边值问题及脉冲效应在生物动力系统中的应用
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摘要
本文主要在连续生物动力学模型的定性和稳定性理论基础上,利用脉冲微分方程及其边值问题、比较定理以及数值模拟等相关理论和方法来研究对种群的近远期脉冲控制策略的可行性及相关问题,其中利用边值问题来考虑种群的近期管理或有限时间控制的可行性,利用定时脉冲及状态脉冲对生物动力系统的影响来考虑对种群的长期控制策略,以期为实践中的种群控制提供一些理论参考.全文共分6章:
第1章介绍了对种群进行有限时间控制和长期管理的背景及意义,并简要叙述了脉冲微分方程及其在生物动力系统应用的研究现状和本文的主要工作.
第2章给出了脉冲微分方程及其边值问题的基本理论和相关知识.
第3章主要讨论了边值问题在具有Allee效应单种群生物动力系统中的应用.按照物种的初始密度,本章提出了两类具有Allee效应的单种群有限时间管理模型:脉冲投放模型和脉冲收获模型.基于比较定理和上下解方法,分别获得了模型有解或无解的充分条件,估计了相关可控参数,即脉冲投放模型中的投放量以及脉冲收获模型中的脉冲收获次数.
第4章考虑了边值问题在两种群脉冲生物动力系统中的应用.第1节提出了用以描述有限时间害虫控制问题的一类带有脉冲收获的Lotka-Volterra捕食系统的边值问题.利用比较定理,通过一系列上解获得了带有边值问题的模型有解的充分条件,并由一系列下解得到了模型无解的充分条件,进而估计了在有限时间内收获害虫的次数.第2节考虑了一类带有脉冲收获的Lotka-Volterra竞争系统的有限时间控制模型.同样地,利用比较定理和上下解方法获得了模型有解或无解的条件,并对这些条件的实际意义进行了解释和说明.作为应用的例子,估计了脉冲控制次数,并给出了数值模拟.
第5章研究了三类定时周期脉冲控制对捕食系统的影响.第1节利用链变换、脉冲微分方程及其Floquet理论研究了脉冲收获控制对一类带有分布时滞的捕食系统的影响,获得了物种灭绝和系统持久性的条件,证明了脉冲控制周期和脉冲收获比例都将最终影响每一个物种的命运.最后通过数值模拟验证了理论结果.第2节讨论了脉冲投放对一类带有Ivlev及Beddington-DeAngelis功能性反应的两捕食者一食饵捕食系统的影响.利用Floquet理论和小振幅扰动方法,证明了当脉冲周期小于某个临界值时,食饵灭绝周期解是局部渐近稳定的.进一步地,获得了系统持久性的条件.最后,利用数值模拟显示了系统具有复杂动力学性质,包括周期解、倍周期分支、混沌等,并给出了关于连续控制系统和脉冲控制系统关系的简短讨论.第3节讨论了利用传染病动力学对一类杂食种群的脉冲控制问题,获得了易感捕食者灭绝和系统中各物种共存的充分条件.数值模拟验证了理论结果,并显示了系统具有复杂的动力学行为.
第6章研究了脉冲状态反馈控制对一类具有Monod增长率的恒浊器模型周期解存在性的影响,通过研究周期解的存在性来讨论微生物连续培养的周期性.在定性分析的基础上,利用脉冲自治系统周期解的存在定理,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、稀释率的控制参数和微生物及营养基的初始浓度.通过研究周期解,给出了周期解的周期及初始位置.最后利用数值模拟验证了理论结果.
Based on the qualitative results of continuous dynamical models in biology,the long-term and short-term managements of species are investigated by means of the theory of impulsive differential equations and boundary value problem.The short-term managements can be described by the systems with boundary value problem,and the long-term managements can be investigated by discussing the effects of the impulse at fixed moments and the state impulse on the dynamical systems in biology.The thesis has 6 chapters.
Chapter 1 gives the biological backgrounds of the long-term and short-term managements of species,briefly states the present studies of impulsive differential equation and its applications to the dynamical systems in biology.
Chapter 2 introduces the basic theories and preliminaries of impulsive differential equation and boundary value problem.
In Chapter 3,the applications of boundary value problem to the system with Allee effect and impulsive effect are given.According to the initial density of a single-species with Allee effect,Chapter 3 presents two kinds of time-limited management models:the model with impulsive release and the model with impulsive harvest.By means of the comparison principle and the method of upper and lower solution,the corresponding sufficient conditions under which the models have a solution or no solution are obtained. If other parameters are given,the controllable parameters such as the population of release and the times of impulsive harvest are estimated respectively.
