两类生态模型解的性质及一类二阶非自治微分方程解的振动性研究
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摘要
生物数学是通过建立数学模型,把复杂的生物学问题转化为数学问题,然后利用数学理论和方法达到对生命现象研究的目的.本文研究了两类生态模型解的性质及一类二阶非自治微分方程解的振动性,其中包括模型正平衡态的存在唯一性、分支周期解的存在性及其近似表达式、解的有界性、正平衡态的全局吸引性以及解的振动性等问题.
近年来关于时滞产生周期解的研究得到了迅速发展,大量文献研究了时滞产生的Hopf分支和分支周期解的近似表达式.本文首先研究了一类含时滞和放养项的广义Logistic单种群模型的稳定性和Hopf分支问题.利用特征值理论,讨论了模型正平衡态的局部稳定性和Hopf分支的存在性；通过周期函数正交性方法得到了分支周期解的近似表达式；给出实例验证了定理的可实现性,且运用Matlab绘出了参数取不同数值时的曲线拟合图,并分析了参数对周期解的周期,振幅及正平衡态的影响.
在自然界中,任何物种都与其它物种存在着相互制约和相互依赖的关系,其中包括捕食,竞争和互惠共存.本文其次研究了具有反馈控制的多时滞两种群竞争模型正解的全局吸引性.本章利用振动性理论和极限思想,证明了该模型解的有界性；通过构造Lyapunov泛函和不等式估值的方法,得到了该模型解全局吸引性的充分条件.
由于微分方程的振动性理论是动力系统研究的重要问题之一.本文最后研究了一类具有多时滞二阶非自治微分方程振动性.通过对一阶系数取不同范围的值进行了讨论,利用Knaster-Tarski不动点原理和微分中值定理得到了方程的线性振动准则,给出了其线性振动的充要条件,使其解的振动性问题得以简化.
Ecological mathematics traslates complex biological problems into a mathemat-ical problem by the establishment of mathematical models.Then ecological mathe-matics studies all kinds of natural phenomenon through the mathematical theory and way.The properties of two ecological systems and the oscillation of a second-order differential equation are investigated.Here,the behavior includes the existence and uniqueness of the positive equilibrium state、the existence and approximate expression of the periodic solution、the boundedness of the solutions、the global attractivity of the solutions and the oscillations of solutions.
The research of the time delay leads to the periodic solutions' has developed rapidly in recent years,and the hopf bifurcation produced by the time delay and the periodic solutions similar expressions of the bifurcation have been studied in lots of papers. Frist,the hopf bifurcation of a class of general Logistic model with a discrete time delay and stocking item is investigated.The part stability of the positive equilibrium state and existence of the hopf branch are discussed through the eigenvalue theory. The form of the approximate periodic solution is obtained by orthogonal conditions.The paper points out the example to confirm the theorem's realizability, and fitted curve figures with different values,in which the influence toperiod, swing, positive equilibrium of period solution are discussed, are achieved by using matlab.
In nature,the relationship between one group and another is restriced and de-pendent,such as predator-prey,competition and reciprocity coexistence. Sceond,the global attractivity of the solutions of a two-species competitive system with feedback controls and mutl-delay is investigated.The boundedness of the solutions of this model has been proved through the oscillation theory and the thinking of limit.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Liapunov functional and inequality valuation.
Because the differential equation's oscillation theory is one of the important issues about the power system study. Third,the linearized oscillation for a second-order differential equation with mutl-delay is studied.Through the first-order coefficient s in the different range was discussed.The linearized oscillation criterion of the equation was obtained by using the Knaster-Tarski fixed- point theorem and the value theorem.The whole sufficient and neces-sary condition of it was derived,and so the oscillatin of the equation's solution was simplified.
近年来关于时滞产生周期解的研究得到了迅速发展,大量文献研究了时滞产生的Hopf分支和分支周期解的近似表达式.本文首先研究了一类含时滞和放养项的广义Logistic单种群模型的稳定性和Hopf分支问题.利用特征值理论,讨论了模型正平衡态的局部稳定性和Hopf分支的存在性；通过周期函数正交性方法得到了分支周期解的近似表达式；给出实例验证了定理的可实现性,且运用Matlab绘出了参数取不同数值时的曲线拟合图,并分析了参数对周期解的周期,振幅及正平衡态的影响.
在自然界中,任何物种都与其它物种存在着相互制约和相互依赖的关系,其中包括捕食,竞争和互惠共存.本文其次研究了具有反馈控制的多时滞两种群竞争模型正解的全局吸引性.本章利用振动性理论和极限思想,证明了该模型解的有界性；通过构造Lyapunov泛函和不等式估值的方法,得到了该模型解全局吸引性的充分条件.
