几类种群脉冲动力系统的动力学分析
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摘要
连续动力系统、离散动力系统和脉冲动力系统是三大主要的动力系统.经过近三十年的研究,脉冲微分方程的理论已经得到深入的发展,但是这些理论在实际问题中很难应用.由于脉冲动力系统的解在两脉冲时刻间具有连续性,而在脉冲时刻处却具有间断性,因而脉冲动力系统的动力学比连续动力系统的动力学更加丰富和复杂.脉冲现象广泛存在于各种应用领域,在种群动力学中有很好的应用前景.因此,研究具脉冲效应的种群动力系统的动力学性质具有很好理论意义和实用价值.
本文综合数学上的连续动力系统、离散动力系统和脉冲动力系统的相关理论,系统地分析了所提出种群模型的各种动力学性质,并借助Maple数学软件,对系统进行了模拟.
第一章,介绍了本文的研究目的及意义、国内外的脉冲微分动力系统动力学性质的研究现状以及本文的主要工作.
第二章,介绍脉冲微分动力系统的基本理论及一些相关的定义.
第三章,基于害虫综合管理策略,利用脉冲比较定理、Floquent理论及微小扰动法,研究了两类具有功能反应和一个固定脉冲时刻的食饵–捕食系统.对于具有Ivlev功能反应的两食饵–一捕食者脉冲动力系统,得到了系统食饵灭绝和持续生存的充分条件,数值模拟验证了结论的正确性,最后讨论了该害虫综合管理策略的有效性.对于具有Holling II功能反应的一食饵–两捕食者系统,给出了投放临界值p2?,得到了系统灭绝和持续生存的充分条件.数值模拟表明,随着投放量p?2的增加,系统出现倍周期分支、对称破裂分支、混沌、半周期分支、吸引子突变等复杂的动力学性质,最后讨论了该综合害虫管理策略的有效性.
第四章,基于害虫综合管理策略研究了三类脉冲发生在两个固定时刻、具有复杂非线性且非单调功能反应的食饵–捕食系统,前两类是具有Holling IV功能反应和脉冲效应的两食饵一捕食者系统和两食饵–两捕食者脉冲系统,给出了投放临界值,证明了系统两食饵灭绝和持续生存的充分条件,而且给出了一食饵种群灭绝其余种群持续生存的两个充分条件.数值模拟表明,随着投放量的增加,系统出现复杂的动力学性质.第三类是具有Beddington–DeAngelis功能反应的一食饵–多捕食者脉冲系统的动力学性质,证明了系统食饵灭绝和系统持续生存的充分条件,给出了相应的数值模拟.
最后,对全文进行了总结,指出了研究中还没有解决的问题,并对以后要研究的工作进行了展望.
Continuous dynamical system, discrete dynamical system and impulsive dynam-ical system are three main kinds of dynamical systems. Fairly rich results have beenmade for the theories of impulsive dynamical system for almost thirty years. However,these theories are hard to be applied in actual problems. The solutions of impulsivesystems are continuous between two impulses and discontinuous at the impulses, whichmakes the dynamics of impulsive systems richer and more complicated than that ofthe corresponding continuous systems. There are impulsive phenomena in all kinds ofapplied fields, especially in population dynamics. Consequently, studying populationdynamics of dynamical systems involving impulse effects is very interesting in boththeory and practice.
In this paper, we use a combined approach of continuous dynamics, discrete dy-namics and impulsive dynamics to investigate various dynamical behavior of the popu-lation systems we consider. The impulsive dynamical systems are simulated by math-ematical software Maple.
In Chapter 1, the goal, significance and recent development of the dynamics re-search in impulsive differential dynamical system, as well as the content of this disser-tation are introduced.
In Chapter 2, the basic theory of impulsive differential dynamical system anddefinitions are presented.
In Chapter 3, considering the strategy of integrated pest management (IPM), twoclasses of prey–predator systems with functional response, impulsive effect at one fixedtime are studied by using impulsive comparison theorem, Floquent theory and smallamplitude perturbation skill. In two–prey one–predator impulsive dynamical systemwith the Ivlev functional response and impulsive effect, the suffcient conditions for thesystem to be extinct of prey and permanence are given. Numerical simulations showthat the conclusion is correct. Finally, it is concluded that the strategy of integratedpest management is more effective than the classical one. In one–prey two–predatorsystem with the Holling II functional response and impulsive effect, the critical value p2ffof impulsive immigration and suffcient conditions for the system to be extinct and per-manence are given by using impulsive comparison theorem, Floquent theory and smallamplitude perturbation skill. Numerical simulations show that with the increasing ofimmigration p2 the system has more complex dynamics including periodic doubling bi-furcation, symmetry–breaking bifurcation, chaos, periodic halving bifurcation, crises,etc. Finally, it is discussed that the strategy of integrated pest management is moreeffective than the classical one.
In Chapter 4, considering the strategy of integrated pest management , threeclasses of prey–predator systems with complex and non–monotonic functional response,impulsive effect at different fixed times are established by using impulsive compar-ison theorem, Floquent theory and small amplitude perturbation skill. Firstly andSecondly, we consider two impulsive systems of two–prey one–predator and two–preytwo–predator with Holling IV functional response. The critical value of impulsiveimmigration is given. Two suffcient conditions for the two prey to be extinct andpermanence of system are proved. Moreover, the two suffcient conditions for the ex-tinction of one of two prey and permanence of other populations are given. Numericalsimulation shows that with the increasing of immigration the system has more com-plex dynamics. Thirdly, we consider one–prey multi–predator impulsive system withBeddington–DeAngelis functional response. The two suffcient conditions for the preyto be extinct and permanence of system are proved. Moreover, the numerical simula-tion is given.