In Chapter 4,the applications of boundary value problem to the system of two species with impulsive control in finite time are investigated.Section 4.1 presents a kind of timelimited pest control of a Lotka-Volterra predator-prey model with impulsive harvest.By the comparison principle,the conditions under which the model has a solution are found by a series of the upper solutions and the conditions under which the model has no solution are also given by a series of the lower solutions.Furthermore,the times of harvesting pest in the given time is estimated.Section 4.2 presents a kind of time-limited control model of a Lotka-Volterra competition system with impulsive harvest.The existence of solution of the model is discussed.Similarly,by the comparison principle,the conditions under which the model has a solution or no solution are found.Finally,the practical meanings of those conditions are explained.As an example of theoretical results,if other parameters are given,the times of impulsive control is estimated and the theoretical results are verified by numerical simulations.
In Chapter 5,the effects of three kinds of impulsive control strategies on predatorprey system are investigated by the Floquet's theory and the comparison theorem of impulsive differential equation.By means of chain transform,Section 5.1 investigates the impulsive effects on a kind of predator-prey system with distributed time delay.The thresholds between permanence and extinction are obtained as functions of model parameters. It is proved that the impulsive period and the proportion of the impulsive harvest will ultimately affect the fate of each species.Section 5.2 discusses the effects of impulsive release on a kind of one-prey two-predator system with Ivlev's and Beddington-DeAngelis' functional response.It is proved that the prey-free periodic solution is locally asymptotically stable when the period of impulsive release is less than a critical value. Furthermore,the conditions for the permanence of the system are obtained.Numerical simulations show that the system has complex properties,including periodic solution, period-doubling bifurcation,chaos,etc.Finally,a brief discussions on the relationships between the continuous system and the impulsive system are given.Section 5.3 discusses a kind of impulsive control to an omnivorous species by means of the dynamics of infectious disease and obtains the sufficient conditions under which the susceptible predator will be extinct or which the species coexist.The numerical simulations verify the theoretical results and show the system has complex dynamical behaviors.
In Chapter 6,the periodicity of continuous culture of microorganism in the turbidostat is discussed by investigating the existence of periodic solution of a kind of trubidostat model with impulsive state feedback control and Monod growth rate.Based on the qualitative analysis,the conditions for the existence of periodic solution of order one are obtained by the existence theorem 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 dilution rate and the initial concentrations of microorganisms and substrate.The period and the initial point of the periodic solution are given.Finally,the theoretical results are verified by numerical simulations.
第1章介绍了对种群进行有限时间控制和长期管理的背景及意义,并简要叙述了脉冲微分方程及其在生物动力系统应用的研究现状和本文的主要工作.
第2章给出了脉冲微分方程及其边值问题的基本理论和相关知识.
第3章主要讨论了边值问题在具有Allee效应单种群生物动力系统中的应用.按照物种的初始密度,本章提出了两类具有Allee效应的单种群有限时间管理模型:脉冲投放模型和脉冲收获模型.基于比较定理和上下解方法,分别获得了模型有解或无解的充分条件,估计了相关可控参数,即脉冲投放模型中的投放量以及脉冲收获模型中的脉冲收获次数.
第4章考虑了边值问题在两种群脉冲生物动力系统中的应用.第1节提出了用以描述有限时间害虫控制问题的一类带有脉冲收获的Lotka-Volterra捕食系统的边值问题.利用比较定理,通过一系列上解获得了带有边值问题的模型有解的充分条件,并由一系列下解得到了模型无解的充分条件,进而估计了在有限时间内收获害虫的次数.第2节考虑了一类带有脉冲收获的Lotka-Volterra竞争系统的有限时间控制模型.同样地,利用比较定理和上下解方法获得了模型有解或无解的条件,并对这些条件的实际意义进行了解释和说明.作为应用的例子,估计了脉冲控制次数,并给出了数值模拟.
第5章研究了三类定时周期脉冲控制对捕食系统的影响.第1节利用链变换、脉冲微分方程及其Floquet理论研究了脉冲收获控制对一类带有分布时滞的捕食系统的影响,获得了物种灭绝和系统持久性的条件,证明了脉冲控制周期和脉冲收获比例都将最终影响每一个物种的命运.最后通过数值模拟验证了理论结果.第2节讨论了脉冲投放对一类带有Ivlev及Beddington-DeAngelis功能性反应的两捕食者一食饵捕食系统的影响.利用Floquet理论和小振幅扰动方法,证明了当脉冲周期小于某个临界值时,食饵灭绝周期解是局部渐近稳定的.进一步地,获得了系统持久性的条件.最后,利用数值模拟显示了系统具有复杂动力学性质,包括周期解、倍周期分支、混沌等,并给出了关于连续控制系统和脉冲控制系统关系的简短讨论.第3节讨论了利用传染病动力学对一类杂食种群的脉冲控制问题,获得了易感捕食者灭绝和系统中各物种共存的充分条件.数值模拟验证了理论结果,并显示了系统具有复杂的动力学行为.