由于微分方程的振动性理论是动力系统研究的重要问题之一.本文最后研究了一类具有多时滞二阶非自治微分方程振动性.通过对一阶系数取不同范围的值进行了讨论,利用Knaster-Tarski不动点原理和微分中值定理得到了方程的线性振动准则,给出了其线性振动的充要条件,使其解的振动性问题得以简化.
Ecological mathematics traslates complex biological problems into a mathemat-ical problem by the establishment of mathematical models.Then ecological mathe-matics studies all kinds of natural phenomenon through the mathematical theory and way.The properties of two ecological systems and the oscillation of a second-order differential equation are investigated.Here,the behavior includes the existence and uniqueness of the positive equilibrium state、the existence and approximate expression of the periodic solution、the boundedness of the solutions、the global attractivity of the solutions and the oscillations of solutions.
The research of the time delay leads to the periodic solutions' has developed rapidly in recent years,and the hopf bifurcation produced by the time delay and the periodic solutions similar expressions of the bifurcation have been studied in lots of papers. Frist,the hopf bifurcation of a class of general Logistic model with a discrete time delay and stocking item is investigated.The part stability of the positive equilibrium state and existence of the hopf branch are discussed through the eigenvalue theory. The form of the approximate periodic solution is obtained by orthogonal conditions.The paper points out the example to confirm the theorem's realizability, and fitted curve figures with different values,in which the influence toperiod, swing, positive equilibrium of period solution are discussed, are achieved by using matlab.
In nature,the relationship between one group and another is restriced and de-pendent,such as predator-prey,competition and reciprocity coexistence. Sceond,the global attractivity of the solutions of a two-species competitive system with feedback controls and mutl-delay is investigated.The boundedness of the solutions of this model has been proved through the oscillation theory and the thinking of limit.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Liapunov functional and inequality valuation.
Because the differential equation's oscillation theory is one of the important issues about the power system study. Third,the linearized oscillation for a second-order differential equation with mutl-delay is studied.Through the first-order coefficient s in the different range was discussed.The linearized oscillation criterion of the equation was obtained by using the Knaster-Tarski fixed- point theorem and the value theorem.The whole sufficient and neces-sary condition of it was derived,and so the oscillatin of the equation's solution was simplified.
引文
[1]May R.M.Time delay versus stability in population models with two or three tropic levels[J].Ecology,1973,54:315-325.
[2]Weng peixuan,Yi yuyin.Permanenme for a model of hemztopoiesis[J]. Journal of Biomathematics,2002,17(3):279-285.
[3]Tanga X.H,Zou Xingfu. 3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feed-backs[J].J.Differential Equations,2002,186:420-439.
[4]El-Owaidy H, Mohamed H.Y. Linearized oscillation for non-linear sys-tems of delay differential equations[J]. Applied Mathematics and Com-putation,2003,142:17-21.
[5]Zhao Jiandong,Jiang Jifa.Permanence in nonautonomous Lotka-Volterra system with predator-prey[J]. Applied Mathematics and Computation, 2004,152:99-109.
[6]Candan T, Dahiya R. Oscillation theorems for nth-order functional dif-ferential equations[J]. Mathematical and Computer Modeling,2006,43: 357-367.
[7]Zhao Hongyong, LI Sun. Periodic oscillatory and global attractivity for chemostat model involving distributed delays[J]. Nonlinear Analysis: Real World Applications,2006,7:385-394.
[8]Zhao Hongyong, Nan Ding.Existence and global attractivity of positive periodic solution for competition-predator system with variable delays[J]. Chaos, Solitons and Fractals,2006,29:162-170.
[9]陈斯养,王东保.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.
[10]黄利航,陈斯养.一类具有时滞的捕食与被捕食模型的Hopf分支[J].西北师范大学学报(自然科学版),2004,4(40)：12-18.
[11]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.
[12]朱道军,陈斯养.分段常数变量反馈控制Logistic模型的吸引性[J].安徽大学学报(自然科学版),2005,29(2)：13-17.
[13]余胜平,陈斯养,黄建科.具有反馈控制的捕食竞争系统的全局性态[J].西南师范大学学报(自然科学版),2005,30(2)：198-201.
[14]伍代勇,陈斯养.具有反馈控制的广义Logistic增长模型[J].广西师范大学学报(自然科学版),2006,24(1)：41-44.
