一类具有竞争项的时滞系统的Hopf分支分析
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摘要
本文运用常微分方程定性理论和常微分方程稳定性理论，研究了两种带有竞争项的时滞系统，首先得到了系统存在正平衡点的条件，然后对该点处的特征方程进行讨论分析，从而得到该系统产生Hopf分支现象时的条件，最后运用中心流形定理和规范型理论进行讨论分析，得到了分支的相关性质.全文分为五章：
第一章是引言，先介绍了捕食者-食饵模型的研究背景和研究现状，随后介绍了本文的主要工作.
第二章介绍了一些理论工具，包括常微分方程定性和稳定性理论和Hopf支理论的相关引理和定理.
第三章讨论了具有竞争项的双时滞系统的Hopf分支模型，利用常微分方程定性和稳定性理论及方法，选取时滞τ为分支参数，讨论正平衡点的稳定性和Hopf分支的存在性.最后给出了一个数值举例.
第四章讨论了具有竞争项的分布时滞系统的Hopf分支模型，并运用常微分方程定性和稳定性理论及方法，得到了平衡点稳定的条件.
第五章对本文研究的内容进行了总结，并对以后的研究方向和生物数学的未来进行了展望.
In this paper, by using qualitative method and stability theory of ordinarydifferential equations, Hopf bifurcation for two classes with time delay in competitionmodels are studied. First of all, we can get the existence of the positive equilibriumsystem conditions. By analyzing the characteristic equation, we can get condition ofthe system occurring Hopf bifurcation. Finally by using the center manifold theoremand normal form method, we can get the related properties of the bifurcation. Thepaper is divided into five parts.
The first part is the introduction, the predator prey model background and theresearch present situation, then introduces the main work of this paper.
The second part is some theoretical tools, including qualitative and stabilitytheory of ordinary differential equations and Hopf bifurcation theory related lemmasand theorems.
The third part is a discussion with two delays in competition Hopf bifurcationmodel, by using qualitative method and stability theory of ordinary differentialequations, selecting delay for bifurcation parameter, and discuss the existence of Hopfand the positive equilibrium of the stability. Finally, we give a numerical example.
The fourth part discusses a distributed delay in competition Hopf bifurcationmodel, and get the balance of the stability conditions.
The last part has an summarize of this paper, and we have a future prospects ofthe future research direction of mathematical biology.
第一章是引言，先介绍了捕食者-食饵模型的研究背景和研究现状，随后介绍了本文的主要工作.
第二章介绍了一些理论工具，包括常微分方程定性和稳定性理论和Hopf支理论的相关引理和定理.
第三章讨论了具有竞争项的双时滞系统的Hopf分支模型，利用常微分方程定性和稳定性理论及方法，选取时滞τ为分支参数，讨论正平衡点的稳定性和Hopf分支的存在性.最后给出了一个数值举例.
第四章讨论了具有竞争项的分布时滞系统的Hopf分支模型，并运用常微分方程定性和稳定性理论及方法，得到了平衡点稳定的条件.
第五章对本文研究的内容进行了总结，并对以后的研究方向和生物数学的未来进行了展望.
In this paper, by using qualitative method and stability theory of ordinarydifferential equations, Hopf bifurcation for two classes with time delay in competitionmodels are studied. First of all, we can get the existence of the positive equilibriumsystem conditions. By analyzing the characteristic equation, we can get condition ofthe system occurring Hopf bifurcation. Finally by using the center manifold theoremand normal form method, we can get the related properties of the bifurcation. Thepaper is divided into five parts.
The first part is the introduction, the predator prey model background and theresearch present situation, then introduces the main work of this paper.
The second part is some theoretical tools, including qualitative and stabilitytheory of ordinary differential equations and Hopf bifurcation theory related lemmasand theorems.
The third part is a discussion with two delays in competition Hopf bifurcationmodel, by using qualitative method and stability theory of ordinary differentialequations, selecting delay for bifurcation parameter, and discuss the existence of Hopfand the positive equilibrium of the stability. Finally, we give a numerical example.
The fourth part discusses a distributed delay in competition Hopf bifurcationmodel, and get the balance of the stability conditions.
The last part has an summarize of this paper, and we have a future prospects ofthe future research direction of mathematical biology.
引文
[1] May R M. Time delay versus stability in population models with two and three trophiclevels[J]. Ecology,1973,4:315-325.
[2] Faria T. Stability and bifurcation for a delayed predator-prey model and the effect ofdiffusion[J]. J Math Appl,2001,254:433-463.
[3] Song Y, Peng Y. Stability and bifurcation analysis on a Logistic model with discrete anddistributed delays[J]. Appl Math Comput,2006,181:1745-1757.
[4]李小玲、胡广平.时滞食饵-捕食系统平衡点的稳定性和周期解[J].兰州大学学报,2009,45(2):81.
[5]常佳佳、张广.具有捕食者相互残杀项时滞系统的Hopf分支[J].数学的实践与认识,2009,39(12):97-102.
