几类离散人口模型的持久性和周期性问题
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摘要
本文在已有的Lotka-Volterra模型的基础上，考虑多个物种并加入常时滞或变时滞，得到了更符合现实的几类离散时滞人口模型。我们主要对这几类模型的持久性和周期解的存在性进行了较深入的研究。本篇论文由四章构成。
第一章概述了数学生态学的发展历史和前人所做的一些相关工作以及本文问题的产生。另外还简单介绍了本文的主要工作。
第二章讨论了具有变时滞的一般n-种群周期Lotka-Volterra互惠差分系统。利用拓扑度理论中的连续定理以及M-矩阵的性质获得了该系统正周期解存在的充分条件。由此结果推得的二维系统正周期解存在的充分条件与相应微分系统的已知结果相同。
第三章考虑了具有时滞的n-种群非自治的Lotka-Volterra竞争差分系统。通过构造一个新的拟李雅谱诺夫函数，得到了判定该系统持久的一些易于检验的准则。这些结果较好地推广了一些已知结论。另外，尽管有的模型覆盖面比该模型要广，但它要求的条件过于苛刻，而我们大大地减弱了这些条件。在此条件下，我们较大程度上改进了这个结果。
第四章研究的仍然是n-种群的Lotka-Volterra竞争差分系统，但是是具有不同变时滞的周期系统。利用重合度理论得到了该系统正周期解存在的充分条件。
Based on the existed Lotka-Volterra population models, we consider multi-species models with constant or varied delays instead of two species ones and obtain several classes of discrete delayed population models which seem more practical than those existed. We mainly make much investigation for the existence of periodic solution and the permanence of these new models. This thesis is composed of four chapters.
In the first chapter, we state the history of Mathematical ecology's development, the existed related work and the origin of the problems we discussed. And the main work of this paper is also simply introduced.
In the second chapter, we consider a periodic Lotka-Volterra cooperative difference system with varied delays. By using the continuation theorem of topology degree theory and properties of nonsingular M-matrix, we obtain sufficient conditions for the existence of positive periodic solutions of this system. As the special case, the results for the two species model are obtained, which are same to the known results of the corresponding differential system.
The purpose of the third chapter is to study the permanence of a nonau-tonomous n-species Lotka-Volterra competitive discrete system with delays. By constructing a new qusi-Liapunov function, we obtain some new criteria which are easily checked. The known results of several two-species competitive models are well generalized. On the other hand, although some models cover this model, the required conditions are too rigorous to fit, and the conditions are weakened in our theorems. In this sense, we improve these results.
Finally, in the fourth chapter, we still consider a n-species Lotka-Volterra competitive difference system, but this model is a periodic system with different varied delays. The existence of positive periodic solutions of this system is established by using coincidence degree theory.
第一章概述了数学生态学的发展历史和前人所做的一些相关工作以及本文问题的产生。另外还简单介绍了本文的主要工作。
第二章讨论了具有变时滞的一般n-种群周期Lotka-Volterra互惠差分系统。利用拓扑度理论中的连续定理以及M-矩阵的性质获得了该系统正周期解存在的充分条件。由此结果推得的二维系统正周期解存在的充分条件与相应微分系统的已知结果相同。
第三章考虑了具有时滞的n-种群非自治的Lotka-Volterra竞争差分系统。通过构造一个新的拟李雅谱诺夫函数，得到了判定该系统持久的一些易于检验的准则。这些结果较好地推广了一些已知结论。另外，尽管有的模型覆盖面比该模型要广，但它要求的条件过于苛刻，而我们大大地减弱了这些条件。在此条件下，我们较大程度上改进了这个结果。
第四章研究的仍然是n-种群的Lotka-Volterra竞争差分系统，但是是具有不同变时滞的周期系统。利用重合度理论得到了该系统正周期解存在的充分条件。
Based on the existed Lotka-Volterra population models, we consider multi-species models with constant or varied delays instead of two species ones and obtain several classes of discrete delayed population models which seem more practical than those existed. We mainly make much investigation for the existence of periodic solution and the permanence of these new models. This thesis is composed of four chapters.
In the first chapter, we state the history of Mathematical ecology's development, the existed related work and the origin of the problems we discussed. And the main work of this paper is also simply introduced.
In the second chapter, we consider a periodic Lotka-Volterra cooperative difference system with varied delays. By using the continuation theorem of topology degree theory and properties of nonsingular M-matrix, we obtain sufficient conditions for the existence of positive periodic solutions of this system. As the special case, the results for the two species model are obtained, which are same to the known results of the corresponding differential system.