[15]田亚品,陈斯养,N-种群Lotka-Volterra扩散竞争反馈控制生态系统的持久性和全局渐近性[J].应用数学(自然科学版),2007,20(3)：485-490.
[16]Hassard B, Kazarionff D, Wan Y. Theory and applications of Hopf bi-furcation[M]. Cambridge:Cambridge University Press,1981.
[17]Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics[M]. Dordrecht:Kluwer Academics Publishers, 1992.
[18]范丽,陈斯养,史忠科.一类广义Logistic单种群实质时滞模型的Hopf分支[J].兰州大学学报(自然科学版),2007,43(6)：97-102.
[19]Ma Zhien. Stability of predation models with time delay[J]. Appl Anal,1986,22:159-192.
[20]Frfedman H.I, Rao V.S.H. Stability of a system involving two time de-lay[J]. Siam J Appl Math,1986,46:552-560.
[21]Kuang Y. Delay differential equations with applications in population dynamics[M].Boston:Academic Press,1993.
[22]田亚品,陈斯养.一类造血模型的全局渐近性及Hopf分支周期解单[J].安徽大学学报(自然科学版),2007,31(6)：19-23.
[23]Song Yongli, PENG Yahong. Stability and bifurcation analysis on a Logistic model with discrete and distributed delays[J]. Applied Mathe-matics and Computation,2006,181(2):1745-1757.
[24]司瑞霞,陈斯养.一类含时滞的广义logistic的Hopf分支[J].西北师范大学学报(自然科学版),2006,42(6)：18-22.
[25]Gopalsamy K,Weng peixue.Global attractivity in a competition sys-tem with feedback controls[J]. Computers and Mathematics with Applications,2003,45:665-676.
[26]LU Zhengyi, Yasuhiro Takeuchi.Permanence and global attractivity for competitive Lotka-Volterra systems with delay.Nonlinear Analysis:The-ory,Methods and Applications,1994,22(7):847-856.
[27]陈凤德.n种群Ltoka-Volterra时滞竞争反馈控制生态系统的全局吸引性[J].数学学报,2006,49(2)：335-346.
[28]Jiandong Zhao, Jifa Jiang and Alan C. Lazer.The permanence and global attractivity in a nonautonomous Lotka-Volterra system.Nonlinear Anal-ysis:Real World Applications,2004,5(2):265-276.
[29]陈斯养.n维时滞Lotka-Volterra捕食模型的全局稳定性[J].陕西师范大学学报(自然科学版),1992,20(1)：6-12.
[30]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.陕西师范大学学报(自然科学版),2001,39(3)：2-30.
[31]Lu Z,Wang W. Global stability for two-species Lotka-Volterra systems with delay[J]. Proc Amer Math Soc,1986,96:75-78.
[32]Yasuhisa Saith,Tadayuki Hara,Ma Wanbiao.Necessary and sufficient con-ditions for permanence and global stability of a Lotka-Volterra competi-tive system wifh two delays[J].J Math Anal,1999,236:557-584.
[33]Grove E. A, Ladas G,Meimaridou A.A necessary and sufficient condi-tion for the oscillation of neutral equations[J]. Journal of Mathematical Analysis and Applications,1987,126(2):341-354.
[34]Grammatikopoulos M.K,Grove E.A,Ladas G. Oscillations of first-order neutral delay differential equations[J].Journal of Mathematical Analysis and Applications,1986,120(2):510-520.
[35]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.
[36]Zhang B.G.Oscillation of first order neutral functional dif-ferential equations[J]. Journal of Mathematical Analysis and Applications.1989,139(2):311-318.
[37]Mihaly Pituk. Linearized oscillation in a nonautonomous scalar delay dif-ferential equation[J].Applied Mathematics Letters,2006,19:320-325.
[38]Kubiaczyk I,Saker S.H.Oscillation and stability in nonlinear delay differ-ential equations of population dynamics[J]. Mathematical and Computer Modelling,2002,35(3-4):295-301.
[39]Leonid Berezanskya,Elena Bravermanb.Linearized oscillati on theory for a nonlinear nonautonomous delay differential equation[J].Journal of Computational and Applied Mathematics,2003,151:119-127.
[40]DUAN Yongrui,WEI Feng,YAN Jurang.Linearized oscillation of nonlin-ear impulsive delay differentia equation[J].Computers and Mathematics with Applications,2002,44:1267-1274.
[41]Leonid berezanskya,Elena Braverman.Linearized oscillation theory for a nonlinear delay impulsive equation[J].Joural of Computational and Ap-plied Mathematics,2003,161:477-495.
