具时滞及脉冲作用的生物动力系统的研究
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摘要
种群动力学是生物数学的一个重要分支.本文在传统常微分模型的基础上,讨论加入了脉冲和时滞作用的种群动力学模型.本文中所讨论的内容简单安排如下：
第一章简单介绍种群动力系统的发展概况,并给出本文所要做的主要工作.
第二章研究一类具阶段结构及捕食者有脉冲扰动的食饵-捕食者模型(其中食饵幼体有父母照顾),利用脉冲比较定理讨论食饵灭绝周期解的稳定性和系统的持续生存,利用数值模拟揭示系统的复杂动力学行为,并分析阶段结构对系统复杂动力行为的影响.
第三章讨论一类具时滞的食饵-捕食者系统的稳定性：时滞广泛存在于生物种群的活动中,时滞在生物动力系统中有着重要的作用,它可能会影响到正平衡点的稳定性.由Roth-Hurwitz判据讨论系统正平衡点的存在性和稳定性.
第四章讨论一类具时滞及修改的Holling-Tanner食饵-捕食者系统的Hopf分支：利用特征方程讨论正平衡点的渐近稳定性.运用Hopf分支理论,以时滞τ为参数给出系统发生Hopf分支的条件；利用规范型定理和中心流形定理,计算出Hopf分支在分支临界值τi处的性质.运用数值模拟揭示系统正平衡点的局部稳定性以及由Hopf分支产生的复杂动力学行为.
Population dynamics is a very important branch of biomathematics. In this paper, the population dynamics models with impulsive and time delay on the base of traditional differential model have been considered. This paper arranged as follows:
In the first chapter, the development of population dynamics system are given, and the major work of this paper is introduced.
In the second chapter, a delayed stage-structured predator-prey model with the immature prey are taken care of by their parents and biological control strategies (that is releasing natural enemies at different fixed time) is considered. By using comparison theorem, the existence of the globally asymptotically stable prey-eradication periodic solution when the number of releasing natural enemies more than some critical value is proved. The conditions for the permanence of the system are given. Numerical simulation proved the academic results.
In the third chapter, the stability of a delayed stage-structured predator-prey model is considered. Delay is existed popularly on the action of populations, and it operated important role with populations, it may be affect the stability of positive equilibrium. By using Roth-Hurwitz criterion discussing the existence and stability of positive equilibrium.
In the last chapter, a modified Holling-Tanner predator-prey model is studied. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs whenτcrosses some critical value. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and center manifold theorem. From calculate, the characters of Hopf bifurcation on the critical valueτj are obtained. By using Numerical simulation explained the stability of a positive equilibrium, and the rich dynamic behaviors of Hopf bifurcation.
第一章简单介绍种群动力系统的发展概况,并给出本文所要做的主要工作.
第二章研究一类具阶段结构及捕食者有脉冲扰动的食饵-捕食者模型(其中食饵幼体有父母照顾),利用脉冲比较定理讨论食饵灭绝周期解的稳定性和系统的持续生存,利用数值模拟揭示系统的复杂动力学行为,并分析阶段结构对系统复杂动力行为的影响.
第三章讨论一类具时滞的食饵-捕食者系统的稳定性：时滞广泛存在于生物种群的活动中,时滞在生物动力系统中有着重要的作用,它可能会影响到正平衡点的稳定性.由Roth-Hurwitz判据讨论系统正平衡点的存在性和稳定性.
第四章讨论一类具时滞及修改的Holling-Tanner食饵-捕食者系统的Hopf分支：利用特征方程讨论正平衡点的渐近稳定性.运用Hopf分支理论,以时滞τ为参数给出系统发生Hopf分支的条件；利用规范型定理和中心流形定理,计算出Hopf分支在分支临界值τi处的性质.运用数值模拟揭示系统正平衡点的局部稳定性以及由Hopf分支产生的复杂动力学行为.
Population dynamics is a very important branch of biomathematics. In this paper, the population dynamics models with impulsive and time delay on the base of traditional differential model have been considered. This paper arranged as follows:
In the first chapter, the development of population dynamics system are given, and the major work of this paper is introduced.
