脉冲微分方程在种群生态学中的一些应用
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摘要
种群动力学中的许多现象和人为的干扰因素都可以用脉冲来描述.本文以脉冲微分方程为基础,建立和研究了在固定时刻喷洒杀虫剂和释放天敌的害虫治理模型、脉冲收获与放养模型、染污环境下周期排放毒素的捕食者–食饵模型.结合连续动力系统,脉冲动力系统的相关理论系统地分析了所提出的各个模型,并用数值模拟的方法研究了模型的复杂动力学行为.全文共由六章构成.
第一章,对捕食-食饵生物系统定性分析问题以及脉冲动力系统在生物数学方面的应用等问题的历史背景和研究现状进行了综述,对本文所做工作作了简单的介绍,并给出本文所要涉及内容的预备知识.
第二章,对一类具HollingⅢ类功能反应模型的平衡点、极限环的存在性以及唯一性进行研究.通过计算推理与理论证明,分析了中心焦点的阶数以及稳定性.给出了极限环不存在、极限环存在和唯一性的各种条件.
第三章,基于IBM策略,建立了两类具类功能反应和一个固定时刻脉冲,捕食者间存在相互干扰的动力系统,并利用脉冲微分方程的Floquet理论、比较定理及微小扰动法对这两个系统进行研究。对于具有HollingⅡ类功能反应的系统,得到了系统灭绝和持续生存的充分条件,并给出了临界投放值pmax,通过数值模拟验证了结论的正确性。对于具有HollingⅣ类功能反应的系统,也得到了系统灭绝和持续生存的充分条件,通过数值模拟,发现该系统具有丰富的动力学性质,随着投放量p的增加,系统出现拟周期振荡、倍周期分支、混沌、半周期分支,对称破裂分支等复杂的动力学性质,最后通过对比,说明了害虫综合治理的有效性.
第四章,我们建立了一类在两个不同的固定时刻分别收获与放养的具有HollingⅡ类功能反应和相互干扰的捕食者–食饵脉冲动力系统,并由脉冲微分方程的Floquet理论和比较定理,得到了系统灭绝和持续生存的充分条件.
第五章,研究了一个在污染环境中的捕食者–食饵模型,该模型考虑外界污染源在固定时刻排放毒素,利用函数积分均值的研究方法,得出了两种群永久生存与灭绝的阈值.
第六章,就全文进行了总结,并就研究中还没有彻底解决的问题进行了说明.
It is well-known that many real world phenomena and human activities do exhibitimpulsive effects.In this thesis , based on impulsive differential equations, we establishand investigate some kinds of pest management models with periodic releasing naturalenemies and spraying pesticides at fixed time, impulsive harvesting and stocking atfixed time and a prey-predator model in a polluted environment with periodic pulsetoxicant input. Mathematically, by use of a combined approach of discrete dynamics,continuous dynamics and impulsive dynamics we study the various dynamical behaviorof the given systems. With the help of mathematical software, we investigate all kindsof bifurcations and complexity of the system we consider.
In Chapter 1, the historical background of the prey-predator functional responsesystem and bifurcations of limit cycles is introduced and the main works of this paperare concluded as well,and some preliminaries are given.
In Chapter 2, An qualitative analysis to a prey-predator with constant prey har-vesting under HollingⅢfunctional response system is given. Firstly, the existence forequilibrium points and their propertities are discussed. Secondly, some conditions forthe nonexistence of limit cycle are given. Finally, some conditions for the existince anduniqueness of limit cycle are derive.
In Chapter 3, Considering biological control and chemical control strategy, twoclasses of predator-prey models with functional response and mutual interference, im-pulsive effect at one fixed time are proposed. In the prey-predator model with HollingⅡfunctional response and impulsive effect, suffcient conditions for the system to beextinct and permanence are obtained by using the Floquet theory of impulsive equa-tion and comparison theorem, the critical value pmax of impulsive is given. Numericalsimulations show that the conclusion is correct.In the system with HollingⅣfunctionalresponse and impulsive effect, suffcient conditions for the system to be extinct and per-manence are obtained too. And numerical simulations show that the system has richdynamical behaviors, such as doubling bifurcation, periodic halving bifurcation, chaos,symmetry-breaking bifurcation, crises and so on. Finally we draw a conclusion thatthe strategy of integrated pest management is more effective than spraying pesticide.
In Chapter 4, a HollingⅡfunctional response predator-prey system with impulsiveimpulsive harvesting and stocking is proposed. Conditions for the system to be extinctare given and permanence conditions are established via the method of comparisoninvolving multiple Lyapunov functions. Further inffuences of the impulsive harvestingand stocking on the system are studied, and numerical simulations show that the systemhas rich dynamical behaviors.
