两类三维系统的稳定性与Hopf分支
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摘要
本文主要运用常微分方程定性与稳定性理论及分支的方法，研究了两类种群生态学模型．全文内容共分为三章．本文主要作了以下工作：
1．第一部分是绪论．先综述了生态学发展的状况，随后介绍了问题的引入以及本文的主要工作．最后介绍了本文所用到的一些有关生态学和稳定性方面的定理和引理．
2．第二部分研究了一类具有目标转移强度的三种群捕食-食饵系统，通过Hurwitz判据和Lyapunov方法，得到了系统正平衡点稳定的条件，当两食饵种群的内秉增长率相同时，该捕食系统将有一个周期解．接着讨论了系统的Hopf分支，得到了当取比例常数k_2为分支参数时，系统发生Hopf分支的条件．进一步讨论了当比例常数k_2、k_1满足函数关系时，即k_2=f(k_1)(k_1>0)时，系统发生Hopf分支的必要条件，并相应给出了例子，当k_2=lk_1，k_2=k_1~α(α>1)，k_2=k_1~α(0<α<1)时，系统在正平衡点处的Hopf分支．
3．第三部分主要研究了一类具有多时滞和阶段结构的捕食-食饵系统，其中食饵具有阶段结构，且捕食者只捕食成年食饵．运用比较定理判断出解的有界性．采用常微分方程稳定性和定性方法，分析了系统的非负不变性、边际平衡点的性质及正平衡点的局部渐近稳定性与Hopf分支，得到了当τ=0时，系统在正平衡点的稳定性的充分条件．同时考虑了时滞对于系统稳定性的影响，当选取时滞τ=τ_1+τ_2作为分支参数时，得到了当时滞τ=τ_1+τ_2由0变化到临界值时，系统在正平衡点附近发生Hopf分支，即当τ增加通过临界值时，从正平衡点分支出周期解，
In this paper, by using the qualitative and stability theories and bifurcation method of ordinary differential equations, two population models in ecology are studied .The whole paper consists of three chapters. Details are as follows:
1.The first chapter is the introduction. The development of ecology and the main works of the thesis are introduced, then some fundamental theories and lemmas about ecology and stability that can be used in this paper are given.
2.A mathematical model of two prey and one predator system which has the switching property of predation is considered. By using the Routh-Hurwitz criteria and Lya-punov method ,the condition of the stable positive equilibrium is obtained. In the special case that two prey species have the same intrinsic growth rates, it is shown that the system asymptotically settles a Volterra's oscillation in three-dimensional space.Then the Hopf bifurcation of the system is discussed.The conditions of the Hopf bifurcation are obtained when k_2 is parameters.Further ,the necessary conditions of the Hopf bifurcation are obtained when k_2 = f(k_1)(k_1 > 0) ,and several examples are given when k_2 = lk_1, k_2 = k_1~α(α> 1), k_2 = k_1~α(0 <α< 1).
3.A prey-predator system of two species with stage structure and time delay is investigated . The boundary of the solution to the system is obtained by using summary theory. By using the stability theory of the differential equation ,the invariance of non-negativity, nature of the boundary equilibrium ,the local stability and Hopf bifurcation of the positive equilibrium are analyzed.The sufficient conditions for local asymptotic stability of the positive equilibrium are given when time delayτ= 0.Furthermore, it shows that positive equilibrium is locally asymptotically stable when time delayτ=τ_1 +τ_2 is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases .That is ,a family of periodic solutions bifurcates from positive equilibrium as r passes through the critical value .
