几类非线性生物模型的持久性
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摘要
随着人类社会经济活动的迅速发展,人类因为短期利益或是对大自然认识的不足,盲目掠夺性地经济活动如大面积砍伐森林和开垦耕地、非法盗猎、不规范的旅游业、盲目开采及工厂”三废”的释放等等,引发了土地沙漠化、水土流失、环境污染和物种迅速消亡等一系列严重恶果,生物多样性丧失和生态环境的破坏使人类及其后代的生存和发展面临严重威胁。人们在研究生物灭绝的过程中,发现许多生物的灭绝过程都是栖息地先行破碎,连续分布的种群裂成斑块状种群,然后逐个斑块种群灭绝,最后导致整个种群的灭绝。为了能长期有效开发和利用资源,就必须具有合理性。自然资源的重要决定了在其开发和利用过程中必须实现可持续性。应该做到科学合理地开发利用自然资源,不断提高资源的开发利用水平及能力,力求形成一个科学合理的资源开发体系。而数学中微分方程中持久性问题正是对生物种群能否长期共存准确而科学的描述。
近来,时滞微分方程在生态模型中得到广泛应用,在这些模型中主要是研究时滞对稳定性的影响。从一些具有时滞的模型中,我们了解到稳定性随着时滞的改变而改变且时滞无限增长时会导致系统最终变成不稳定。但是在一些时滞模型中,随着时滞的改变,系统的一致持久并没有发生改变。多物种种群动力学模型也层出不穷,尤其是两物种种群模型中的捕食-食饵系统,它是一种很重要的生物数学模型。随着功能性反映函数的出现,解决了捕食-食饵系统存在的不合理之处,使模型更加完善。此外,阶段结构种群模型得到广泛关注,因为阶段结构模型不仅比偏微分方程形式更简单,而且它们所显示的现象和一些偏微分方程相类似。在第二章利用比较定理得到了一类带有Beddington-DeAngelis功能性反应函数和无限时滞的周期捕食-食饵模型持久性的充分必要条件。
脉冲微分方程描述了某些运动状态在固定或不固定时刻的快速变化或跳跃。它是对自然界的发展过程更真实的反映。脉冲微分方程的理论和方法在近三十年的研究中得到不断的完善,已经形成一个比较完善的体系,被广泛应用于种群动力学、传染病动力学、药物动力学、生物控制论、生物统计、数量遗传、化学反应等方面。尽管许多学者为脉冲微分方程在各领域的应用做了很多工作,但脉冲微分方程在种群动力学和传染病动力学的应用研究中,尚存在某些研究的空白。因此,本文第三章研究了一类具有非单调功能性反应函数脉冲捕食者-食饵模型的持久性,通过利用Floquet定理和小参数扰动技巧得到了保证该模型持久和灭亡的条件。
传染病历来就是危害人群健康的大敌,历史上传染病一次又一次的流行给人类生存的国计民生带来了巨大的灾难。长期以来,人类与各种传染病进行了不屈不挠的斗争。回顾斗争历程,应该说20世纪是人类征服传染病取得最辉煌成果的时期。近20年来,国际上传染病动力学的研究进展迅速,大量的数学模型被用于各种各样的传染病问题。这些数学模型大多是适用于各种传染病的一般规律的研究,也有部分是针对诸如麻疹、疟疾、肺结核、性病、艾滋病等诸多具体的疾病。建立传染病模型的主要理由是可利用模型对影响疾病传播的生物学和社会机理作清晰描述,通过模型的研究来揭示疾病流行规律,预测流行趋势,为发现、预防和控制疾病的流行提供理论根据和策略。本文第四章构建了一个脉冲治理害虫的病毒模型,即通过脉冲投放病虫致使易感害虫种群感染病毒,成为患病的害虫,从而达到消灭害虫的目的。利用比较定理及解的一致有界,得到了易感害虫灭绝周期解稳定的临界值条件。
Along with the rapid development of social economic activity,because of short-term interests of humanity and lack of awareness of nature,blind predatory economic activity such as large-scale deforestation and reclamation of cultivated land,illegal poaching,non-standard tourism,mining and factories blindly "three wastes" the release and so on,have caused desertification,soil erosion,environmental pollution and the rapid disappearance of species,such as a series of serious consequences. As we know,loss of biodiversity and ecological have damaged to the environment and human's survival.In the process of studying biological extinction,people found that the extinction of many biological processes are habitat broken firstly,species continuous distribution of species fragmented into patches,then patch-by-species extinction,and finally cause the extinction of entire species.The persistent problem of biological species for the differential equations in mathematics is a long-term coexistence of accurate and scientific description.Recently,the delay differential equation in the ecological model widely is applied widely to study the impact of delay on the stability.From some of the model with time delay,we know that the stability will change with the changing time delay and will cause unlimited growth of the system eventually becoming unstable.In addition,the stage-structured population models are widely concerned about the phase structure of the model because of its form different from partial differential equations