In Chapter 5, A predator-prey model with toxicant emitted from external sourcesimpulsively in a polluted environment is proposed, and the threshold of persistence inmean and extinction for each species are obtained by use the method of integral meanvalue.
In the last chapter, the whole paper is summarized and the problems which arestill unsolved in the research are showed.
第一章,对捕食-食饵生物系统定性分析问题以及脉冲动力系统在生物数学方面的应用等问题的历史背景和研究现状进行了综述,对本文所做工作作了简单的介绍,并给出本文所要涉及内容的预备知识.
第二章,对一类具HollingⅢ类功能反应模型的平衡点、极限环的存在性以及唯一性进行研究.通过计算推理与理论证明,分析了中心焦点的阶数以及稳定性.给出了极限环不存在、极限环存在和唯一性的各种条件.
第三章,基于IBM策略,建立了两类具类功能反应和一个固定时刻脉冲,捕食者间存在相互干扰的动力系统,并利用脉冲微分方程的Floquet理论、比较定理及微小扰动法对这两个系统进行研究。对于具有HollingⅡ类功能反应的系统,得到了系统灭绝和持续生存的充分条件,并给出了临界投放值pmax,通过数值模拟验证了结论的正确性。对于具有HollingⅣ类功能反应的系统,也得到了系统灭绝和持续生存的充分条件,通过数值模拟,发现该系统具有丰富的动力学性质,随着投放量p的增加,系统出现拟周期振荡、倍周期分支、混沌、半周期分支,对称破裂分支等复杂的动力学性质,最后通过对比,说明了害虫综合治理的有效性.
第四章,我们建立了一类在两个不同的固定时刻分别收获与放养的具有HollingⅡ类功能反应和相互干扰的捕食者–食饵脉冲动力系统,并由脉冲微分方程的Floquet理论和比较定理,得到了系统灭绝和持续生存的充分条件.
第五章,研究了一个在污染环境中的捕食者–食饵模型,该模型考虑外界污染源在固定时刻排放毒素,利用函数积分均值的研究方法,得出了两种群永久生存与灭绝的阈值.
第六章,就全文进行了总结,并就研究中还没有彻底解决的问题进行了说明.
It is well-known that many real world phenomena and human activities do exhibitimpulsive effects.In this thesis , based on impulsive differential equations, we establishand investigate some kinds of pest management models with periodic releasing naturalenemies and spraying pesticides at fixed time, impulsive harvesting and stocking atfixed time and a prey-predator model in a polluted environment with periodic pulsetoxicant input. Mathematically, by use of a combined approach of discrete dynamics,continuous dynamics and impulsive dynamics we study the various dynamical behaviorof the given systems. With the help of mathematical software, we investigate all kindsof bifurcations and complexity of the system we consider.
In Chapter 1, the historical background of the prey-predator functional responsesystem and bifurcations of limit cycles is introduced and the main works of this paperare concluded as well,and some preliminaries are given.
In Chapter 2, An qualitative analysis to a prey-predator with constant prey har-vesting under HollingⅢfunctional response system is given. Firstly, the existence forequilibrium points and their propertities are discussed. Secondly, some conditions forthe nonexistence of limit cycle are given. Finally, some conditions for the existince anduniqueness of limit cycle are derive.
In Chapter 3, Considering biological control and chemical control strategy, twoclasses of predator-prey models with functional response and mutual interference, im-pulsive effect at one fixed time are proposed. In the prey-predator model with HollingⅡfunctional response and impulsive effect, suffcient conditions for the system to beextinct and permanence are obtained by using the Floquet theory of impulsive equa-tion and comparison theorem, the critical value pmax of impulsive is given. Numericalsimulations show that the conclusion is correct.In the system with HollingⅣfunctionalresponse and impulsive effect, suffcient conditions for the system to be extinct and per-manence are obtained too. And numerical simulations show that the system has richdynamical behaviors, such as doubling bifurcation, periodic halving bifurcation, chaos,symmetry-breaking bifurcation, crises and so on. Finally we draw a conclusion thatthe strategy of integrated pest management is more effective than spraying pesticide.
In Chapter 4, a HollingⅡfunctional response predator-prey system with impulsiveimpulsive harvesting and stocking is proposed. Conditions for the system to be extinctare given and permanence conditions are established via the method of comparisoninvolving multiple Lyapunov functions. Further inffuences of the impulsive harvestingand stocking on the system are studied, and numerical simulations show that the systemhas rich dynamical behaviors.
In Chapter 5, A predator-prey model with toxicant emitted from external sourcesimpulsively in a polluted environment is proposed, and the threshold of persistence inmean and extinction for each species are obtained by use the method of integral meanvalue.
In the last chapter, the whole paper is summarized and the problems which arestill unsolved in the research are showed.
引文
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