In the second chapter, a delayed stage-structured predator-prey model with the immature prey are taken care of by their parents and biological control strategies (that is releasing natural enemies at different fixed time) is considered. By using comparison theorem, the existence of the globally asymptotically stable prey-eradication periodic solution when the number of releasing natural enemies more than some critical value is proved. The conditions for the permanence of the system are given. Numerical simulation proved the academic results.
In the third chapter, the stability of a delayed stage-structured predator-prey model is considered. Delay is existed popularly on the action of populations, and it operated important role with populations, it may be affect the stability of positive equilibrium. By using Roth-Hurwitz criterion discussing the existence and stability of positive equilibrium.
In the last chapter, a modified Holling-Tanner predator-prey model is studied. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs whenτcrosses some critical value. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and center manifold theorem. From calculate, the characters of Hopf bifurcation on the critical valueτj are obtained. By using Numerical simulation explained the stability of a positive equilibrium, and the rich dynamic behaviors of Hopf bifurcation.
引文
[1]马知恩.种群生态学的数学建模与研究[M].合肥：安徽教育出版社,1996.
[2]徐克学.生物数学[M].北京：科学出版社,1999.
[3]陆征一,王稳地.生物数学前沿[M].北京：科学出版社,2008.
[4]Lotka A J. Elements of physical biology[M].Baltimore:Williams and Wilkins,1925.
[5]Volterra V, Variations and fluctuations of a number of individuals in animal species living together[M]. New York:Megrawhill,1931.
[6]. Holling C S, The functional-response of predator to prey density and its role in mimicry and population regulation[J].Men. Ent. Sec. Can.1965,45:1-60.
[7]Beddington J. Mutual interference between parasites or predators and its effect on searching efficiency[J]. J. Animal. Ecok.1975,44:331-340.
[8]DeAngelis D R, Goldstein, O'Neill R. A model for trophic interaction [J]. Ecology,1975,56: 881-892.
[9]Liang Z Q, Pan H W. Qualitative analysis of a ratio-dependent Holling-Tanner model[J]. J. Math. Anal. Appl.2007,334:954-964.
[10]Haquea M, Venturino E. The role of transmissible diseases in the Holling-Tanner predator-prey model[J]. Theoretical Population Biology 2006,70:273-288.
[11]Peng R, Wang M X. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model[J]. Applied Mathematics Letters,2007,20:664-670.
[12]Hale J K,Waltman P. Persistence in infinite-dimensional systems[J]. SIAM J. Math. Anal., 1989,20:388-395.
[13]王稳地.一个竞争模型的一致持续生存[J]. Biomath.,1991,4:164-169.
[14]Wang W, Ma Z. Harmless delays for uniform persistence[J]. Math. Anal. Appl.,1991,158: 256-268.
[15]Bainov D D, Simenonov P S. Impulsive differential equations:Periodic solutions and application[M]. England:Longman Scientific and Technical,1993.
[16]Bainov D D, Simenonov P S. System with impulsive effect:stability, theory and applications[M], New York John Wiley and Sons,1989.
[17]Lakshmikantham V, Bainov D D, Simenonov P S. Theory of impulsive differential equations[M]. Singapore:World Scientific,1989.
[18]Samoilenko A M, Perestyuk N A. Impulsive differential equations[M]. Singapore:World Scientific,1995.
[19]Bainov D D, Hristova S G. Existence of periodic solutions of nonlinear systems of differential equations with impulse effect[J]. J Math. Anal. Appl.,1987,125:192-202.
[20]Erbe L H, Liu X X. Existence of periodic solutions of impulsive differential systems[J]. J Appl. Math. Stoc Anal.,1991,4:137-146.
[21]Gutu V I. Periodic solutions of linear differential equations with impulses in a Banach space[M]. Differential Equations and mathematical Physics,1989:59-66.
[22]Hristova S G, Bainov D D. Numerical-analytic method for finding the periodic solutions of nonlinear differential-difference equations with impulses[M]. Computing,1987,38:363-368.
[23]Hristova S G, Bainov D D. Application of Lyapunov's functions to finding periodic solutions of systems of differential equations with impulses[J]. Bol. Soc. Paran. Mat.,1988,9: 151-163.