and more simple,In the second chapter,a class of predator-prey model with Beddington-DeAngelis functional response function and the infinite delay is concerned,necessary and sufficient conditions of permanence are obtained by using the comparison theorem.Impulsive differential equation describes the status of certain sports at fixed or rapidly changing moment.It is widely used in species dynamics,dynamics of infectious diseases, pharmacokinetics,bio-cybernetics,bio-statistics,the number of genetic,chemical reactions and so on.However,the application of impulsive differential equations in Population dynamics and Infectious Diseases remains gaps in study.Therefore, this article in ChapterⅢstudies a class of impulse prey-predator model with nonmonotonic functional response function of sustainability and obtains the conditons of permanence and the extinction of the model by Using Floquet theorem and small parameter perturbation technique.Infectious diseases have always been against the great enemy of population health,history time and time again on the prevalence of infectious diseases to human,the people's livelihood has brought enormous disaster. So,chapterⅣof this article constructed a pulsed pest control virus disease model, that is,pest species and pulse are injected through the worm susceptible to achieve the purpose of the eradication of pests.
近来,时滞微分方程在生态模型中得到广泛应用,在这些模型中主要是研究时滞对稳定性的影响。从一些具有时滞的模型中,我们了解到稳定性随着时滞的改变而改变且时滞无限增长时会导致系统最终变成不稳定。但是在一些时滞模型中,随着时滞的改变,系统的一致持久并没有发生改变。多物种种群动力学模型也层出不穷,尤其是两物种种群模型中的捕食-食饵系统,它是一种很重要的生物数学模型。随着功能性反映函数的出现,解决了捕食-食饵系统存在的不合理之处,使模型更加完善。此外,阶段结构种群模型得到广泛关注,因为阶段结构模型不仅比偏微分方程形式更简单,而且它们所显示的现象和一些偏微分方程相类似。在第二章利用比较定理得到了一类带有Beddington-DeAngelis功能性反应函数和无限时滞的周期捕食-食饵模型持久性的充分必要条件。
脉冲微分方程描述了某些运动状态在固定或不固定时刻的快速变化或跳跃。它是对自然界的发展过程更真实的反映。脉冲微分方程的理论和方法在近三十年的研究中得到不断的完善,已经形成一个比较完善的体系,被广泛应用于种群动力学、传染病动力学、药物动力学、生物控制论、生物统计、数量遗传、化学反应等方面。尽管许多学者为脉冲微分方程在各领域的应用做了很多工作,但脉冲微分方程在种群动力学和传染病动力学的应用研究中,尚存在某些研究的空白。因此,本文第三章研究了一类具有非单调功能性反应函数脉冲捕食者-食饵模型的持久性,通过利用Floquet定理和小参数扰动技巧得到了保证该模型持久和灭亡的条件。
传染病历来就是危害人群健康的大敌,历史上传染病一次又一次的流行给人类生存的国计民生带来了巨大的灾难。长期以来,人类与各种传染病进行了不屈不挠的斗争。回顾斗争历程,应该说20世纪是人类征服传染病取得最辉煌成果的时期。近20年来,国际上传染病动力学的研究进展迅速,大量的数学模型被用于各种各样的传染病问题。这些数学模型大多是适用于各种传染病的一般规律的研究,也有部分是针对诸如麻疹、疟疾、肺结核、性病、艾滋病等诸多具体的疾病。建立传染病模型的主要理由是可利用模型对影响疾病传播的生物学和社会机理作清晰描述,通过模型的研究来揭示疾病流行规律,预测流行趋势,为发现、预防和控制疾病的流行提供理论根据和策略。本文第四章构建了一个脉冲治理害虫的病毒模型,即通过脉冲投放病虫致使易感害虫种群感染病毒,成为患病的害虫,从而达到消灭害虫的目的。利用比较定理及解的一致有界,得到了易感害虫灭绝周期解稳定的临界值条件。
Along with the rapid development of social economic activity,because of short-term interests of humanity and lack of awareness of nature,blind predatory economic activity such as large-scale deforestation and reclamation of cultivated land,illegal poaching,non-standard tourism,mining and factories blindly "three wastes" the release and so on,have caused desertification,soil erosion,environmental pollution and the rapid disappearance of species,such as a series of serious consequences. As we know,loss of biodiversity and ecological have damaged to the environment and human's survival.In the process