[6] Song Y L, Wei J J. Local Hopf bifurcation and global periodic solutions in a delayedpredator-prey system [J]. Math Appl,2005,301:1-21.
[7] Yan X P, Li W T. Hopf bifurcation and global periodic solutions in a delayed predator-preysystem [J]. Appl Math Comput,2006,177:427-445.
[8] Faria T, Oliveira J J. Local and global stability for Lotka-Volterra systems with distributeddelays and instantaneous negative feedbacks[J]. J of Differential Equations,2008,244(5):1049-1079.
[9] Wu F, Hu S. Stochastic functional Kolmogorov-type population dynamics[J]. J Math AnalAppl,2008,347:534-549.
[10] Zhao J, Fu L, Cheng R, Ruan J. Permanence and extinction in a nonautonomous T-periodiccompetitive Lotka-Volterra systems[J]. Dynamics of continuous, Discrete&ImpulsiveSystems, Series B,2010,17:449-456.
[11] Xiang L V, Lu S P, Yan P. Existence and global attractivity of positive periodic solutions ofLotka-Volterra Predator-prey systems with deviating arguments[J]. Real World Appl,2010,11:574-583.
[12] Zhang J Z, Jin Z, Yan J R, Sun G Q. Stability and Hopf bifurcation in a delayed competitionsystem [J]. Nonlinear Anal,2009,70:658-670.
[13]王新秀,窦雾虹,王艳霜.一类具有两个时滞的捕食—食饵系统的Hopf分支[J].延安大学学报,2009,28(3):10-13.
[14]彭煌,窦雾虹,王新秀.一类具功能反应的食饵与捕食系统的极限环[J].陕西科技大学学报,2009,27(6):129-132.
[15]刘晓娜,陈斯养.具有时滞的捕食-被捕食模型的稳定性及Hopf分支[J].陕西科技大学学报,2011,29(2):153-160.
[16]段文英.关于具时滞捕食-被捕食系统的稳定性与Hopf[J].生物数学学报,2004,19(1):87-92.
[17]魏章志,张祖峰.一类时滞系统的全局Hopf分支的存在性[J].合肥学院学报,2007,17(4):4-8.
[18] Xu C J, Chen D X. Bifurcations in a Delayed Predator-prey Model[J].重庆师范大学学报,2011,28(3):43-48.
[19] Cushing J M. Integro-differential Equations and Delay Models in Population Dynamics.Springer[M],1977.
[20]贾艳丽,陈斯养.具有时滞的Lotka-Volterra模型的Hopf分支与数值模拟[J].科学技术与工程,2011,11(15):3366-3371.
[21]张天平,徐瑞.一类具有时滞的捕食系统的稳定性与Hopf分支[J].军械工程学院学报,2007,19(3):76-78.
[22] Li K, Wei J J. Stability and Hopf bifurcation analysis of a prey-predator system with twodelays[J]. Chaos, Solitons and Fractals,2009,42:2606-2613.
[23] Hu G P, Li W T, Yan X P. Hopf bifurcations in a predator-prey system with multipledelays[J]. Chaos, Solitons and Fractals,2009,42:1273-1285.
[24] Nie L, Peng J, Teng Z. Permanence and stability in multi-species non-Lotak-Volterracompetitive systems with delays and feedback controls[J]. Mathematical and ComputerModelling,2009,49:295-306.
[25]马战云.捕食-食饵模型的稳定性和Hopf分支[J].昆明冶金高等专科学校学报,2010,26(1):65-69.
[26]张萍,王雪,薛锡亭.一类时滞系统的Hopf分支计算及稳定性分析[J].长春工业大学学报,2010,31(4):374-378.
[27]王玲书,徐瑞,冯光辉.一类具有时滞的捕食模型的稳定性和Hopf分支[J].数学的实践与认识,2010,40(5):124-128.
[28]徐秀艳.一类具有双时滞的食饵-捕食系统的Hopf分支分析[J].科技导报,2011,29(25):75-79.
[29]李大虎,陈淑苹,童欢,朱文文.具有捕食者相互残杀项双时滞系统的Hopf分支[J].湖北师范学院学报,2011,31(3):76-80.
[30]廖晓昕.稳定性的理论、方法和应用[M].武汉:华中理工大学出版社.,1999.
[31]廖晓昕.动力系统的稳定性理论和应用[M].北京:国防工业出版社.,2000.
[32]王新秀.具有时滞的Volterra捕食系统的Hopf分支分析[D].西北大学硕士学位论文，2010.
[33]刘春燕.两类生物数学模型的稳定性及Hopf分支[D].广西师范大学硕士学位论文，2010.
[2] Faria T. Stability and bifurcation for a delayed predator-prey model and the effect ofdiffusion[J]. J Math Appl,2001,254:433-463.
[3] Song Y, Peng Y. Stability and bifurcation analysis on a Logistic model with discrete anddistributed delays[J]. Appl Math Comput,2006,181:1745-1757.