[24]Ailleo W G, Freedman H I. A time-delay model of single-species growth with stage-structured [J]. Math. Bio. sci.1990,101-139.
[25]Wang W, Nakaoka S, Takeuchi Y, Invest conflicts of adult predators[J]. J. Theor. Biol. vol. 2008,253:12-23.
[26]Lakshmikantham V, Bainov D D, Simeonov P. Theory of impulsive differential equations [M]. Singapor:World Scientific,1989.
[27]Jiao J, Chen L, Cai S. A delayed stage-structured Holling Ⅱ predato-prey model with mutual interference and impulsive perturbations on predator[J]. Chaos, Solitons & Fractals,2009,40: 1946-1955.
[28]Dong L Z, Chen L S, Sun L H. Extinction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively math[J]. Math. Appl. Sci.,2006,29: 415-425.
[29]Wang H L, Wang W M. The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect[J]. Chaos, Solitons & Fractals,2008,38(4):1168-1176.
[30]Kooij R, Zegeling A. A predator-prey model with Ivlev's functions Response[J]. J. Math. Anal. Appl.,1996,198:473-489.
[31]Sugie J. Two parameter bifurcation in a predator-prey system of Ivlev type[J]. J. Math. Anal. Appl.,1998,217:350-371.
[32]Torsten L. Qualitative analysis of a predator-prey system with limit cycles[J]. J. Math. Biol. 1993,31:541-561.
[33]Leslie P H. Some further notes on the use of matrices in population.mathematics[J]. Biometrika,1948,35:213-245.
[34]Leslie P H, Gower J C, The properties of a stochastic model for the predator-prey type of interaction between two species[J]. Biometrika,1960,47:219-234.
[35]Pielou E C. An Introduction to Mathematical Ecology[M]. New York:Wiley-Interscience, 1969.
[36]Korobeinikov A, A Lyapunov function for Leslie-Gower predator-prey model[M]. Appl. Math. Lett.,2001,14:697-699.
[37]Hsu S B, Hwang T W. Global stability for a class of predator-prey systems[J]. SIAM J. Appl. Math.1995,55:763-783.
[38]Saez E, Gonzalez-Olivares E. Dynamics of predator-prey model[J]. SIAM J Appl. Math., 1999,59:1867-1878.
[39]Gasull A, Kooij R E, Torregrosa J. Limit cycles in the Holling-Tanner model[J]. Publ. Mat. 1997,41:149-167.
[40]Aziz-Alaoui M A, Daher-Okiye M. Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes[J]. Appl. Math. Lett.,2003, 16:1069-1075.
[41]Liang Z Q, Pan H W. Qualitative analysis of a ratio-dependent Holling-Tanner model[J]. J. Math. Anal. Appl.,2007,334:954-964.
[42]Haquea M, Venturino E. The role of transmissible diseases in the Holling-Tanner predator-prey model[J]. Theoretical Population Biology,2006,70:273-288.
[43]Peng R, Wang M X. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model[J]. Applied Mathematics Letters,2007,20:664-670.
[44]Hale J, Lunel S. Introduction to functional differential equations[M]. New York: Springer-Verlag,1993.
[45]Hassard B, Kazarinoff D, Wang Y. Theory and applications of Hopf bifurcation[M]. Cambridge:Cambridge University Press,1981.
[46]刘秉止,彭建华.非线性动力学[M].北京：高等教育出版社,2004.
[47]桂占吉.生物动力学模型与计算机仿真[M].北京：科学出版社,2005.
[48]陈兰荪,宋新宇,陆征一.数学生态学模型与研究方法[M].成都：四川科学技术出版社,2003.
[49]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,2001.
[50]陆征一,周义仓.生物数学进展[M].北京：科学出版社,2006.
[51]傅希林,闫宝强,刘衍强.脉冲微分系统引论[M].北京：科学出版社,2005.
[52]陈兰荪.生物数学引论[M].北京：科学出版社,1998.
[53]陈兰荪,陈建.非线性生物动力系统[M].北京：科学出版社,1993.
[54]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京：科学出版社,1985.
[55]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.