The purpose of the third chapter is to study the permanence of a nonau-tonomous n-species Lotka-Volterra competitive discrete system with delays. By constructing a new qusi-Liapunov function, we obtain some new criteria which are easily checked. The known results of several two-species competitive models are well generalized. On the other hand, although some models cover this model, the required conditions are too rigorous to fit, and the conditions are weakened in our theorems. In this sense, we improve these results.
Finally, in the fourth chapter, we still consider a n-species Lotka-Volterra competitive difference system, but this model is a periodic system with different varied delays. The existence of positive periodic solutions of this system is established by using coincidence degree theory.
引文
[1] 文贤章．多种群生态竞争．捕食时滞系统正周期解的全局吸引性．数学学报，2002，45(1)：83-94
[2] Ray R. Lotka-Volterra System with Constant Interaction Coeffcients. Nonlinear Analysis, 2001, 46(8): 1151-1164
[3] Chen L S, Teng Z D. Global Asymptotic Stability of Periodic Lotka-Volterra Systems with Delays. Nonl Anal, 2001, 45(8): 1081-1095
[4] 范猛，王克．一类积分微分方程系统正周期解的存在性．数学学报，2001，44(3)：437-444
[5] Yan J R, Feng Q X. Global Attractivity and Oscillation in a Nonlinear Delay Equation. Nonl Anal, 2001, 43(1): 101-108
[6] He X Z. Stability and Delays in a Predator-Prey System. J Math Anal Appl, 1996, 198(2): 355-370
[7] Karakostas G, Philos C G, Sficas Y G. The Dynamics of Some Discrete Population Models. Nonl Anal, 1991, 17(11): 1069-1084
[8] Chen Y M, Zhou Z. Stable Periodic Solution of a Discrete Lotka-Volterra Competition System. J Math Anal Appl, 2003, 277(1): 358-366
[9] 马知恩．种群生态学的数学建模与研究．第一版．安徽：安徽教育出版社，1996：5-369
[10] 姜启源．数学模型．北京：高等教育出版社，1993：185-300
[11] 汪礼仍．环境数学模型．上海：华东师范大学出版社，1997：14-46
[12] Gopalsamy K, He X Z. Persistence, Attractivity, and Delay in Facultative Mutualism. J Math Anal Appl, 1997, 215(1): 154-173
[13] 杨帆，蒋达清，万阿英．多时滞Lotka-Volterra互惠系统周期正解的存在性．工程数学学报，2002，19(3)：64-68
[14] Li Y K, Kuang Y. Periodic Solutions of Periodic Delay Lotka-Volterra Equations and Systems. J Math Anal Appl, 2001, 255(1): 260-280
[15] Lu Z Y, Takeuchi Y. Permanence and Global Attractiveity for Competitive LotkaVolterra Systems with Delay. Nonl Anal, 1994, 22(7): 847-856
[16] 滕志东．具有时滞的非自治Lotka-Volterra竞争系统的持久与灭绝．数学学报，2001，44(2)：293-306
[17] Lu Z Y, Wang W D. Permanence and Global Attractivity for Lotka-Volterra Differece sYSTEMS. J Math Biol, 1999, 39(2): 269-282
[18] Saito Y, Ma W B, Hara T. A Necessary and Sufficient Condition for Permanence
of a Lotka-Volterra Discrete System with Delays. J Math Anal Appl, 2001, 256(1): 162-174
[19] Zhang Q Q, Zhou Z. Permanence of a Nonautononous Lotka-Volterra Differece System with Delays. Math Sci Res J, 2003, 7(3): 99-106
[20] Li Y K. Periodic Solution for Delay Lotka-Volterra Competition Systems. J Math Anal Appl, 2000, 246(1): 230-244
[21] Gai M J, Shi B, Yang S J. The Existence of Positive Periodic Solution for a LotkaVolterra Differece System. Ann Differ Equation, 2002, 18(4): 323-329
[22] Gains R E, Mawhin J L. Coincidence Degree and Nonlinear Differential Equations. Berlin: Springer-Verlag, 1977:38-43
[23] 廖晓昕．稳定性的理论、方法和应用．武汉：华中理工大学出版社，1999：1-226
[24] Gyori I, Ladas G. Oscillation Theory of Delay Differential Equations with Aplications. Oxford: Clarendon Press, 1991:1-199
[25] Agarwal R P. Difference Equtions and Inequalities: Theory, Methods and Applications. New York: Marcel Dekker, 2000:1-115