[4]李小玲、胡广平.时滞食饵-捕食系统平衡点的稳定性和周期解[J].兰州大学学报,2009,45(2):81.
[5]常佳佳、张广.具有捕食者相互残杀项时滞系统的Hopf分支[J].数学的实践与认识,2009,39(12):97-102.
[6] Song Y L, Wei J J. Local Hopf bifurcation and global periodic solutions in a delayedpredator-prey system [J]. Math Appl,2005,301:1-21.
[7] Yan X P, Li W T. Hopf bifurcation and global periodic solutions in a delayed predator-preysystem [J]. Appl Math Comput,2006,177:427-445.
[8] Faria T, Oliveira J J. Local and global stability for Lotka-Volterra systems with distributeddelays and instantaneous negative feedbacks[J]. J of Differential Equations,2008,244(5):1049-1079.
[9] Wu F, Hu S. Stochastic functional Kolmogorov-type population dynamics[J]. J Math AnalAppl,2008,347:534-549.
[10] Zhao J, Fu L, Cheng R, Ruan J. Permanence and extinction in a nonautonomous T-periodiccompetitive Lotka-Volterra systems[J]. Dynamics of continuous, Discrete&ImpulsiveSystems, Series B,2010,17:449-456.
[11] Xiang L V, Lu S P, Yan P. Existence and global attractivity of positive periodic solutions ofLotka-Volterra Predator-prey systems with deviating arguments[J]. Real World Appl,2010,11:574-583.
[12] Zhang J Z, Jin Z, Yan J R, Sun G Q. Stability and Hopf bifurcation in a delayed competitionsystem [J]. Nonlinear Anal,2009,70:658-670.
[13]王新秀,窦雾虹,王艳霜.一类具有两个时滞的捕食—食饵系统的Hopf分支[J].延安大学学报,2009,28(3):10-13.
[14]彭煌,窦雾虹,王新秀.一类具功能反应的食饵与捕食系统的极限环[J].陕西科技大学学报,2009,27(6):129-132.
[15]刘晓娜,陈斯养.具有时滞的捕食-被捕食模型的稳定性及Hopf分支[J].陕西科技大学学报,2011,29(2):153-160.
[16]段文英.关于具时滞捕食-被捕食系统的稳定性与Hopf[J].生物数学学报,2004,19(1):87-92.
[17]魏章志,张祖峰.一类时滞系统的全局Hopf分支的存在性[J].合肥学院学报,2007,17(4):4-8.
[18] Xu C J, Chen D X. Bifurcations in a Delayed Predator-prey Model[J].重庆师范大学学报,2011,28(3):43-48.
[19] Cushing J M. Integro-differential Equations and Delay Models in Population Dynamics.Springer[M],1977.
[20]贾艳丽,陈斯养.具有时滞的Lotka-Volterra模型的Hopf分支与数值模拟[J].科学技术与工程,2011,11(15):3366-3371.
[21]张天平,徐瑞.一类具有时滞的捕食系统的稳定性与Hopf分支[J].军械工程学院学报,2007,19(3):76-78.
[22] Li K, Wei J J. Stability and Hopf bifurcation analysis of a prey-predator system with twodelays[J]. Chaos, Solitons and Fractals,2009,42:2606-2613.
[23] Hu G P, Li W T, Yan X P. Hopf bifurcations in a predator-prey system with multipledelays[J]. Chaos, Solitons and Fractals,2009,42:1273-1285.
[24] Nie L, Peng J, Teng Z. Permanence and stability in multi-species non-Lotak-Volterracompetitive systems with delays and feedback controls[J]. Mathematical and ComputerModelling,2009,49:295-306.
[25]马战云.捕食-食饵模型的稳定性和Hopf分支[J].昆明冶金高等专科学校学报,2010,26(1):65-69.
[26]张萍,王雪,薛锡亭.一类时滞系统的Hopf分支计算及稳定性分析[J].长春工业大学学报,2010,31(4):374-378.
[27]王玲书,徐瑞,冯光辉.一类具有时滞的捕食模型的稳定性和Hopf分支[J].数学的实践与认识,2010,40(5):124-128.
[28]徐秀艳.一类具有双时滞的食饵-捕食系统的Hopf分支分析[J].科技导报,2011,29(25):75-79.
[29]李大虎,陈淑苹,童欢,朱文文.具有捕食者相互残杀项双时滞系统的Hopf分支[J].湖北师范学院学报,2011,31(3):76-80.
[30]廖晓昕.稳定性的理论、方法和应用[M].武汉:华中理工大学出版社.,1999.
[31]廖晓昕.动力系统的稳定性理论和应用[M].北京:国防工业出版社.,2000.
[32]王新秀.具有时滞的Volterra捕食系统的Hopf分支分析[D].西北大学硕士学位论文，2010.
[33]刘春燕.两类生物数学模型的稳定性及Hopf分支[D].广西师范大学硕士学位论文，2010.