[56]Hofbauer J, Sigmund K著.陆征一,罗勇 译.进化对策与种群动力学[M].四川科学技术出版社,2002.
[57]LaSalle J P著.陆征一译.动力系统的稳定性[M].四川科学技术出版社,2002.
[58]陆征一,周义仓.数学生物学进展[M].北京：科学出版社,2006.
[59]王顺庆,王万雄,徐海根.数学生态学稳定性理论与方法[M].北京：科学出版社,2004.
[60]陆启韶.常微分方程的定性方法和分叉[M].北京：北京航空航天大学出版社,1989.
[2]徐克学.生物数学[M].北京：科学出版社,1999.
[3]陆征一,王稳地.生物数学前沿[M].北京：科学出版社,2008.
[4]Lotka A J. Elements of physical biology[M].Baltimore:Williams and Wilkins,1925.
[5]Volterra V, Variations and fluctuations of a number of individuals in animal species living together[M]. New York:Megrawhill,1931.
[6]. Holling C S, The functional-response of predator to prey density and its role in mimicry and population regulation[J].Men. Ent. Sec. Can.1965,45:1-60.
[7]Beddington J. Mutual interference between parasites or predators and its effect on searching efficiency[J]. J. Animal. Ecok.1975,44:331-340.
[8]DeAngelis D R, Goldstein, O'Neill R. A model for trophic interaction [J]. Ecology,1975,56: 881-892.
[9]Liang Z Q, Pan H W. Qualitative analysis of a ratio-dependent Holling-Tanner model[J]. J. Math. Anal. Appl.2007,334:954-964.
[10]Haquea M, Venturino E. The role of transmissible diseases in the Holling-Tanner predator-prey model[J]. Theoretical Population Biology 2006,70:273-288.
[11]Peng R, Wang M X. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model[J]. Applied Mathematics Letters,2007,20:664-670.
[12]Hale J K,Waltman P. Persistence in infinite-dimensional systems[J]. SIAM J. Math. Anal., 1989,20:388-395.
[13]王稳地.一个竞争模型的一致持续生存[J]. Biomath.,1991,4:164-169.
[14]Wang W, Ma Z. Harmless delays for uniform persistence[J]. Math. Anal. Appl.,1991,158: 256-268.
[15]Bainov D D, Simenonov P S. Impulsive differential equations:Periodic solutions and application[M]. England:Longman Scientific and Technical,1993.
[16]Bainov D D, Simenonov P S. System with impulsive effect:stability, theory and applications[M], New York John Wiley and Sons,1989.
[17]Lakshmikantham V, Bainov D D, Simenonov P S. Theory of impulsive differential equations[M]. Singapore:World Scientific,1989.
[18]Samoilenko A M, Perestyuk N A. Impulsive differential equations[M]. Singapore:World Scientific,1995.
[19]Bainov D D, Hristova S G. Existence of periodic solutions of nonlinear systems of differential equations with impulse effect[J]. J Math. Anal. Appl.,1987,125:192-202.
[20]Erbe L H, Liu X X. Existence of periodic solutions of impulsive differential systems[J]. J Appl. Math. Stoc Anal.,1991,4:137-146.
[21]Gutu V I. Periodic solutions of linear differential equations with impulses in a Banach space[M]. Differential Equations and mathematical Physics,1989:59-66.
[22]Hristova S G, Bainov D D. Numerical-analytic method for finding the periodic solutions of nonlinear differential-difference equations with impulses[M]. Computing,1987,38:363-368.
[23]Hristova S G, Bainov D D. Application of Lyapunov's functions to finding periodic solutions of systems of differential equations with impulses[J]. Bol. Soc. Paran. Mat.,1988,9: 151-163.
[24]Ailleo W G, Freedman H I. A time-delay model of single-species growth with stage-structured [J]. Math. Bio. sci.1990,101-139.
[25]Wang W, Nakaoka S, Takeuchi Y, Invest conflicts of adult predators[J]. J. Theor. Biol. vol. 2008,253:12-23.
[26]Lakshmikantham V, Bainov D D, Simeonov P. Theory of impulsive differential equations [M]. Singapor:World Scientific,1989.