[42]Peng D.H,Han M.A,Wang H.Y.Linearized oscillation of first-order non-linear neutral delay difference equations[J].Computers and Mathematics with Applications,2003,45:1785-1796.
[43]Tang X.H.Linearized oscillation of first-order nonlinear neutral delay dif-ferential equations[J].journal of Mathematical analysis and Applications ,2001,258:194-208.
[44]Tang X.H.Linearized oscillation of odd order nonlinear neutral delay dif-ferential equations[J].J.Math. Anal. Appl.,2006,322:864-872.
[45]Zhang B.G,LiuB.M.Linearized oscillation theorems for certain nonlin-ear difference equations with continuous arguments[J].Mathematical and Computer Modelling,1999,33:89-96.
[46]Kulenovic,Ladas. Oscillation of the sunfiower equation [J]. Quarterly of Applied Mathematics.1988,46:8-23.
[47]Gyori I,Ladas G.Oscillation theory of delay differential equations with application[M].New York Oxfork University Press,1992.
[48]Tarski A.A lattice theoretical fixed-piont theorem and its applica-tions[J]. Pacific Journal of Mathematicis,1995,5:285-309.
[49]马知恩,周义仓.常微分定性与稳定性分析[M].北京：科学出版社,2001.
[50]陈兰荪.数学生态学模型与研究方法[M].北京：科学出版社,1988：55-88.
[51]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京：北京大学出版社,2000,26-36.
[2]Weng peixuan,Yi yuyin.Permanenme for a model of hemztopoiesis[J]. Journal of Biomathematics,2002,17(3):279-285.
[3]Tanga X.H,Zou Xingfu. 3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feed-backs[J].J.Differential Equations,2002,186:420-439.
[4]El-Owaidy H, Mohamed H.Y. Linearized oscillation for non-linear sys-tems of delay differential equations[J]. Applied Mathematics and Com-putation,2003,142:17-21.
[5]Zhao Jiandong,Jiang Jifa.Permanence in nonautonomous Lotka-Volterra system with predator-prey[J]. Applied Mathematics and Computation, 2004,152:99-109.
[6]Candan T, Dahiya R. Oscillation theorems for nth-order functional dif-ferential equations[J]. Mathematical and Computer Modeling,2006,43: 357-367.
[7]Zhao Hongyong, LI Sun. Periodic oscillatory and global attractivity for chemostat model involving distributed delays[J]. Nonlinear Analysis: Real World Applications,2006,7:385-394.
[8]Zhao Hongyong, Nan Ding.Existence and global attractivity of positive periodic solution for competition-predator system with variable delays[J]. Chaos, Solitons and Fractals,2006,29:162-170.
[9]陈斯养,王东保.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.
[10]黄利航,陈斯养.一类具有时滞的捕食与被捕食模型的Hopf分支[J].西北师范大学学报(自然科学版),2004,4(40)：12-18.
[11]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.
[12]朱道军,陈斯养.分段常数变量反馈控制Logistic模型的吸引性[J].安徽大学学报(自然科学版),2005,29(2)：13-17.
[13]余胜平,陈斯养,黄建科.具有反馈控制的捕食竞争系统的全局性态[J].西南师范大学学报(自然科学版),2005,30(2)：198-201.
[14]伍代勇,陈斯养.具有反馈控制的广义Logistic增长模型[J].广西师范大学学报(自然科学版),2006,24(1)：41-44.
[15]田亚品,陈斯养,N-种群Lotka-Volterra扩散竞争反馈控制生态系统的持久性和全局渐近性[J].应用数学(自然科学版),2007,20(3)：485-490.
[16]Hassard B, Kazarionff D, Wan Y. Theory and applications of Hopf bi-furcation[M]. Cambridge:Cambridge University Press,1981.
[17]Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics[M]. Dordrecht:Kluwer Academics Publishers, 1992.
[18]范丽,陈斯养,史忠科.一类广义Logistic单种群实质时滞模型的Hopf分支[J].兰州大学学报(自然科学版),2007,43(6)：97-102.
[19]Ma Zhien. Stability of predation models with time delay[J]. Appl Anal,1986,22:159-192.
[20]Frfedman H.I, Rao V.S.H. Stability of a system involving two time de-lay[J]. Siam J Appl Math,1986,46:552-560.
[21]Kuang Y. Delay differential equations with applications in population dynamics[M].Boston:Academic Press,1993.
[22]田亚品,陈斯养.一类造血模型的全局渐近性及Hopf分支周期解单[J].安徽大学学报(自然科学版),2007,31(6)：19-23.