1．第一部分是绪论．先综述了生态学发展的状况，随后介绍了问题的引入以及本文的主要工作．最后介绍了本文所用到的一些有关生态学和稳定性方面的定理和引理．
2．第二部分研究了一类具有目标转移强度的三种群捕食-食饵系统，通过Hurwitz判据和Lyapunov方法，得到了系统正平衡点稳定的条件，当两食饵种群的内秉增长率相同时，该捕食系统将有一个周期解．接着讨论了系统的Hopf分支，得到了当取比例常数k_2为分支参数时，系统发生Hopf分支的条件．进一步讨论了当比例常数k_2、k_1满足函数关系时，即k_2=f(k_1)(k_1>0)时，系统发生Hopf分支的必要条件，并相应给出了例子，当k_2=lk_1，k_2=k_1~α(α>1)，k_2=k_1~α(0<α<1)时，系统在正平衡点处的Hopf分支．
3．第三部分主要研究了一类具有多时滞和阶段结构的捕食-食饵系统，其中食饵具有阶段结构，且捕食者只捕食成年食饵．运用比较定理判断出解的有界性．采用常微分方程稳定性和定性方法，分析了系统的非负不变性、边际平衡点的性质及正平衡点的局部渐近稳定性与Hopf分支，得到了当τ=0时，系统在正平衡点的稳定性的充分条件．同时考虑了时滞对于系统稳定性的影响，当选取时滞τ=τ_1+τ_2作为分支参数时，得到了当时滞τ=τ_1+τ_2由0变化到临界值时，系统在正平衡点附近发生Hopf分支，即当τ增加通过临界值时，从正平衡点分支出周期解，
In this paper, by using the qualitative and stability theories and bifurcation method of ordinary differential equations, two population models in ecology are studied .The whole paper consists of three chapters. Details are as follows:
1.The first chapter is the introduction. The development of ecology and the main works of the thesis are introduced, then some fundamental theories and lemmas about ecology and stability that can be used in this paper are given.
2.A mathematical model of two prey and one predator system which has the switching property of predation is considered. By using the Routh-Hurwitz criteria and Lya-punov method ,the condition of the stable positive equilibrium is obtained. In the special case that two prey species have the same intrinsic growth rates, it is shown that the system asymptotically settles a Volterra's oscillation in three-dimensional space.Then the Hopf bifurcation of the system is discussed.The conditions of the Hopf bifurcation are obtained when k_2 is parameters.Further ,the necessary conditions of the Hopf bifurcation are obtained when k_2 = f(k_1)(k_1 > 0) ,and several examples are given when k_2 = lk_1, k_2 = k_1~α(α> 1), k_2 = k_1~α(0 <α< 1).
3.A prey-predator system of two species with stage structure and time delay is investigated . The boundary of the solution to the system is obtained by using summary theory. By using the stability theory of the differential equation ,the invariance of non-negativity, nature of the boundary equilibrium ,the local stability and Hopf bifurcation of the positive equilibrium are analyzed.The sufficient conditions for local asymptotic stability of the positive equilibrium are given when time delayτ= 0.Furthermore, it shows that positive equilibrium is locally asymptotically stable when time delayτ=τ_1 +τ_2 is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases .That is ,a family of periodic solutions bifurcates from positive equilibrium as r passes through the critical value .
引文
[1]陆志奇，李静．竞争数学模型的理论研究[M]．北京：科学出版社，2008
[2] Lotka A J.Undamped oscillations derived from the law of mass action[J]. Journal Am Chemical Sociology, 1920, 42: 1595-1599
[3]马知恩，周义仓．常微分方程定性与稳定性方法[M]．北京：科学出版社，2005， 244-259
[4] Holling C S. The functional resPonse of Predator to Prey density and its role in mimicry and PoPulation regulation[J]. Men. Ent Soe. Can, 1965, 45: 1-60