of studying biological extinction,people found that the extinction of many biological processes are habitat broken firstly,species continuous distribution of species fragmented into patches,then patch-by-species extinction,and finally cause the extinction of entire species.The persistent problem of biological species for the differential equations in mathematics is a long-term coexistence of accurate and scientific description.Recently,the delay differential equation in the ecological model widely is applied widely to study the impact of delay on the stability.From some of the model with time delay,we know that the stability will change with the changing time delay and will cause unlimited growth of the system eventually becoming unstable.In addition,the stage-structured population models are widely concerned about the phase structure of the model because of its form different from partial differential equations and more simple,In the second chapter,a class of predator-prey model with Beddington-DeAngelis functional response function and the infinite delay is concerned,necessary and sufficient conditions of permanence are obtained by using the comparison theorem.Impulsive differential equation describes the status of certain sports at fixed or rapidly changing moment.It is widely used in species dynamics,dynamics of infectious diseases, pharmacokinetics,bio-cybernetics,bio-statistics,the number of genetic,chemical reactions and so on.However,the application of impulsive differential equations in Population dynamics and Infectious Diseases remains gaps in study.Therefore, this article in ChapterⅢstudies a class of impulse prey-predator model with nonmonotonic functional response function of sustainability and obtains the conditons of permanence and the extinction of the model by Using Floquet theorem and small parameter perturbation technique.Infectious diseases have always been against the great enemy of population health,history time and time again on the prevalence of infectious diseases to human,the people's livelihood has brought enormous disaster. So,chapterⅣof this article constructed a pulsed pest control virus disease model, that is,pest species and pulse are injected through the worm susceptible to achieve the purpose of the eradication of pests.
引文
[1]P.H.Leslie,Some further notes on the use of matrics in population mathematics,Biometrika.35 213-246(1948).
[2]S.W.Zhang,D.J.Tan,L.S.Chen,Chaos in periodically forced Holling type Ⅱ predatorprey system with impulsive perturbations,Chaos.Solitons Fractals,28(2006) 367-370.
[3]R S Cantrel,C Cosner,On the dynamics of predatory-pery models with the Beddington-DeAngelis function response,J.Meth.Anal.Appl.,257(1)(2001) 201-222.
[4]Liu B,Teng Z,Chen L,Analysis of a predator-prey model with Holling Ⅱ functional response concerning impulsive control strategy,J.Comp.Appl.Math.,193(2006) 347-62.
[5]G Ballinger,X Liu,Permanence of population growth models with impulsive effects,Math.Comput.Modell,26(1997) 59-72.
