非线性微分动力系统的定性分析
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摘要
本文主要研究了一类非线性微分动力系统模型，主要分为三个部分。
第一章建立并分析了食饵种群具有传染病且有垂直传染的生态—流行病模型，讨论了解的有界性，应用特征根法、Hurwitz判别法得到了平衡点局部渐近稳定的充分条件，并进一步讨论了正平衡点的全局稳定性，得到了正平衡点全局稳定的充分条件。并将垂直传染率q=1时的模型作为特例，同样做了如上的讨论。得到结论：当传染病的垂直传染率为1时，疾病一旦流行就会成为地方性传染病。而当传染病的垂直传染率小于1时，可以通过控制对易感类食饵种群和染病类食饵种群的捕食率来影响地方性平衡点的全局稳定性，这样就可以防止疾病的流行。
第二章研究了捕食率是具有非单调性功能反应函数的脉冲捕食—食饵系统，运用脉冲比较定理、Floquet乘子理论、Liapunov函数等方法讨论了系统的灭绝性和持续生存性，得到了相应的充分条件。
第三章研究了具有脉冲预防接种且传染率是函数β(N)的SIRS传染病模型，利用脉冲比较原理，证明了无病周期解的存在性和全局稳定性。得到结论：可以通过对脉冲接种比例的调整来控制阈值(?)_2的数值，从而达到控制传染病蔓延的效果。
The main research of this text is about the models of nonlinear differential dynamic system. The full text can be divided into three parts.
In the first part, we formulated and analyzed the predator-prey model of prey with epidemic and vertical transmission. The boundness of solutions are studied, and the sufficient condition of locally asymptotically stability of the equilibrium are studied by latent root method and Hurwitz method. Furthermore, the global stability of the equilibrium, and the sufficient condition of global stability of positive equilibrium are also obtained. We look the model with vertical transmission
rate q = 1 as an especial example. We do the same studying. Finally, we get the conclusion:
When the vertical transmission rate equal to 1, the disease will be to local infection. When the vertical transmission is less than 1, we can predominate the global stability of the local equilibrium by controlling predatory rate of infective prey and susceptible prey. Then we can prevent prevalence of disease.
In the second part, we study an impulse predator-prey system with nonmonotonic predatory rate. We discuss the extinction and permanent existence by the impulse comparison theorem、 Floquet multiplier and Liapunov function. Finally, we get the relevant sufficient condition. In the third part, The SIRS epidemical model with impulsive vaccinations is discussed, and
the infective rate is function β(N). We proved the existence and global stability of the disease-free periodic solution by impulsive comparison theorem and get the conclusion: we can control the threshold R|-_2 by adjusting impulsive vaccinations, then we can prevent the
spreading of disease.
Wei Zou (Application mathematics) Directed by: Zuoliang Xiong Professor
第一章建立并分析了食饵种群具有传染病且有垂直传染的生态—流行病模型，讨论了解的有界性，应用特征根法、Hurwitz判别法得到了平衡点局部渐近稳定的充分条件，并进一步讨论了正平衡点的全局稳定性，得到了正平衡点全局稳定的充分条件。并将垂直传染率q=1时的模型作为特例，同样做了如上的讨论。得到结论：当传染病的垂直传染率为1时，疾病一旦流行就会成为地方性传染病。而当传染病的垂直传染率小于1时，可以通过控制对易感类食饵种群和染病类食饵种群的捕食率来影响地方性平衡点的全局稳定性，这样就可以防止疾病的流行。
第二章研究了捕食率是具有非单调性功能反应函数的脉冲捕食—食饵系统，运用脉冲比较定理、Floquet乘子理论、Liapunov函数等方法讨论了系统的灭绝性和持续生存性，得到了相应的充分条件。
第三章研究了具有脉冲预防接种且传染率是函数β(N)的SIRS传染病模型，利用脉冲比较原理，证明了无病周期解的存在性和全局稳定性。得到结论：可以通过对脉冲接种比例的调整来控制阈值(?)_2的数值，从而达到控制传染病蔓延的效果。
The main research of this text is about the models of nonlinear differential dynamic system. The full text can be divided into three parts.
In the first part, we formulated and analyzed the predator-prey model of prey with epidemic and vertical transmission. The boundness of solutions are studied, and the sufficient condition of locally asymptotically stability of the equilibrium are studied by latent root method and Hurwitz method. Furthermore, the global stability of the equilibrium, and the sufficient condition of global stability of positive equilibrium are also obtained. We look the model with vertical transmission
rate q = 1 as an especial example. We do the same studying. Finally, we get the conclusion:
When the vertical transmission rate equal to 1, the disease will be to local infection. When the vertical transmission is less than 1, we can predominate the global stability of the local equilibrium by controlling predatory rate of infective prey and susceptible prey. Then we can prevent prevalence of disease.