[26] Elaydi S N. An Introduction to Difference Equations. New York: Springer-Verlag, 1999:1-211
[27] Kocic V L, Ladas G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Boston: Kluwer Academic Publishers, 1993:1-199
[28] Jack H. Theory of Functional Differential Equations. Berlin: Springer-Verlag, 1977: 1-254
[29] 郑祖庥．泛函微分方程理论．安徽：安徽教育出版社，1994：1-463
[30] Fan M, Wang K. Global Existence of Positive Periodic Solution of Periodic PredatorPrey System with Infinite Delays. J Math Anal Appl, 2001, 262(1): 1-11
[31] 桂占吉，陈兰荪．具有时滞的周期Logistic方程的持续性与周期性．数学研究与评论，2003，23(1)：109-114
[32] Li Y K. Periodic Solution of a Periodic Delay Perdator-Prey System, Amer Math Soc, 1999, 127(5): 1331-1335
[33] 蒋达清，魏俊杰．非自治时滞微分方程周期正解的存在性．数学年刊，1999，20(6)：715-720
[34] Li Y K. Periodic Solution of Periodic Delay Lotka-Volterra Equations and Systems. J Math Anal Appl, 2001, 255(1): 260-280
[35] Fan M, Wang K. Periodic in a Delayed Ratio-Dependent Predator-Prey System. J Math Anal Appl, 2001, 262(1): 179-190
[36] 陈公宁．矩阵理论与应用．北京：高等教育出版社，1990：99-178
[37] Lu Z Y, Takeuchi Y. Permanence and Global Stability for Cooperative LotkaVolterra Diffusion Systems. Nonl Anal, 1992, 19(10): 963-975
[38] Xu R,Chen L S. Persistence and Global Stablity for a Delayed Nonautonomous Predator-Prey System without Dominating Instantaneous Negative Feedback. J Math Anal Appl, 2001, 262(1): 50-61
[39] Wang W D. Harmless Delays for Uniform Persistence. J Math Anal Appl, 1991, 158(1): 256-268
[40] Wang W D. Global Stability of Discrete Competition Model. Comput Math Appl, 2001, 42(6): 773-782
[41] Wang W D, Lu Z Y. Global Stability of Discrete Models of Lotka-Volterra Type. Nonl Anal, 1999, 35(8): 1019-1030
[42] Mohamad S, Gopalsamy K. Extreme Stability and Almost Periodicity in a Discrete Logistic Equation. Tohoku Math J, 2000, 52(1) : 107-115
[43] Tang B R and Kuang Y. Permanence in Kolmoborov-Type System of Nonautonomous Functional Differential Equations. J Math Anal Appl, 1996, 197(2): 427-447
[44] Wang W D. Global Stability of Discrete Population Model with Time Delays and Fluctuating Environment. J Math Anal Appl, 2002, 264(1): 147-167
[2] Ray R. Lotka-Volterra System with Constant Interaction Coeffcients. Nonlinear Analysis, 2001, 46(8): 1151-1164
[3] Chen L S, Teng Z D. Global Asymptotic Stability of Periodic Lotka-Volterra Systems with Delays. Nonl Anal, 2001, 45(8): 1081-1095
[4] 范猛，王克．一类积分微分方程系统正周期解的存在性．数学学报，2001，44(3)：437-444
[5] Yan J R, Feng Q X. Global Attractivity and Oscillation in a Nonlinear Delay Equation. Nonl Anal, 2001, 43(1): 101-108
[6] He X Z. Stability and Delays in a Predator-Prey System. J Math Anal Appl, 1996, 198(2): 355-370
[7] Karakostas G, Philos C G, Sficas Y G. The Dynamics of Some Discrete Population Models. Nonl Anal, 1991, 17(11): 1069-1084
[8] Chen Y M, Zhou Z. Stable Periodic Solution of a Discrete Lotka-Volterra Competition System. J Math Anal Appl, 2003, 277(1): 358-366
[9] 马知恩．种群生态学的数学建模与研究．第一版．安徽：安徽教育出版社，1996：5-369
[10] 姜启源．数学模型．北京：高等教育出版社，1993：185-300
[11] 汪礼仍．环境数学模型．上海：华东师范大学出版社，1997：14-46
[12] Gopalsamy K, He X Z. Persistence, Attractivity, and Delay in Facultative Mutualism. J Math Anal Appl, 1997, 215(1): 154-173
[13] 杨帆，蒋达清，万阿英．多时滞Lotka-Volterra互惠系统周期正解的存在性．工程数学学报，2002，19(3)：64-68
[14] Li Y K, Kuang Y. Periodic Solutions of Periodic Delay Lotka-Volterra Equations and Systems. J Math Anal Appl, 2001, 255(1): 260-280