[6]S.Zhang,D.Tan,L.Chen,Chaos in periodically forced Holling type Ⅱ predator-prey system with impulsive perturbations,Chaos.Solitons.Fractals,28(2006) 307-376.
[7]J.Yan,Stability for impuisive delay differential equations,Nonl.Anal.TAM,63(2005)66-80.
[8]B.Leonid,B.Elena,Linearized oscillation theory for a nonlinear delay impulsive equation,J.Comp.Appl.Math..161(2003) 477-495.
[9]V.Lakshmikantham.D.D.Bainov,P.S.Simeonov,Theory of Impulsive Differential Equations,World Scientific Publishing Co.,Inc,Teaneck,NJ,1989.
[10]Yang Kuang.Delay differential equation with application in population dynamics,NY:Academic Press;1987.p.67-70.
[11]W.Wang,L.Chen,A predator-prey system with stage-structure for predator,Comp.Math.Appl.,33(1997) 83-91.
[12]S.A.Gourley,Y.Kuang,A stage structured predator-prey model and its dependence through-stage delay and death rate,J.Math.Biol.,49(2004) 188-200.
[13]S.Liu,L.Chen,Extinction and permanence in competitive stage-structured system with time-delay,Nonl.Anal.TMA,51(2002) 1347-1361.
[14]X.Song,J.Cui,The stage-structured predator-prey system with delay and harvesting,Appl.Anal.,81(2002) 1127-1142.
[15]W.G.Aiello,H.I.Freedman,A time-delay model of single-species growth with stage structure,Math.Biosci.,101(1990) 139-153.
[16]X.Y.Song,L.S.Chen.Optimal harvesting policy and stability for single-species growth model with stage structure,Journal of Systems.Complexity.15(2002)194-201.
[17]Z.H.Lu,L.S.Chen.Global attractivity of nonautonomous inshore fishing models with stage-structure,Applicable Analysis.81(2002) 589-605.
[18]H.Fang,J.Li.On the existence of periodic solutions of a neutral delay model of singlespecies population growth,J.Math.Anal.Appl.,259(2001) 8-17.
[19]F.D.Chen.Permanence of periodic Holling type predator-prey system with stage Structure for prey,Appl.Comput.182(2006)1849-1860.
[20]S.Q,Liu,L.S.Chen,G.L.Luo.Extinction and permance in competitive stage-structure system with time delays,Nonlinear Analysis.51(2002)1347-1361.
[21]Z.Q.Zhang,Periodic solutions of a predator-prey system with stage-structures for predator and prey,J.Math.Anal.Appl.302(2)(2005)291-305.
[22]J.M.Cushing,Periodic time dependent predator-prey system,SIAM J.Math.32(1977)82-95.
[23]Y.N.Xiao,L.S.Chen,On an SIS epidemic model with stage structure,J.Complexity.16(2003)275-288.
[24]Y.N.Xiao,L.S.Chen,F.V.D.Bosh,Dynamical behavior for stage-structared SIR infectious disease model,Nonlinear Analysis.3(2002)175-190.
[25]J.B.Bence,R.M.Nisbet,Space limited recruitment in open systems:the importance of time delays,Ecology 70(1989) 1434-1441.
[26]J.A.Cui.X.Y.Song,Permanence of a predator-prey system with stage structure,Discret.Contin.Dyn.Ser.B 4(3)(2004)547-554.
[27]J.A.Cui,Y.Sun,Permanence of predator-prey system with infinite delay,Electron.J.Differ.Equat.81(2004) 1-12.
[28]J R Beddington,Mutrual interference between parasites or predator and its effect on searching sfficiency,J.Animal Ecol,197,5.44:331-340.
[29]D L Deangelis,Goldstein R A and O'Lveill R V.A model for trophic interaction,Ecology,1975,56:881-892.
[30]A.A.Berryman,The origins and evolution of predator-prey theory,Ecology 75(1992)1530-1535.
[31]J.Cui,L.Chen,W.Wang,The effect of dispersal on population growth with stage-Structure,Comput.Math.Appl.,39(2002)91-102.