In the second part, we study an impulse predator-prey system with nonmonotonic predatory rate. We discuss the extinction and permanent existence by the impulse comparison theorem、 Floquet multiplier and Liapunov function. Finally, we get the relevant sufficient condition. In the third part, The SIRS epidemical model with impulsive vaccinations is discussed, and
the infective rate is function β(N). We proved the existence and global stability of the disease-free periodic solution by impulsive comparison theorem and get the conclusion: we can control the threshold R|-_2 by adjusting impulsive vaccinations, then we can prevent the
spreading of disease.
Wei Zou (Application mathematics) Directed by: Zuoliang Xiong Professor
引文
[1] Hadeler K P, Freedman H I. Predator-prey population with parasitic infection.[J]. J Math Biol, 1989,27:609-631.
[2] Venturino E. The influence of disease on Lotka-Volterra system. Rockymount. [J]. J Math, 1994, 24:389-402.
[3] Chattopadhyay J, Arino O. A predator-prey model with disease in the prey. [J]. Nonlinear Anal. 1999, 36:749-766.
[4] Xiao Y N, Chen L S. Modeling and analysis of a predator-prey model with disease in the prey. [J]. Math Biosci.2001.171 (1):59-82.
[5] 陈晓鹰，朱婉珍．具有Holling Ⅱ型功能性反应的捕食者—食饵种群SIS模型定性分析．[J]．厦门大学学报(自然科学版)．2005，44(1)：16-19．
[6] 宋新宇，肖燕妮，陈兰荪．具有时滞的生态—流行病模型的稳定性和Hopf分支[J]．数学物理学报，2005，25A(1)：57-66
[7] 陈兰荪．数学生态学模型与研究方法[M]．北京：科学出版社，1988；58—59，396．
[8] 马知恩．常微分方程定性与稳定性方法[M]．北京：科学出版社，2001；12—13
[9] 马知恩，周义仓，王稳地，靳祯．传染病动力学的数学建模与研究[M]．北京：科学出版社，2004；
[10] Shigui Ruan and Wendi Wang. Dynamical behavior of an epidemic model with a nonlinear incidence rate [J]. Differential Equations, 188(2003): 135-163
[11] Jianquan Li, Zhien Ma. Global Analysis of SIS Epidemic Models with Variable Total Population Size [J]. Mathematical and Computer Modeling, 39(2004): 1231-1242
[12] Yuliya N. Kyrychko, Konstantin B. Blyuss. Global properties of a delayed SIR model with temporary immunity and.nonlinear incidence rate [J] Nonlinear Analysis:6(2005):495-507
[13] G. Sabin and D. Summers Chaos in a periodically forced predator-prey eco-system modeL[J]. Mathematical Biosciences, 1993, (113):91-113.
[14] Z. Amine and R. Ortega. A periodic prey-predator system. [J]. Journal of Mathematical Analysis and Applications, 1994,(185):477-489.
[15] 滕志东，陈兰荪．周期捕食被捕食系统正周期解存在的充要条件．[J]．数学物理学报，1998，18(4)：402-406．
[16] 许斌，陈狄岚，孙继涛．一类具有功能反应的生物捕食系统的脉冲控制．[J]．生物数学学报．2004，19(1)：77-81．
[17] 杨志春．具有脉冲和HollingⅢ类功能反应的捕食系统的持续生存和周期解．[J]．生物数学学报．2004，19(4)：439-444．
[18] 谭德君．具有非单调功能反应和脉冲扰动的捕食系统的研究．[J]．数学杂志．2005，25(2)：210-216．
[19] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations [M]. World Scientific, 1989.
[20] Mena-lorca and J, Hethcote H W. Dynamic models of infectious diseases as regulations of populationsizes. [J]. J. Math. Biol, 1992, 30(6):693-716.
[21] Hethcote H W. The mathematics of infectious diseases. [J]. SIAM Rev, 2000, 42:599-653.
[22] Lajmanovich A, Yorke J A. A deterministic model for gonorrhea in a nonhomogeneous population. [J]. Math. Biosci, 1976,28(3):221-236.