[15] Lu Z Y, Takeuchi Y. Permanence and Global Attractiveity for Competitive LotkaVolterra Systems with Delay. Nonl Anal, 1994, 22(7): 847-856
[16] 滕志东．具有时滞的非自治Lotka-Volterra竞争系统的持久与灭绝．数学学报，2001，44(2)：293-306
[17] Lu Z Y, Wang W D. Permanence and Global Attractivity for Lotka-Volterra Differece sYSTEMS. J Math Biol, 1999, 39(2): 269-282
[18] Saito Y, Ma W B, Hara T. A Necessary and Sufficient Condition for Permanence
of a Lotka-Volterra Discrete System with Delays. J Math Anal Appl, 2001, 256(1): 162-174
[19] Zhang Q Q, Zhou Z. Permanence of a Nonautononous Lotka-Volterra Differece System with Delays. Math Sci Res J, 2003, 7(3): 99-106
[20] Li Y K. Periodic Solution for Delay Lotka-Volterra Competition Systems. J Math Anal Appl, 2000, 246(1): 230-244
[21] Gai M J, Shi B, Yang S J. The Existence of Positive Periodic Solution for a LotkaVolterra Differece System. Ann Differ Equation, 2002, 18(4): 323-329
[22] Gains R E, Mawhin J L. Coincidence Degree and Nonlinear Differential Equations. Berlin: Springer-Verlag, 1977:38-43
[23] 廖晓昕．稳定性的理论、方法和应用．武汉：华中理工大学出版社，1999：1-226
[24] Gyori I, Ladas G. Oscillation Theory of Delay Differential Equations with Aplications. Oxford: Clarendon Press, 1991:1-199
[25] Agarwal R P. Difference Equtions and Inequalities: Theory, Methods and Applications. New York: Marcel Dekker, 2000:1-115
[26] Elaydi S N. An Introduction to Difference Equations. New York: Springer-Verlag, 1999:1-211
[27] Kocic V L, Ladas G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Boston: Kluwer Academic Publishers, 1993:1-199
[28] Jack H. Theory of Functional Differential Equations. Berlin: Springer-Verlag, 1977: 1-254
[29] 郑祖庥．泛函微分方程理论．安徽：安徽教育出版社，1994：1-463
[30] Fan M, Wang K. Global Existence of Positive Periodic Solution of Periodic PredatorPrey System with Infinite Delays. J Math Anal Appl, 2001, 262(1): 1-11
[31] 桂占吉，陈兰荪．具有时滞的周期Logistic方程的持续性与周期性．数学研究与评论，2003，23(1)：109-114
[32] Li Y K. Periodic Solution of a Periodic Delay Perdator-Prey System, Amer Math Soc, 1999, 127(5): 1331-1335
[33] 蒋达清，魏俊杰．非自治时滞微分方程周期正解的存在性．数学年刊，1999，20(6)：715-720
[34] Li Y K. Periodic Solution of Periodic Delay Lotka-Volterra Equations and Systems. J Math Anal Appl, 2001, 255(1): 260-280
[35] Fan M, Wang K. Periodic in a Delayed Ratio-Dependent Predator-Prey System. J Math Anal Appl, 2001, 262(1): 179-190
[36] 陈公宁．矩阵理论与应用．北京：高等教育出版社，1990：99-178
[37] Lu Z Y, Takeuchi Y. Permanence and Global Stability for Cooperative LotkaVolterra Diffusion Systems. Nonl Anal, 1992, 19(10): 963-975
[38] Xu R,Chen L S. Persistence and Global Stablity for a Delayed Nonautonomous Predator-Prey System without Dominating Instantaneous Negative Feedback. J Math Anal Appl, 2001, 262(1): 50-61
[39] Wang W D. Harmless Delays for Uniform Persistence. J Math Anal Appl, 1991, 158(1): 256-268
[40] Wang W D. Global Stability of Discrete Competition Model. Comput Math Appl, 2001, 42(6): 773-782
[41] Wang W D, Lu Z Y. Global Stability of Discrete Models of Lotka-Volterra Type. Nonl Anal, 1999, 35(8): 1019-1030
[42] Mohamad S, Gopalsamy K. Extreme Stability and Almost Periodicity in a Discrete Logistic Equation. Tohoku Math J, 2000, 52(1) : 107-115
[43] Tang B R and Kuang Y. Permanence in Kolmoborov-Type System of Nonautonomous Functional Differential Equations. J Math Anal Appl, 1996, 197(2): 427-447
[44] Wang W D. Global Stability of Discrete Population Model with Time Delays and Fluctuating Environment. J Math Anal Appl, 2002, 264(1): 147-167