第6章研究了脉冲状态反馈控制对一类具有Monod增长率的恒浊器模型周期解存在性的影响,通过研究周期解的存在性来讨论微生物连续培养的周期性.在定性分析的基础上,利用脉冲自治系统周期解的存在定理,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、稀释率的控制参数和微生物及营养基的初始浓度.通过研究周期解,给出了周期解的周期及初始位置.最后利用数值模拟验证了理论结果.
Based on the qualitative results of continuous dynamical models in biology,the long-term and short-term managements of species are investigated by means of the theory of impulsive differential equations and boundary value problem.The short-term managements can be described by the systems with boundary value problem,and the long-term managements can be investigated by discussing the effects of the impulse at fixed moments and the state impulse on the dynamical systems in biology.The thesis has 6 chapters.
Chapter 1 gives the biological backgrounds of the long-term and short-term managements of species,briefly states the present studies of impulsive differential equation and its applications to the dynamical systems in biology.
Chapter 2 introduces the basic theories and preliminaries of impulsive differential equation and boundary value problem.
In Chapter 3,the applications of boundary value problem to the system with Allee effect and impulsive effect are given.According to the initial density of a single-species with Allee effect,Chapter 3 presents two kinds of time-limited management models:the model with impulsive release and the model with impulsive harvest.By means of the comparison principle and the method of upper and lower solution,the corresponding sufficient conditions under which the models have a solution or no solution are obtained. If other parameters are given,the controllable parameters such as the population of release and the times of impulsive harvest are estimated respectively.
In Chapter 4,the applications of boundary value problem to the system of two species with impulsive control in finite time are investigated.Section 4.1 presents a kind of timelimited pest control of a Lotka-Volterra predator-prey model with impulsive harvest.By the comparison principle,the conditions under which the model has a solution are found by a series of the upper solutions and the conditions under which the model has no solution are also given by a series of the lower solutions.Furthermore,the times of harvesting pest in the given time is estimated.Section 4.2 presents a kind of time-limited control model of a Lotka-Volterra competition system with impulsive harvest.The existence of solution of the model is discussed.Similarly,by the comparison principle,the conditions under which the model has a solution or no solution are found.Finally,the practical meanings of those conditions are explained.As an example of theoretical results,if other parameters are given,the times of impulsive control is estimated and the theoretical results are verified by numerical simulations.
In Chapter 5,the effects of three kinds of impulsive control strategies on predatorprey system are investigated by the Floquet's theory and the comparison theorem of impulsive differential equation.By means of chain transform,Section 5.1 investigates the impulsive effects on a kind of predator-prey system with distributed time delay.The thresholds between permanence and extinction are obtained as functions of model parameters. It is proved that the impulsive period and the proportion of the impulsive harvest will ultimately affect the fate of each species.Section 5.2 discusses the effects of impulsive release on a kind of one-prey two-predator system with Ivlev's and Beddington-DeAngelis' functional response.It is proved that the prey-free periodic solution is locally asymptotically stable when the period of impulsive release is less than a critical value. Furthermore,the conditions for the permanence of the system are obtained.Numerical simulations show that the system has complex properties,including periodic solution, period-doubling bifurcation,chaos,etc.Finally,a brief discussions on the relationships between the continuous system and the impulsive system are given.Section 5.3 discusses a kind of impulsive control to an omnivorous species by means of the dynamics of infectious disease and obtains the sufficient conditions under which the susceptible predator will be extinct or which the species coexist.The numerical simulations verify the theoretical results and show the system has complex dynamical behaviors.
In Chapter 6,the periodicity of continuous culture of microorganism in the turbidostat is discussed by investigating the existence of periodic solution of a kind of trubidostat model with impulsive state feedback control and Monod growth rate.Based on the qualitative analysis,the conditions for the existence of periodic solution of order one are obtained by the existence theorem 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 dilution rate and the initial concentrations of microorganisms and substrate.The period and the initial point of the periodic solution are given.Finally,the theoretical results are verified by numerical simulations.
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