingⅡ型。研究了当脉冲式的捕获食饵时对天敌生存的影响,得到了捕食者灭绝周期解全局吸引的充分条件和系统持续生存的条件。并且数值模拟了种群的动力学行为,分析了在害虫的种群密度低于经济危害水平的前提下,需要害虫和天敌共存,以维护生态平衡。
第四章基于当前“以虫治虫,以菌治虫”的可持续发展思想,讨论了害虫管理策略的数学模型。第一节研究了一个SI流行病模型。具体地研究了一个连续控制模型和一个脉冲控制模型,目的是利用流行病来控制害虫的数量。对于连续控制模型研究了系统的平衡点的存在性和全局渐近稳定性;对于脉冲控制模型研究了易感害虫灭绝周期解的存在条件和系统的持久性。并且利用数值模拟验证了所得结果,第二节研究一个利用投放病毒来控制害虫数量的害虫-病毒模型。害虫分为两类:易感害虫和染病害虫,染病害虫可以释放病毒细胞,从而感染更多的易感害虫,建立了在染病害虫中引入有限染病年龄结构的害虫-病毒模型,利用合理的假设将偏微分模型转化成了相应的具有分布时滞的常微分方程模型,利用分析的技巧和初始条件的非负性,研究系统的渐近行为和线性化的稳定性,得到了易感害虫灭绝周期解全局吸引的充分条件。
第五章讨论一个具有阶段结构和周期时间依赖的种群模型。由于自然界中很多因素都是呈现周期变化的,因此在这一章讨论了周期变化环境下非自治种群模型的动力学行为,食饵种群分为两个阶段:幼年和成年,捕食者种群是杂食的,并且按不同的捕食率捕获幼年食饵和成年食饵,同时对捕食者种群进行脉冲式的收获,脉冲时间和脉冲量也是周期变化的,得到了系统持久的充要条件,并且得到了因过度捕获使捕食者种群灭绝的阈值。
Mathematical models of differential equations play an important role in describing population dynamic behavior. Mathematically, these models explain all kinds of population dynamic behavior, which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control. Delayed impulsive equations, which consider the effects of both the present state and the passed state on the behavior of dynamical system, is more suitable to describe the population dynamical system. In this dissertation, population dynamic models are established to consider several problems in pathogen controls and population controls by means of the theory and method of delayed functional differential equations and impulsive differential equations. We investigate dynamic behavior including the stability of equilibrium, the existence of periodic solution, the permanence and extinction of system. The main results of this dissertation may be summarized as follows:
In Chapter 2, some basic theories are provided for delayed equations and impulsive equations.
In Chapter 3, three predator-prey models with time delay and functional response are investigated. In section 3.1, a Ivlev predator-prey model with time delay and impulsive interruption in the predator is studied. The prey population is divided into two classes, the immature prey and mature prey. The time from immature to mature is a constant, and is expressed with a time delay. The functional response is Ivlev type, and the predator only capture the mature prey. We get the sufficient condition for the global attractivity of the pest-eradication periodic solution, and the condition for the permanence of the system. In section 3.2, a predator-prey model with time delay and impulsive harvesting strategy is studied. We impulsively release the prey at fixed time, and harvest the predator continuously. We get the condition for the global attractivity of the predator-eradication periodic solution, and we also obtain the condition for permanence of the system. Our result provide some theoretical base for exploitation of biological resources. In section 3.3, a stage-structured Gomportz predator-prey model with time delay and impulsive interruption in the prey is studied. The predator population is divided into two classes, the immature predator and the mature predator. Only the mature predator has the ability of capture the prey. We impulsively capture the prey, and the growth rate for the prey is Gomportz type, while the predation response is type Holling II. Our main purpose is to studied the effect of impulsive capturing of prey on the predator population. The sufficient condition for the global attractivity of the predator-eradication periodic solution is obtained, and we also get the condition for the permanence of the system. Some numerical results are done to show the dynamical behaviors of the system.
In Chapter 4, pest management models are studied. In section 4.1, two SI epidemic models are investigated. One continuous control system and one impulsive control system are used to control the number of pest, by using endemic. To the continuous system, we get the sufficient condition for the existence and stability condition for the equilibriums. To the impulsive system, a sufficient condition for the existence and stability of periodic solution are obtained. The results are also verified by simulations. In section 4.2, a pest-pathogen model is formulated. The pest is divided into two classes, the susceptible and infective. Infected pest can release pathogen cells to infect the susceptible. By some suitable assumptions, the partial differential equation system is transformed to ordinary differential equation system. The sufficient condition for the susceptible pest eradication periodic solution is obtained as our main result.
In Chapter 5, a stage-structured and periodic time dependent predator-prey model is studied. Since there are many factors in the natural world are periodic, it is necessary to study the effect of periodic variable environment on the behaviors of population model. In our model, the prey population is divided into two classes, the immature prey and mature prey. The predator is omnivorous, and it can capture both of the immature prey and the mature prey at different predation rate. The predator population is also being impulsively harvested. The condition for the permanence of the system is obtained. We get the threshold, above which the predator population will die out due to excessive harvesting.
引文
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