[23]Song Yongli, PENG Yahong. Stability and bifurcation analysis on a Logistic model with discrete and distributed delays[J]. Applied Mathe-matics and Computation,2006,181(2):1745-1757.
[24]司瑞霞,陈斯养.一类含时滞的广义logistic的Hopf分支[J].西北师范大学学报(自然科学版),2006,42(6)：18-22.
[25]Gopalsamy K,Weng peixue.Global attractivity in a competition sys-tem with feedback controls[J]. Computers and Mathematics with Applications,2003,45:665-676.
[26]LU Zhengyi, Yasuhiro Takeuchi.Permanence and global attractivity for competitive Lotka-Volterra systems with delay.Nonlinear Analysis:The-ory,Methods and Applications,1994,22(7):847-856.
[27]陈凤德.n种群Ltoka-Volterra时滞竞争反馈控制生态系统的全局吸引性[J].数学学报,2006,49(2)：335-346.
[28]Jiandong Zhao, Jifa Jiang and Alan C. Lazer.The permanence and global attractivity in a nonautonomous Lotka-Volterra system.Nonlinear Anal-ysis:Real World Applications,2004,5(2):265-276.
[29]陈斯养.n维时滞Lotka-Volterra捕食模型的全局稳定性[J].陕西师范大学学报(自然科学版),1992,20(1)：6-12.
[30]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.陕西师范大学学报(自然科学版),2001,39(3)：2-30.
[31]Lu Z,Wang W. Global stability for two-species Lotka-Volterra systems with delay[J]. Proc Amer Math Soc,1986,96:75-78.
[32]Yasuhisa Saith,Tadayuki Hara,Ma Wanbiao.Necessary and sufficient con-ditions for permanence and global stability of a Lotka-Volterra competi-tive system wifh two delays[J].J Math Anal,1999,236:557-584.
[33]Grove E. A, Ladas G,Meimaridou A.A necessary and sufficient condi-tion for the oscillation of neutral equations[J]. Journal of Mathematical Analysis and Applications,1987,126(2):341-354.
[34]Grammatikopoulos M.K,Grove E.A,Ladas G. Oscillations of first-order neutral delay differential equations[J].Journal of Mathematical Analysis and Applications,1986,120(2):510-520.
[35]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.
[36]Zhang B.G.Oscillation of first order neutral functional dif-ferential equations[J]. Journal of Mathematical Analysis and Applications.1989,139(2):311-318.
[37]Mihaly Pituk. Linearized oscillation in a nonautonomous scalar delay dif-ferential equation[J].Applied Mathematics Letters,2006,19:320-325.
[38]Kubiaczyk I,Saker S.H.Oscillation and stability in nonlinear delay differ-ential equations of population dynamics[J]. Mathematical and Computer Modelling,2002,35(3-4):295-301.
[39]Leonid Berezanskya,Elena Bravermanb.Linearized oscillati on theory for a nonlinear nonautonomous delay differential equation[J].Journal of Computational and Applied Mathematics,2003,151:119-127.
[40]DUAN Yongrui,WEI Feng,YAN Jurang.Linearized oscillation of nonlin-ear impulsive delay differentia equation[J].Computers and Mathematics with Applications,2002,44:1267-1274.
[41]Leonid berezanskya,Elena Braverman.Linearized oscillation theory for a nonlinear delay impulsive equation[J].Joural of Computational and Ap-plied Mathematics,2003,161:477-495.
[42]Peng D.H,Han M.A,Wang H.Y.Linearized oscillation of first-order non-linear neutral delay difference equations[J].Computers and Mathematics with Applications,2003,45:1785-1796.
[43]Tang X.H.Linearized oscillation of first-order nonlinear neutral delay dif-ferential equations[J].journal of Mathematical analysis and Applications ,2001,258:194-208.
[44]Tang X.H.Linearized oscillation of odd order nonlinear neutral delay dif-ferential equations[J].J.Math. Anal. Appl.,2006,322:864-872.
[45]Zhang B.G,LiuB.M.Linearized oscillation theorems for certain nonlin-ear difference equations with continuous arguments[J].Mathematical and Computer Modelling,1999,33:89-96.
[46]Kulenovic,Ladas. Oscillation of the sunfiower equation [J]. Quarterly of Applied Mathematics.1988,46:8-23.
[47]Gyori I,Ladas G.Oscillation theory of delay differential equations with application[M].New York Oxfork University Press,1992.
[48]Tarski A.A lattice theoretical fixed-piont theorem and its applica-tions[J]. Pacific Journal of Mathematicis,1995,5:285-309.
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