[5] Beddington J R. Mutual interference between parasites or predators and its effect on searching efficiency[J]. Journal Animal Ecology, 1975, 44: 331-341
[6] DeAngelis D L,Goldstein R A , O'Neill R V. A model for trophic interaction [J] Ecologyogy, 1975, 56: 881-892
[7] Kuang Y, Beretta E.Global qualitative analysis of a ratio-dependent predator-prey system[J]. J Math Biol, 1998, 36: 389-406
[8]柏灵，王克．一类具有比率型功能反应的食物链系统周期解的存在性[J]．东北师大学 报(自然科学版)，2003，(01)：1-9
[9]黄建科，吴筱宁．基于比率的三种群捕食模型的稳定性[J]．江西师范大学学报(自然 科学版)．2007，31(6)：587-590
[10]徐瑞，陈兰荪．具有时滞和基于比率的三种群捕食系统的持久性与稳定性[J]．系统科 学与数学，2001，21(2)：204-212
[11]谭德君．基于比率的三种群捕食系统的持久生存[J]．生物数学学报，2003，18(1)： 50-56
[12] W.Sokol W, Howell.J.A.Kinetics of pheol oxidation by washed cells[J].Biotechnol. Bioeng, 1980(23): 2039-2049
[13]蔡礼明．时滞非线性动力系统的建模与研究[D]．北京：北京信息控制研究所博士学 位论文，2008
[14] R.M.Nisbet, W.Gurne. The systematie formulation of population models for insects with dynamically varying instar duration[J]. Theor. Pop. Biol, 1983, 23: 114-135
[15] R.M.Nisbet, W.Gurney. "Stage-Strueture" Models of UniformL arval Competition [J]. Mathematical Eeology(Trieste, 1982), Lecture Notes in Biomath. 54, Springer, Berlin, 1984, PP. 97-113
[16] W.Gurney, R.M.Nisbet, S.Blythe. The Systematic Formulation of Models of Stage-Struetured Populations[J]. The Dynamics of Physiologically Struetured Populations (Amsterdam, 1983), Lecture-notes in Biomath, 68, Springer, Berlin, 1986. pp. 474-494
[17] Aiello W G,Freedman H I. A time-delay model of single species growth with stage structure[J]. Math Biosci, 1990, 01: 139-156
[18] Wang W, Chen L. A predator-prey system with stage-structure for predator[J]. Compute Math Appl, 1997, 33: 83-91
[19]庄科俊．一类阶段结构捕食系统的周期解[J]．重庆工学院学报：自然科学版，2008， 22(3)：133-136
[20]杨文生，李学鹏．一类具有阶段结构和功能性反应的捕食系统[J]．莆田学院学报， 2008，15(2)：18-21
[21] Jiao Jianjun, Pang Guo ping, Chen Lan sun. A delayed stage-structured predatorprey model with impulsive stocking on prey and continuous harvesting on predator[J]. Applied Mathematies and Computation, 2008, 195(1): 316-325
[22]安存斌，马小箭．具有阶段结构和扩散的一类自治捕食系统全局渐近稳定性研究[J]． 山西大学学报：自然科学版，2008，24(3)：8-10
[23] Miao Baojun, Rong Yuetang, Zhou Lun. Local analysis of a food-chain model with stage structure and time delay[J]. Basic Sciences Journal of textile universities, 2008, 21(2): 182-187
[24]Chen Y M，Huang Y N，zhang J X．Switching Effect in three-dimensional prey- predator system[J]．郑州大学学报：理学版，2007，39(1)：20-23
[25]陈元明，黄永年．三种群捕食系统的Hopf分支[J]．丽水学院学报，2007，29(5)：18-20
[26]宋永利，韩茂安，魏俊杰．多时滞捕食食饵系统的正平衡点的稳定性及Hopf分支[J]． 数学年刊，2004，25：783-790
[27]朱道军，兰叶霞．具有多时滞捕食被捕食流行病模型的稳定性[J]．安徽大学学报． 2007，31(4)：5-8
[28] Cao Y, Fan J, Card T C. The effects of stage-structured population growth models[J]. Nonli Anal: Th Math Applic(S0362-546X), 1992, 16(2): 95-105
[29] Cui J A, Chen L S, Wang W D.The effect of dispersal on population Growth with Stage-structure[J]. Computers Math Applic(S0898-1221), 2000, 39: 91-102. 2002, 17(1): 5-11
