两类生态模型解的定性分析及一类微分不等式解的渐近性理论
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摘要
在生态学中,外界的干扰、时滞、生物的不定期迁徙和种内竞争等因素对生物种群数量有很大影响,且时滞和反馈控制对种群数量具有有效的控制作用.本文利用特征值理论,构造Lyapunov泛函,比较原理等讨论了一类广义Logistic模型的Hopf分支、具有混合时滞反馈控制Logistic模型的全局吸引性和多时滞的二阶非线性微分积分不等式解的振动性、其中包括正解的存在唯一性、正平衡态的局部渐近稳定性、正平衡态的全局吸引性与解的振动性等问题.
生物种群数量的增长不仅受外界因素与时滞的影响,亦会受到种群自身数量增长的制约.本文首先研究了一类具有时滞且含扰动参数和常数收获率的广义Logistic模型的Hopf分支周期解问题.利用函数单调性得到模型正平衡态存在唯一的充分条件；应用特征值理论探讨模型产生Hopf分支的条件,运用周期函数正交性方法得到其近似周期解的表达式,通过Matlab给出了参数取不同数值时的曲线拟合图,并讨论了参数对周期解的周期,振幅及正平衡态的影响.
为了保护物种的多样性及保持生态平衡,可人为地采取必要的方法以控制种群的数量.本文其次研究了具有混合时滞的自治Logistic反馈控制系统解的全局吸引性.利用比较原理,证明了系统解的有界性,通过构造Lyapunov函数的方法,得到系统解全局吸引的充分条件.
生物种群自身的变化发展及人类对其不断地影响,可能使得生态系统中的某些种群数量,在某时间段内迅速增加或减少,或在某时刻灭绝.种群密度的变化与当前及以前的任意时刻的数量都有关系,也就是泛函微分方程中的连续时滞.因此,连续时滞对微分方程或不等式振动性的研究具有重要的理论与现实意义.本文最后讨论了具有多时滞的二阶变系数微分积分不等式解的振动性理论,利用Lebesgue控制收敛定理得到正解的存在性条件,并运用反证法得到其正解不存在的充分条件,通过讨论给出了其解振动的充要条件.
The interference of external and the time delay and irrgular migration and com-petition between populations have more influence and effective control on population density for most species in ecology. Hopf bifurcation of a class of general logistic model and the global attractivity of feedback control logistic model with discrete and continuous delay and the oscillation for a second-order differential integral inequal-ity with muti-delay are investigated by the eigenvalue theory and using constructing Lyapunov functions and comparison principl,including the existence and uniquity of the positive solution, local asymptotic stability of the positive equlibria.the global attractivity of the positive solution,oscillation of the solutions.
The population of species is affected by external factors, time delay and the number of its own.Firstly,the hopf bifurcation of a class of general logistic model with time delay, disturb and yield rate is investigated.The sufficient conditions of the existence and uniquity of the pos-itive equlibria by applying functional derivative is obtained,the condition for the existence of bifuration period solution is obtained by the eigenvalue theory,the form of approximate period solution is derived by the orthogonal condition,fitted curve figures are achieved by Matlab,when asign the differents, the effect of parameteras on the period,swing,position equilibrium of the periodic solution are discussed.
In order to protect diversity of species and ecological balance,Using necessary artificial methods control the number of species.secondly, the global attractivity of feedback control logistic model with discrete and continuous delay is discussed.The boundedness of the solutions of this model has been proved through the Comparison principle.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Lyapunov functional.
The number of some species is affected by species itself and the continuous impact of human. The number of species maybe increase rapidly or reduce rapidly at some times, or become extinct at a moment. Changes in species density is re-lated to current time and any time before.In other words, this is continuous delay of functional differential equations. Researching continuous delay of functional differ-ential equations has important theoretical and practical significance on Differential Equations and the oscillation of inequality. Finally,the oscillation for a second-order differential integral inequality with muti-delay is discussed. Using Lebesgue's deminated convergence theorem,sufficient condition to the existence of the positive solution for inequality is gained, by the method of disproof, the sufficient condition to the inexistence of the positive solution for inequality is obtained,making use of discussion, the sufficient and necessary condition to oscillation of it's solution is derived.