[23] Hethcote H W, Van Ark J W. Epidemiological models with heterogeneous populations: Proportionate mixing, parameter estimation and immunization programs. [J]. Math.??Biosci,1987, 84(1)：85_118．
[24] 马知恩，靳祯．总人口在变化的流行病动力学模型．[J]．华北工学院学报，2001，22(4)：262-271．
[25] 陈军杰．一类具有常数迁入且总人口在变化的SIRI传染病模型的稳定性．[J]．生物数学学报．2004，19(3)：310-316．
[26] 马之恩，周义仓，王稳地，靳祯．传染病动力学的数学建模与研究．[M]．北京．科学出版社．2004：313—314．
[27] 张玉娟，陈兰荪，孙丽华．一类具有脉冲效应的捕食者—食饵系统分析．[J]．大连理工大学学报．2004，44(5)：769-774．
[28] 胡宝安，陈博文，原存德．具有阶段结构的SIS传染病模型．[J]．2005，20(1)：58-64．
[29] 张江山，孙树林．捕食者有病的生态—流行病模型的分析．[J]．2005，20(2)：157-164．
[30] 张树文，陈兰荪．具有脉冲效应和综合害虫控制的捕食系统．[J]．系统科学与数学．2005，25(3)：264-275．
[31] 陈军杰．若干具有非线性传染力的传染病模型的稳定性分析．[J]．2005，20(3)：286-296．
[32] Mimmo lannelli, Maia Martcheva, Xue-Zhi Li. Strain replacement in an epidemic model with super-infection and perfect vaccination. [J]. Mathematical Bioscience. 2005. 195:23-46.
[33] 柳合龙，徐厚宝，于景元，朱广田．带有脉冲免疫和传染年龄的传染病模型的全局吸引性．[J]．应用泛函分析学报．2005，7(3)：280-288．
[34] Guang Zhao Zeng, Lan Sun Chen, Li Hua Sun. Complexity of an SIR epidemics model with impulsive vaccination control. [J]. Chaos, Solitons and Fractals, 2005, 26: 495-505.
1．《一类时滞三种群捕食—食饵系统周期正解的存在性》，熊佐亮，邹娓，南昌大学学报理科版，2004年9月。
2. 《Qualitative Analysis on a Holling Type-Ⅳ Functional Response Model with Rate Prey Investing》, Xiong zuoliang, Zou wei, 2005 Seminar on Engineering Education Cooperation & Academic Research for Chinese-French Universities, 2005年5月。
3. 《Qualitative Analysis of SI Epidemic Model with Constant Growth Rate》, Zou wei, Xlong zuoliang, 2005 Seminar on Engineering Education Cooperation & Academic Research for Chinese-French Universities, 2005年5月。
4．《具功能性反应的食饵—捕食者两种群模型的定性分析》，邹娓，熊佐亮，南昌大学学报工科版，2005年9月。
5．《Study on Dynamic M0del of a Ratio-Dependent two Competing Predator-one prey》，Zou wei，Xiong zuoliang，生物数学学报2005年第20卷第5期。
[2] Venturino E. The influence of disease on Lotka-Volterra system. Rockymount. [J]. J Math, 1994, 24:389-402.
[3] Chattopadhyay J, Arino O. A predator-prey model with disease in the prey. [J]. Nonlinear Anal. 1999, 36:749-766.
[4] Xiao Y N, Chen L S. Modeling and analysis of a predator-prey model with disease in the prey. [J]. Math Biosci.2001.171 (1):59-82.
[5] 陈晓鹰，朱婉珍．具有Holling Ⅱ型功能性反应的捕食者—食饵种群SIS模型定性分析．[J]．厦门大学学报(自然科学版)．2005，44(1)：16-19．
[6] 宋新宇，肖燕妮，陈兰荪．具有时滞的生态—流行病模型的稳定性和Hopf分支[J]．数学物理学报，2005，25A(1)：57-66
[7] 陈兰荪．数学生态学模型与研究方法[M]．北京：科学出版社，1988；58—59，396．
[8] 马知恩．常微分方程定性与稳定性方法[M]．北京：科学出版社，2001；12—13
[9] 马知恩，周义仓，王稳地，靳祯．传染病动力学的数学建模与研究[M]．北京：科学出版社，2004；
[10] Shigui Ruan and Wendi Wang. Dynamical behavior of an epidemic model with a nonlinear incidence rate [J]. Differential Equations, 188(2003): 135-163
[11] Jianquan Li, Zhien Ma. Global Analysis of SIS Epidemic Models with Variable Total Population Size [J]. Mathematical and Computer Modeling, 39(2004): 1231-1242
[12] Yuliya N. Kyrychko, Konstantin B. Blyuss. Global properties of a delayed SIR model with temporary immunity and.nonlinear incidence rate [J] Nonlinear Analysis:6(2005):495-507
[13] G. Sabin and D. Summers Chaos in a periodically forced predator-prey eco-system modeL[J]. Mathematical Biosciences, 1993, (113):91-113.