Lastly, we make a summarization of the whole paper and point out the prob-lems that are still unsolved in the research are showed. Moreover, the future study isprospected.
本文综合数学上的连续动力系统、离散动力系统和脉冲动力系统的相关理论,系统地分析了所提出种群模型的各种动力学性质,并借助Maple数学软件,对系统进行了模拟.
第一章,介绍了本文的研究目的及意义、国内外的脉冲微分动力系统动力学性质的研究现状以及本文的主要工作.
第二章,介绍脉冲微分动力系统的基本理论及一些相关的定义.
第三章,基于害虫综合管理策略,利用脉冲比较定理、Floquent理论及微小扰动法,研究了两类具有功能反应和一个固定脉冲时刻的食饵–捕食系统.对于具有Ivlev功能反应的两食饵–一捕食者脉冲动力系统,得到了系统食饵灭绝和持续生存的充分条件,数值模拟验证了结论的正确性,最后讨论了该害虫综合管理策略的有效性.对于具有Holling II功能反应的一食饵–两捕食者系统,给出了投放临界值p2?,得到了系统灭绝和持续生存的充分条件.数值模拟表明,随着投放量p?2的增加,系统出现倍周期分支、对称破裂分支、混沌、半周期分支、吸引子突变等复杂的动力学性质,最后讨论了该综合害虫管理策略的有效性.
第四章,基于害虫综合管理策略研究了三类脉冲发生在两个固定时刻、具有复杂非线性且非单调功能反应的食饵–捕食系统,前两类是具有Holling IV功能反应和脉冲效应的两食饵一捕食者系统和两食饵–两捕食者脉冲系统,给出了投放临界值,证明了系统两食饵灭绝和持续生存的充分条件,而且给出了一食饵种群灭绝其余种群持续生存的两个充分条件.数值模拟表明,随着投放量的增加,系统出现复杂的动力学性质.第三类是具有Beddington–DeAngelis功能反应的一食饵–多捕食者脉冲系统的动力学性质,证明了系统食饵灭绝和系统持续生存的充分条件,给出了相应的数值模拟.
最后,对全文进行了总结,指出了研究中还没有解决的问题,并对以后要研究的工作进行了展望.
Continuous dynamical system, discrete dynamical system and impulsive dynam-ical system are three main kinds of dynamical systems. Fairly rich results have beenmade for the theories of impulsive dynamical system for almost thirty years. However,these theories are hard to be applied in actual problems. The solutions of impulsivesystems are continuous between two impulses and discontinuous at the impulses, whichmakes the dynamics of impulsive systems richer and more complicated than that ofthe corresponding continuous systems. There are impulsive phenomena in all kinds ofapplied fields, especially in population dynamics. Consequently, studying populationdynamics of dynamical systems involving impulse effects is very interesting in boththeory and practice.
In this paper, we use a combined approach of continuous dynamics, discrete dy-namics and impulsive dynamics to investigate various dynamical behavior of the popu-lation systems we consider. The impulsive dynamical systems are simulated by math-ematical software Maple.
In Chapter 1, the goal, significance and recent development of the dynamics re-search in impulsive differential dynamical system, as well as the content of this disser-tation are introduced.
In Chapter 2, the basic theory of impulsive differential dynamical system anddefinitions are presented.
In Chapter 3, considering the strategy of integrated pest management (IPM), twoclasses of prey–predator systems with functional response, impulsive effect at one fixedtime are studied by using impulsive comparison theorem, Floquent theory and smallamplitude perturbation skill. In two–prey one–predator impulsive dynamical systemwith the Ivlev functional response and impulsive effect, the suffcient conditions for thesystem to be extinct of prey and permanence are given. Numerical simulations showthat the conclusion is correct. Finally, it is concluded that the strategy of integratedpest management is more effective than the classical one. In one–prey two–predatorsystem with the Holling II functional response and impulsive effect, the critical value p2ffof impulsive immigration and suffcient conditions for the system to be extinct and per-manence are given by using impulsive comparison theorem, Floquent theory and smallamplitude perturbation skill. Numerical simulations show that with the increasing ofimmigration p2 the system has more complex dynamics including periodic doubling bi-furcation, symmetry–breaking bifurcation, chaos, periodic halving bifurcation, crises,etc. Finally, it is discussed that the strategy of integrated pest management is moreeffective than the classical one.
In Chapter 4, considering the strategy of integrated pest management , threeclasses of prey–predator systems with complex and non–monotonic functional response,impulsive effect at different fixed times are established by using impulsive compar-ison theorem, Floquent theory and small amplitude perturbation skill. Firstly andSecondly, we consider two impulsive systems of two–prey one–predator and two–preytwo–predator with Holling IV functional response. The critical value of impulsiveimmigration is given. Two suffcient conditions for the two prey to be extinct andpermanence of system are proved. Moreover, the two suffcient conditions for the ex-tinction of one of two prey and permanence of other populations are given. Numericalsimulation shows that with the increasing of immigration the system has more com-plex dynamics. Thirdly, we consider one–prey multi–predator impulsive system withBeddington–DeAngelis functional response. The two suffcient conditions for the preyto be extinct and permanence of system are proved. Moreover, the numerical simula-tion is given.
Lastly, we make a summarization of the whole paper and point out the prob-lems that are still unsolved in the research are showed. Moreover, the future study isprospected.
引文
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