生物种群数量的增长不仅受外界因素与时滞的影响,亦会受到种群自身数量增长的制约.本文首先研究了一类具有时滞且含扰动参数和常数收获率的广义Logistic模型的Hopf分支周期解问题.利用函数单调性得到模型正平衡态存在唯一的充分条件；应用特征值理论探讨模型产生Hopf分支的条件,运用周期函数正交性方法得到其近似周期解的表达式,通过Matlab给出了参数取不同数值时的曲线拟合图,并讨论了参数对周期解的周期,振幅及正平衡态的影响.
为了保护物种的多样性及保持生态平衡,可人为地采取必要的方法以控制种群的数量.本文其次研究了具有混合时滞的自治Logistic反馈控制系统解的全局吸引性.利用比较原理,证明了系统解的有界性,通过构造Lyapunov函数的方法,得到系统解全局吸引的充分条件.
生物种群自身的变化发展及人类对其不断地影响,可能使得生态系统中的某些种群数量,在某时间段内迅速增加或减少,或在某时刻灭绝.种群密度的变化与当前及以前的任意时刻的数量都有关系,也就是泛函微分方程中的连续时滞.因此,连续时滞对微分方程或不等式振动性的研究具有重要的理论与现实意义.本文最后讨论了具有多时滞的二阶变系数微分积分不等式解的振动性理论,利用Lebesgue控制收敛定理得到正解的存在性条件,并运用反证法得到其正解不存在的充分条件,通过讨论给出了其解振动的充要条件.
The interference of external and the time delay and irrgular migration and com-petition between populations have more influence and effective control on population density for most species in ecology. Hopf bifurcation of a class of general logistic model and the global attractivity of feedback control logistic model with discrete and continuous delay and the oscillation for a second-order differential integral inequal-ity with muti-delay are investigated by the eigenvalue theory and using constructing Lyapunov functions and comparison principl,including the existence and uniquity of the positive solution, local asymptotic stability of the positive equlibria.the global attractivity of the positive solution,oscillation of the solutions.
The population of species is affected by external factors, time delay and the number of its own.Firstly,the hopf bifurcation of a class of general logistic model with time delay, disturb and yield rate is investigated.The sufficient conditions of the existence and uniquity of the pos-itive equlibria by applying functional derivative is obtained,the condition for the existence of bifuration period solution is obtained by the eigenvalue theory,the form of approximate period solution is derived by the orthogonal condition,fitted curve figures are achieved by Matlab,when asign the differents, the effect of parameteras on the period,swing,position equilibrium of the periodic solution are discussed.
In order to protect diversity of species and ecological balance,Using necessary artificial methods control the number of species.secondly, the global attractivity of feedback control logistic model with discrete and continuous delay is discussed.The boundedness of the solutions of this model has been proved through the Comparison principle.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Lyapunov functional.
The number of some species is affected by species itself and the continuous impact of human. The number of species maybe increase rapidly or reduce rapidly at some times, or become extinct at a moment. Changes in species density is re-lated to current time and any time before.In other words, this is continuous delay of functional differential equations. Researching continuous delay of functional differ-ential equations has important theoretical and practical significance on Differential Equations and the oscillation of inequality. Finally,the oscillation for a second-order differential integral inequality with muti-delay is discussed. Using Lebesgue's deminated convergence theorem,sufficient condition to the existence of the positive solution for inequality is gained, by the method of disproof, the sufficient condition to the inexistence of the positive solution for inequality is obtained,making use of discussion, the sufficient and necessary condition to oscillation of it's solution is derived.
引文
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[2]Weng Pei-xuan, Yi Yu-yin. Permanenme for a model of hemztopoiesis[J]. Journal of Biomathematics,2002,17(3):279-285.
[3]Zhao Jian-dong. Jiang Ji-fa. Permanence in nonautonomous Lotka-Volterra system with predator-prey[J]. Applied Mathematical and Com-putation,2004,152,99-109.
[4]Owaidy H, Mohamed H Y. Lincaried oscillation for non-linear systems of delay differential equations[J]. Applied Mathematics and Computa-tion.2003,142:17-21.
[5]Gopalsamy K, Zhang B G. Oscillation and nonoscillation in first order neutral differential equations[J]. Journal of Mathematical Analysis and Applications,1990,151(1):42-57.
[6]Mackey M C. Osccillation and chaos in physiological control systems[J]. Science,1997.197:287-289.
[7]Song Xin-yu. Chen Lan-sun. Persistence and global stability for nonau-tonomous predator-prey system with diffusion and time delay [J]. Com-puters and Mathematics with Applications,1998.35,6:33-40.