[30]王豪，郑丽丽．一类具有阶段结构和时滞的捕食与被捕食系统[J]．信阳师范学院学 报：自然科学版，2004，17(2)：140-145
[31] Dickmann O,Nisbet R M, Gumey W S C.A simple mathematical models for cannibalism : a tique and a new approach[J]. Math Biosci(S0025-5564), 1986, 78: 21-46
[32] Hastings A, Cycles in cannibalistic egg-larval interactions[J]. J Math Biol(S0303-6812), 1987, 24: 651-666
[33]师向云，郭振．一类具有时滞和阶段结构的捕食系统的全局分析[J]．信阳师范学院学 报：自然科学版，2008，21(1)：32-35
[34] T.K.Kar, U.K.Pahari. Non-selective harvesting in prey-predator models with delay [J]. Nonliner Science and Numerical Simulation. 2006, 11: 499-509
[35] Ruan, S, Wei J. On the zero of transcendental functions with applications to stability of delay differential equations with to delay [J]. Dynma, Contin. Discrete Implus. System, 2003, (10): 863
[36] Freedman HI, Sree Hari Rao V. The Trade-off between mutual interference and time lags in predator-prey systems[J]. Bull Math Biol, 1983, 45(6): 991-1003
[37]陈兰荪，孙键．非线性生物动力系统[M]．北京：科学出版社，1993
[38]陆征一，周义仓．数学生物学进展[M]．北京：科学出版社，2006
[39] Tansky M. Switching effect in prey-predator system[J]. J Theor Biol, 1978, 70: 263-271
[40] Hastings A. Cycles in Cannibalistic Egg-larval Interaction [J]. J Math Biol(S0303-6812), 1987, 24, 17(2): 140-145
[41]李光云．几类Beddington-DeAngelis型捕食-食饵模型的定性分析[D]．湖南：湖南 大学，2006
[2] Lotka A J.Undamped oscillations derived from the law of mass action[J]. Journal Am Chemical Sociology, 1920, 42: 1595-1599
[3]马知恩，周义仓．常微分方程定性与稳定性方法[M]．北京：科学出版社，2005， 244-259
[4] Holling C S. The functional resPonse of Predator to Prey density and its role in mimicry and PoPulation regulation[J]. Men. Ent Soe. Can, 1965, 45: 1-60
[5] Beddington J R. Mutual interference between parasites or predators and its effect on searching efficiency[J]. Journal Animal Ecology, 1975, 44: 331-341
[6] DeAngelis D L,Goldstein R A , O'Neill R V. A model for trophic interaction [J] Ecologyogy, 1975, 56: 881-892
[7] Kuang Y, Beretta E.Global qualitative analysis of a ratio-dependent predator-prey system[J]. J Math Biol, 1998, 36: 389-406
[8]柏灵，王克．一类具有比率型功能反应的食物链系统周期解的存在性[J]．东北师大学 报(自然科学版)，2003，(01)：1-9
[9]黄建科，吴筱宁．基于比率的三种群捕食模型的稳定性[J]．江西师范大学学报(自然 科学版)．2007，31(6)：587-590
[10]徐瑞，陈兰荪．具有时滞和基于比率的三种群捕食系统的持久性与稳定性[J]．系统科 学与数学，2001，21(2)：204-212
[11]谭德君．基于比率的三种群捕食系统的持久生存[J]．生物数学学报，2003，18(1)： 50-56
[12] W.Sokol W, Howell.J.A.Kinetics of pheol oxidation by washed cells[J].Biotechnol. Bioeng, 1980(23): 2039-2049
[13]蔡礼明．时滞非线性动力系统的建模与研究[D]．北京：北京信息控制研究所博士学 位论文，2008
[14] R.M.Nisbet, W.Gurne. The systematie formulation of population models for insects with dynamically varying instar duration[J]. Theor. Pop. Biol, 1983, 23: 114-135
[15] R.M.Nisbet, W.Gurney. "Stage-Strueture" Models of UniformL arval Competition [J]. Mathematical Eeology(Trieste, 1982), Lecture Notes in Biomath. 54, Springer, Berlin, 1984, PP. 97-113