[32]H.Zhang,L.S.Chen,Permanence and extinction of a periodic predator-prey delay system with functional response and stage structure for prey,Math.Comput.184(2007)931-944.
[33]A.B.Gumel et al.,Modelling strategies for controlling SARS outbreaks Proc.R.Soc.Lond.B 271(2004) 2223.
[34]J.J.Jiao,X.Z.Meng,L.S.Chen,Global attractivity and permanence of a stagestructured pest management SI model with time delay and diseased pest impulsive transmission,Chaos,Solitons and Fractals 38(2008) 658-668.
[35]Van Lanteren JC,Integrated pest management in protected crops,In:Dent D,editor.Integrated pest management.London:Chapman and Hall:1995.
[36]W.Wang,S.Ruan,Simulating the SARS outbreak in Beijing with limited data,J.Theoret.Biol.227(2004) 369.
[37]D.M.Xiao,S.Ruan,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosci.208(2007) 419-429.
[38]Y.Kuang,Delay differential equation with application in population dynamics,NY:Academic Press;1987.p.67-70.
[39]C.Huang,L.Huang,Existence and global exponential stability of two-neuron networks with time-varying delays,Appl.Math.Lett.,19(2006) 126-134.
[40]M.Baptisini,P.Taboas,On the existence and global bifurcation of periodic solutions to planar differential delay equations,J.Differential Equations 127(1996) 391-425.
[41]V.Lakshmikantham,DD.Bainov,P.Simeonov,Theory of impulsive differential equation,Singapor:World Scientific;1989.
[2]S.W.Zhang,D.J.Tan,L.S.Chen,Chaos in periodically forced Holling type Ⅱ predatorprey system with impulsive perturbations,Chaos.Solitons Fractals,28(2006) 367-370.
[3]R S Cantrel,C Cosner,On the dynamics of predatory-pery models with the Beddington-DeAngelis function response,J.Meth.Anal.Appl.,257(1)(2001) 201-222.
[4]Liu B,Teng Z,Chen L,Analysis of a predator-prey model with Holling Ⅱ functional response concerning impulsive control strategy,J.Comp.Appl.Math.,193(2006) 347-62.
[5]G Ballinger,X Liu,Permanence of population growth models with impulsive effects,Math.Comput.Modell,26(1997) 59-72.
[6]S.Zhang,D.Tan,L.Chen,Chaos in periodically forced Holling type Ⅱ predator-prey system with impulsive perturbations,Chaos.Solitons.Fractals,28(2006) 307-376.
[7]J.Yan,Stability for impuisive delay differential equations,Nonl.Anal.TAM,63(2005)66-80.
[8]B.Leonid,B.Elena,Linearized oscillation theory for a nonlinear delay impulsive equation,J.Comp.Appl.Math..161(2003) 477-495.
[9]V.Lakshmikantham.D.D.Bainov,P.S.Simeonov,Theory of Impulsive Differential Equations,World Scientific Publishing Co.,Inc,Teaneck,NJ,1989.
[10]Yang Kuang.Delay differential equation with application in population dynamics,NY:Academic Press;1987.p.67-70.
[11]W.Wang,L.Chen,A predator-prey system with stage-structure for predator,Comp.Math.Appl.,33(1997) 83-91.
[12]S.A.Gourley,Y.Kuang,A stage structured predator-prey model and its dependence through-stage delay and death rate,J.Math.Biol.,49(2004) 188-200.
[13]S.Liu,L.Chen,Extinction and permanence in competitive stage-structured system with time-delay,Nonl.Anal.TMA,51(2002) 1347-1361.
[14]X.Song,J.Cui,The stage-structured predator-prey system with delay and harvesting,Appl.Anal.,81(2002) 1127-1142.
[15]W.G.Aiello,H.I.Freedman,A time-delay model of single-species growth with stage structure,Math.Biosci.,101(1990) 139-153.