[14] Z. Amine and R. Ortega. A periodic prey-predator system. [J]. Journal of Mathematical Analysis and Applications, 1994,(185):477-489.
[15] 滕志东，陈兰荪．周期捕食被捕食系统正周期解存在的充要条件．[J]．数学物理学报，1998，18(4)：402-406．
[16] 许斌，陈狄岚，孙继涛．一类具有功能反应的生物捕食系统的脉冲控制．[J]．生物数学学报．2004，19(1)：77-81．
[17] 杨志春．具有脉冲和HollingⅢ类功能反应的捕食系统的持续生存和周期解．[J]．生物数学学报．2004，19(4)：439-444．
[18] 谭德君．具有非单调功能反应和脉冲扰动的捕食系统的研究．[J]．数学杂志．2005，25(2)：210-216．
[19] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations [M]. World Scientific, 1989.
[20] Mena-lorca and J, Hethcote H W. Dynamic models of infectious diseases as regulations of populationsizes. [J]. J. Math. Biol, 1992, 30(6):693-716.
[21] Hethcote H W. The mathematics of infectious diseases. [J]. SIAM Rev, 2000, 42:599-653.
[22] Lajmanovich A, Yorke J A. A deterministic model for gonorrhea in a nonhomogeneous population. [J]. Math. Biosci, 1976,28(3):221-236.
[23] Hethcote H W, Van Ark J W. Epidemiological models with heterogeneous populations: Proportionate mixing, parameter estimation and immunization programs. [J]. Math.??Biosci,1987, 84(1)：85_118．
[24] 马知恩，靳祯．总人口在变化的流行病动力学模型．[J]．华北工学院学报，2001，22(4)：262-271．
[25] 陈军杰．一类具有常数迁入且总人口在变化的SIRI传染病模型的稳定性．[J]．生物数学学报．2004，19(3)：310-316．
[26] 马之恩，周义仓，王稳地，靳祯．传染病动力学的数学建模与研究．[M]．北京．科学出版社．2004：313—314．
[27] 张玉娟，陈兰荪，孙丽华．一类具有脉冲效应的捕食者—食饵系统分析．[J]．大连理工大学学报．2004，44(5)：769-774．
[28] 胡宝安，陈博文，原存德．具有阶段结构的SIS传染病模型．[J]．2005，20(1)：58-64．
[29] 张江山，孙树林．捕食者有病的生态—流行病模型的分析．[J]．2005，20(2)：157-164．
[30] 张树文，陈兰荪．具有脉冲效应和综合害虫控制的捕食系统．[J]．系统科学与数学．2005，25(3)：264-275．
[31] 陈军杰．若干具有非线性传染力的传染病模型的稳定性分析．[J]．2005，20(3)：286-296．
[32] Mimmo lannelli, Maia Martcheva, Xue-Zhi Li. Strain replacement in an epidemic model with super-infection and perfect vaccination. [J]. Mathematical Bioscience. 2005. 195:23-46.
[33] 柳合龙，徐厚宝，于景元，朱广田．带有脉冲免疫和传染年龄的传染病模型的全局吸引性．[J]．应用泛函分析学报．2005，7(3)：280-288．
[34] Guang Zhao Zeng, Lan Sun Chen, Li Hua Sun. Complexity of an SIR epidemics model with impulsive vaccination control. [J]. Chaos, Solitons and Fractals, 2005, 26: 495-505.
1．《一类时滞三种群捕食—食饵系统周期正解的存在性》，熊佐亮，邹娓，南昌大学学报理科版，2004年9月。
2. 《Qualitative Analysis on a Holling Type-Ⅳ Functional Response Model with Rate Prey Investing》, Xiong zuoliang, Zou wei, 2005 Seminar on Engineering Education Cooperation & Academic Research for Chinese-French Universities, 2005年5月。
3. 《Qualitative Analysis of SI Epidemic Model with Constant Growth Rate》, Zou wei, Xlong zuoliang, 2005 Seminar on Engineering Education Cooperation & Academic Research for Chinese-French Universities, 2005年5月。
4．《具功能性反应的食饵—捕食者两种群模型的定性分析》，邹娓，熊佐亮，南昌大学学报工科版，2005年9月。
5．《Study on Dynamic M0del of a Ratio-Dependent two Competing Predator-one prey》，Zou wei，Xiong zuoliang，生物数学学报2005年第20卷第5期。