[16] W.Gurney, R.M.Nisbet, S.Blythe. The Systematic Formulation of Models of Stage-Struetured Populations[J]. The Dynamics of Physiologically Struetured Populations (Amsterdam, 1983), Lecture-notes in Biomath, 68, Springer, Berlin, 1986. pp. 474-494
[17] Aiello W G,Freedman H I. A time-delay model of single species growth with stage structure[J]. Math Biosci, 1990, 01: 139-156
[18] Wang W, Chen L. A predator-prey system with stage-structure for predator[J]. Compute Math Appl, 1997, 33: 83-91
[19]庄科俊．一类阶段结构捕食系统的周期解[J]．重庆工学院学报：自然科学版，2008， 22(3)：133-136
[20]杨文生，李学鹏．一类具有阶段结构和功能性反应的捕食系统[J]．莆田学院学报， 2008，15(2)：18-21
[21] Jiao Jianjun, Pang Guo ping, Chen Lan sun. A delayed stage-structured predatorprey model with impulsive stocking on prey and continuous harvesting on predator[J]. Applied Mathematies and Computation, 2008, 195(1): 316-325
[22]安存斌，马小箭．具有阶段结构和扩散的一类自治捕食系统全局渐近稳定性研究[J]． 山西大学学报：自然科学版，2008，24(3)：8-10
[23] Miao Baojun, Rong Yuetang, Zhou Lun. Local analysis of a food-chain model with stage structure and time delay[J]. Basic Sciences Journal of textile universities, 2008, 21(2): 182-187
[24]Chen Y M，Huang Y N，zhang J X．Switching Effect in three-dimensional prey- predator system[J]．郑州大学学报：理学版，2007，39(1)：20-23
[25]陈元明，黄永年．三种群捕食系统的Hopf分支[J]．丽水学院学报，2007，29(5)：18-20
[26]宋永利，韩茂安，魏俊杰．多时滞捕食食饵系统的正平衡点的稳定性及Hopf分支[J]． 数学年刊，2004，25：783-790
[27]朱道军，兰叶霞．具有多时滞捕食被捕食流行病模型的稳定性[J]．安徽大学学报． 2007，31(4)：5-8
[28] Cao Y, Fan J, Card T C. The effects of stage-structured population growth models[J]. Nonli Anal: Th Math Applic(S0362-546X), 1992, 16(2): 95-105
[29] Cui J A, Chen L S, Wang W D.The effect of dispersal on population Growth with Stage-structure[J]. Computers Math Applic(S0898-1221), 2000, 39: 91-102. 2002, 17(1): 5-11
[30]王豪，郑丽丽．一类具有阶段结构和时滞的捕食与被捕食系统[J]．信阳师范学院学 报：自然科学版，2004，17(2)：140-145
[31] Dickmann O,Nisbet R M, Gumey W S C.A simple mathematical models for cannibalism : a tique and a new approach[J]. Math Biosci(S0025-5564), 1986, 78: 21-46
[32] Hastings A, Cycles in cannibalistic egg-larval interactions[J]. J Math Biol(S0303-6812), 1987, 24: 651-666
[33]师向云，郭振．一类具有时滞和阶段结构的捕食系统的全局分析[J]．信阳师范学院学 报：自然科学版，2008，21(1)：32-35
[34] T.K.Kar, U.K.Pahari. Non-selective harvesting in prey-predator models with delay [J]. Nonliner Science and Numerical Simulation. 2006, 11: 499-509
[35] Ruan, S, Wei J. On the zero of transcendental functions with applications to stability of delay differential equations with to delay [J]. Dynma, Contin. Discrete Implus. System, 2003, (10): 863
[36] Freedman HI, Sree Hari Rao V. The Trade-off between mutual interference and time lags in predator-prey systems[J]. Bull Math Biol, 1983, 45(6): 991-1003
[37]陈兰荪，孙键．非线性生物动力系统[M]．北京：科学出版社，1993
[38]陆征一，周义仓．数学生物学进展[M]．北京：科学出版社，2006
[39] Tansky M. Switching effect in prey-predator system[J]. J Theor Biol, 1978, 70: 263-271
[40] Hastings A. Cycles in Cannibalistic Egg-larval Interaction [J]. J Math Biol(S0303-6812), 1987, 24, 17(2): 140-145
[41]李光云．几类Beddington-DeAngelis型捕食-食饵模型的定性分析[D]．湖南：湖南 大学，2006