[16]X.Y.Song,L.S.Chen.Optimal harvesting policy and stability for single-species growth model with stage structure,Journal of Systems.Complexity.15(2002)194-201.
[17]Z.H.Lu,L.S.Chen.Global attractivity of nonautonomous inshore fishing models with stage-structure,Applicable Analysis.81(2002) 589-605.
[18]H.Fang,J.Li.On the existence of periodic solutions of a neutral delay model of singlespecies population growth,J.Math.Anal.Appl.,259(2001) 8-17.
[19]F.D.Chen.Permanence of periodic Holling type predator-prey system with stage Structure for prey,Appl.Comput.182(2006)1849-1860.
[20]S.Q,Liu,L.S.Chen,G.L.Luo.Extinction and permance in competitive stage-structure system with time delays,Nonlinear Analysis.51(2002)1347-1361.
[21]Z.Q.Zhang,Periodic solutions of a predator-prey system with stage-structures for predator and prey,J.Math.Anal.Appl.302(2)(2005)291-305.
[22]J.M.Cushing,Periodic time dependent predator-prey system,SIAM J.Math.32(1977)82-95.
[23]Y.N.Xiao,L.S.Chen,On an SIS epidemic model with stage structure,J.Complexity.16(2003)275-288.
[24]Y.N.Xiao,L.S.Chen,F.V.D.Bosh,Dynamical behavior for stage-structared SIR infectious disease model,Nonlinear Analysis.3(2002)175-190.
[25]J.B.Bence,R.M.Nisbet,Space limited recruitment in open systems:the importance of time delays,Ecology 70(1989) 1434-1441.
[26]J.A.Cui.X.Y.Song,Permanence of a predator-prey system with stage structure,Discret.Contin.Dyn.Ser.B 4(3)(2004)547-554.
[27]J.A.Cui,Y.Sun,Permanence of predator-prey system with infinite delay,Electron.J.Differ.Equat.81(2004) 1-12.
[28]J R Beddington,Mutrual interference between parasites or predator and its effect on searching sfficiency,J.Animal Ecol,197,5.44:331-340.
[29]D L Deangelis,Goldstein R A and O'Lveill R V.A model for trophic interaction,Ecology,1975,56:881-892.
[30]A.A.Berryman,The origins and evolution of predator-prey theory,Ecology 75(1992)1530-1535.
[31]J.Cui,L.Chen,W.Wang,The effect of dispersal on population growth with stage-Structure,Comput.Math.Appl.,39(2002)91-102.
[32]H.Zhang,L.S.Chen,Permanence and extinction of a periodic predator-prey delay system with functional response and stage structure for prey,Math.Comput.184(2007)931-944.
[33]A.B.Gumel et al.,Modelling strategies for controlling SARS outbreaks Proc.R.Soc.Lond.B 271(2004) 2223.
[34]J.J.Jiao,X.Z.Meng,L.S.Chen,Global attractivity and permanence of a stagestructured pest management SI model with time delay and diseased pest impulsive transmission,Chaos,Solitons and Fractals 38(2008) 658-668.
[35]Van Lanteren JC,Integrated pest management in protected crops,In:Dent D,editor.Integrated pest management.London:Chapman and Hall:1995.
[36]W.Wang,S.Ruan,Simulating the SARS outbreak in Beijing with limited data,J.Theoret.Biol.227(2004) 369.
[37]D.M.Xiao,S.Ruan,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosci.208(2007) 419-429.
[38]Y.Kuang,Delay differential equation with application in population dynamics,NY:Academic Press;1987.p.67-70.
[39]C.Huang,L.Huang,Existence and global exponential stability of two-neuron networks with time-varying delays,Appl.Math.Lett.,19(2006) 126-134.
[40]M.Baptisini,P.Taboas,On the existence and global bifurcation of periodic solutions to planar differential delay equations,J.Differential Equations 127(1996) 391-425.
[41]V.Lakshmikantham,DD.Bainov,P.Simeonov,Theory of impulsive differential equation,Singapor:World Scientific;1989.