[27]Jiao J, Chen L, Cai S. A delayed stage-structured Holling Ⅱ predato-prey model with mutual interference and impulsive perturbations on predator[J]. Chaos, Solitons & Fractals,2009,40: 1946-1955.
[28]Dong L Z, Chen L S, Sun L H. Extinction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively math[J]. Math. Appl. Sci.,2006,29: 415-425.
[29]Wang H L, Wang W M. The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect[J]. Chaos, Solitons & Fractals,2008,38(4):1168-1176.
[30]Kooij R, Zegeling A. A predator-prey model with Ivlev's functions Response[J]. J. Math. Anal. Appl.,1996,198:473-489.
[31]Sugie J. Two parameter bifurcation in a predator-prey system of Ivlev type[J]. J. Math. Anal. Appl.,1998,217:350-371.
[32]Torsten L. Qualitative analysis of a predator-prey system with limit cycles[J]. J. Math. Biol. 1993,31:541-561.
[33]Leslie P H. Some further notes on the use of matrices in population.mathematics[J]. Biometrika,1948,35:213-245.
[34]Leslie P H, Gower J C, The properties of a stochastic model for the predator-prey type of interaction between two species[J]. Biometrika,1960,47:219-234.
[35]Pielou E C. An Introduction to Mathematical Ecology[M]. New York:Wiley-Interscience, 1969.
[36]Korobeinikov A, A Lyapunov function for Leslie-Gower predator-prey model[M]. Appl. Math. Lett.,2001,14:697-699.
[37]Hsu S B, Hwang T W. Global stability for a class of predator-prey systems[J]. SIAM J. Appl. Math.1995,55:763-783.
[38]Saez E, Gonzalez-Olivares E. Dynamics of predator-prey model[J]. SIAM J Appl. Math., 1999,59:1867-1878.
[39]Gasull A, Kooij R E, Torregrosa J. Limit cycles in the Holling-Tanner model[J]. Publ. Mat. 1997,41:149-167.
[40]Aziz-Alaoui M A, Daher-Okiye M. Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes[J]. Appl. Math. Lett.,2003, 16:1069-1075.
[41]Liang Z Q, Pan H W. Qualitative analysis of a ratio-dependent Holling-Tanner model[J]. J. Math. Anal. Appl.,2007,334:954-964.
[42]Haquea M, Venturino E. The role of transmissible diseases in the Holling-Tanner predator-prey model[J]. Theoretical Population Biology,2006,70:273-288.
[43]Peng R, Wang M X. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model[J]. Applied Mathematics Letters,2007,20:664-670.
[44]Hale J, Lunel S. Introduction to functional differential equations[M]. New York: Springer-Verlag,1993.
[45]Hassard B, Kazarinoff D, Wang Y. Theory and applications of Hopf bifurcation[M]. Cambridge:Cambridge University Press,1981.
[46]刘秉止,彭建华.非线性动力学[M].北京：高等教育出版社,2004.
[47]桂占吉.生物动力学模型与计算机仿真[M].北京：科学出版社,2005.
[48]陈兰荪,宋新宇,陆征一.数学生态学模型与研究方法[M].成都：四川科学技术出版社,2003.
[49]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,2001.
[50]陆征一,周义仓.生物数学进展[M].北京：科学出版社,2006.
[51]傅希林,闫宝强,刘衍强.脉冲微分系统引论[M].北京：科学出版社,2005.
[52]陈兰荪.生物数学引论[M].北京：科学出版社,1998.
[53]陈兰荪,陈建.非线性生物动力系统[M].北京：科学出版社,1993.
[54]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京：科学出版社,1985.
[55]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.
[56]Hofbauer J, Sigmund K著.陆征一,罗勇 译.进化对策与种群动力学[M].四川科学技术出版社,2002.
[57]LaSalle J P著.陆征一译.动力系统的稳定性[M].四川科学技术出版社,2002.
[58]陆征一,周义仓.数学生物学进展[M].北京：科学出版社,2006.
[59]王顺庆,王万雄,徐海根.数学生态学稳定性理论与方法[M].北京：科学出版社,2004.
[60]陆启韶.常微分方程的定性方法和分叉[M].北京：北京航空航天大学出版社,1989.