[8]Lu Zheng-yi. Global stability for two-species Lotka-Volterra systems with delay[J]. Proc Amer Soc,1986,96:75-78.
[9]王爱丽,陈斯养,王东宝.广义Logistic单种群时滞生态模型的渐近性[J].兰州大学学报(自然科学版),2004,40(2)：8-12.
[10]陈斯养,王东宝.一类具有时滞的离散Lotka-Volterra捕食系统的一致持久性[J].西南师范大学学报(自然科学版).2003,4(28)：518-522.
[11]武秀丽,陈斯养,汤红吉.具有多时滞二维Lotka-Volterra捕食系统的渐近性[J].陕西师范大学学报(自然科学版),2001,39(3)：27-30.
[12]Song Yong-li. Peng Ya-hong. Stability and bifurcation analysis on a Lo-gistic model with discrete and distributed delays[J]. Applied Mathematics and Computation.2006.181(2):1745-1757.
[13]Ma Zhi-en. Stability of predation models with time delay[J]. Appl Anal. 1986.22:159-192.
[14]陈兰荪.生态数学模型及研究方法[M].北京,科学出版社,1988.
[15]Kuang Yang. Delay differential equations with application dynamics[M]. Academic press limited publisher.1993.
[16]Gopalsamy K, Ladas G. Oscillations and global attractivity in a model of hematopoiesis[J]. Dynamics and Differential Equations,1990.35(2):117-132.
[17]Hassard B. Kazarinoff D. Theory and applications of Hopf bifurcation[M]. Cambridge:Cambridge university Press,1981.
[18]Gopalsamy K. Stability and oscillation in delay differential equations of popnlation dynamics[M]. Kluwer:Dordrecht Academic Publishers,1992: 148-159.
[19]司瑞霞,陈斯养.一类含时滞的广义logistic模型的Hopf分支[J].西北师范大学学报(自然科学版),2006,42(6)：18-22.
[20]杨颖茶,陈斯养.一类具有时滞的广义Logistic模型的hopf分支[J].云南师范大学学报(自然科学版).2006,04：1-6.
[21]Yang Fan. Jiang Da-qing. Existence and global attractivity of postive periodic solution of a Logistic growth system with feedback control ar-gumenta[J]. Ann of Diff Eqs,2001.17(4):337-384.
[22]Gopalsamy K. Wcng P X. Feedback regulation of Logistic growth[J]. Internat J Math Sci.1993.16(1):177-192.
[23]Meng Fan, Wang Ke. Periodic solutions of single species models with hereditary effects[J]. Mathematica Aplicata,2000,13(2):58-61.
[24]Li Yong-kun. On a periodic Logistic equation with several delay [J]. Ad-vances in Mathematics,1999,28(2):135-142.
[25]Chen Feng-de, Chen Xiao-xing. The n-competing Voltrra-Lotka almost periodic systems with, grazing rates[J]. J Biomathematics,2003.18(3): 52-58.
[26]桂占吉,陈兰荪.具有时滞的周期Logistic方程的持久性与周期解[J].数学研究与评论,2003.23(1)：109-114.
[27]Gopalsamy K. Stability and oscillation in delay differential equations of population dynamics[M]. Kluwer:Dordrecht Academic Publishers,1992: 148-159.
[28]朱道军.分段时滞反馈控制广义Logistic模型的吸引性[J].安徽大学学报(自然科学版),2005,29(2)：13-17.
[29]郑秀亮,陈斯养.多时滞反馈控制广义Logistic模型的吸引性[J].云南师范大学学报(自然科学版),2008,28(5)：11-15.
[30]Dimitri Breda. Solution operator approximations for characteristic roots of delay differential equations[J]. Applied Numerical Mathematics.2006. 56:305-317.
[31]Minoru Funakubo, Tadayuki Hara, Sadahisa. On the uniform asymp-totic stability for a linear integro-differential equation of Volterra type[J].J.Math.Anal.Appl,2006,324(2):1036-1049.
[32]王全义.具有无限时滞的微积分方程的周期解的存在性与唯一性[J].华侨大学学报(自然科学版),1996,25(04)：336-340.
[33]林诗仲,俞元洪.一类泛函微分方程解的振动定理[J].纯粹数学与应用数学,1996,16(01)：18-25.
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